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", FontWeight->"Bold"], "We propose an implementation of fuzzy and classical logic programming \ inspired to Clark's completion method of a program (see[2]). Also, since we \ can consider fuzzy control as a chapter of fuzzy logic programming (see [6] \ and [7]), we show that the same algorithms are useful tools for fuzzy \ control. We confine ourselves to function-free first order languages." }], "Text", CellMargins->{{1.4375, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, FontSize->12, FontWeight->"Plain"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction", FontWeight->"Bold"]], "Subtitle", FontSize->12], Cell[TextData[{ "Fuzzy logic programming is defined as a suitable extension of classical \ logic programming. As in the classical case, we start from a \"fuzzy \ program\", i.e. a fuzzy subset ", StyleBox[" p : Cl", FontSlant->"Italic"], " \[RightArrow][0,1] of the set ", StyleBox["Cl ", FontSlant->"Italic"], "of program clauses and we associate ", StyleBox["p", FontSlant->"Italic"], " with the related \"last fuzzy Herbrand model\", i.e. the fuzzy subset of \ ground atomic formulas we can prove from ", StyleBox["p ", FontSlant->"Italic"], "(see [3], [4], [7], [12], [16] and [17]). In turn, as proposed in [6] and \ [7], fuzzy programming logic enables us to give a rigorous approach to fuzzy \ control. The idea is that the fuzzy function associated with a given system \ of IF-THEN fuzzy rules is the interpretation of a (vague) predicate in the \ least fuzzy Herbrand model of a suitable fuzzy program. The intended meaning \ of such a predicate is that ", StyleBox["\"the control y is correct given the situation x\".", FontSlant->"Italic"], " \n\tIn this paper we show how to implement these ideas in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Namely, in Section 1 we consider classical logic programming by applying \ some rewriting rules inspired to Clark's completion of a program (see [2]). \ In order to give a general framework for fuzzy logic programming, Section 2 \ we sketch the main definitions in the theory of the fuzzy deduction systems. \ In Section 3 we propose an implementation of fuzzy logic programming as a \ simple extension of the one proposed in Section 1. In Section 4 we recall the \ main definitions in fuzzy control and in Section 5 we show how to consider \ fuzzy control as a chapter of fuzzy logic programming. Finally, in Section 6 \ we show that the proposed logical interpretation gives useful suggestions and \ new tools for fuzzy control. In all the cases, we confine ourselves to \ function-free first order languages. For a theoretical justification of the \ proposed algorithms see the quoted papers." }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12], Cell[CellGroupData[{ Cell[TextData[StyleBox["1. Classical logic Programming", FontSize->12, FontWeight->"Bold", FontSlant->"Plain", FontVariations->{"CompatibilityType"->0}]], "Section", FontSize->12], Cell[TextData[{ "To implement the ideas of logic programming we use the translation \ techniques for Clarke's completion of a program (see [2]). The purpose is to \ associate any logic program with a set of definitions of functions", StyleBox[".", FontSlant->"Italic"], " To this aim, we say that a first order formula like ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", StyleBox["body ", FontSlant->"Italic"], "is in ", StyleBox["functional", FontSlant->"Italic"], " ", StyleBox["normal form ", FontSlant->"Italic"], " provided that: \n - ", StyleBox["r", FontSlant->"Italic"], "(", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") is an atomic formula with no constant and whose variables are pairwise \ different\n - ", StyleBox["body", FontSlant->"Italic"], " is a formula in which the only free variables are among ", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ", ... , ", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ".\nAs usual, since the Generalization Inference Rule is intended, we \ identify a formula in functional normal form with the related universal \ closure. The expression \"functional\" means that the formula is ready to be \ translate into a function with values in the Boolean algebra {", StyleBox["False", FontSlant->"Italic"], ",", StyleBox["True", FontSlant->"Italic"], "}. It is easy to prove that we can transform any formula like ", StyleBox["r", FontSlant->"Italic"], "(", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", StyleBox["body,", FontSlant->"Italic"], " where ", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], " are terms, into a logically equivalent formula in functional normal form. \ In fact, a suitable application of the following two rewriting rules enables \ us to transform any formula into a formula in functional normal form.\n\n", StyleBox["R1.", FontWeight->"Bold"], " if ", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], " is a term in ", StyleBox["r", FontSlant->"Italic"], "(", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], "), and ", StyleBox["z", FontSlant->"Italic"], " is a variable not occurring in ", StyleBox["r", FontSlant->"Italic"], "(", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", StyleBox["body, ", FontSlant->"Italic"], "then we can rewrite ", StyleBox["r", FontSlant->"Italic"], "(", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", StyleBox["body", FontSlant->"Italic"], " into the logically equivalent formula ", StyleBox["r", FontSlant->"Italic"], "(", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["z", FontSlant->"Italic"], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] (", StyleBox["z", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["=", FontSlant->"Italic"], StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ")\[And]", StyleBox["body,", FontSlant->"Italic"], "\n", StyleBox["R2.", FontWeight->"Bold"], " if ", StyleBox["y", FontSlant->"Italic"], " occurs in ", StyleBox["body", FontSlant->"Italic"], " but not in ", StyleBox["r", FontSlant->"Italic"], "(", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], "), then we can rewrite ", StyleBox["r", FontSlant->"Italic"], "(", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", StyleBox["body", FontSlant->"Italic"], " into the logically equivalent formula ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Italic"], "1"], TraditionalForm]]], ",...,", StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] \[Exists]", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["body", FontSlant->"Italic"], "). \n\nLet ", StyleBox["P", FontSlant->"Italic"], " be a definite program, i.e. a finite set of program clauses, and denote \ by ", StyleBox["P'", FontSlant->"Italic"], " the set of formulas in functional normal form obtained from ", StyleBox["P", FontSlant->"Italic"], ". Also, given any predicate ", StyleBox["r", FontSlant->"Italic"], " occurring in the head of a formula in ", StyleBox["P'", FontSlant->"Italic"], ", let\n\t", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], "1"]]], ", ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], "2"]]], ", . . . , ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow]", StyleBox[" ", FontFamily->"Times New Roman"], Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["h", FontSlant->"Italic"]]], FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], "\nbe the set of formulas in ", StyleBox["P'", FontSlant->"Italic"], " whose head is ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], "). Then this set of formulas is equivalent with the formula \n\t", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], "1"]], FontFamily->"Times New Roman"], StyleBox["\[Or]...\[Or]", FontFamily->"Times New Roman"], Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["h", FontSlant->"Italic"]]], FontFamily->"Times New Roman"], StyleBox[".", FontFamily->"Times New Roman"], " \nIf ", StyleBox["P\" ", FontSlant->"Italic"], "is the set of formulas obtained from ", StyleBox["P' ", FontSlant->"Italic"], "in this way, then it is immediate that ", StyleBox["P\" ", FontSlant->"Italic"], "is logically equivalent to ", StyleBox["P. ", FontSlant->"Italic"], "Again, we call ", StyleBox["Clark's completion", FontSlant->"Italic"], " of ", StyleBox["P ", FontSlant->"Italic"], "the set ", StyleBox["compl", FontSlant->"Italic"], "(", StyleBox["P", FontSlant->"Italic"], ") of definitions of predicates occurring in ", StyleBox["P ", FontSlant->"Italic"], "where a ", StyleBox["definition", FontSlant->"Italic"], " of a predicate ", StyleBox["r", FontSlant->"Italic"], " is the formula \n\t", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftRightArrow] ", Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], "1"]], FontFamily->"Times New Roman"], StyleBox["\[Or]...\[Or]", FontFamily->"Times New Roman"], Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["h", FontSlant->"Italic"]]], FontFamily->"Times New Roman"], "\nprovided that ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftArrow] ", Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], "1"]], FontFamily->"Times New Roman"], StyleBox["\[Or]...\[Or]", FontFamily->"Times New Roman"], Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["h", FontSlant->"Italic"]]], FontFamily->"Times New Roman"], "belongs to ", StyleBox["P\" ", FontSlant->"Italic"], "and the formula \n\t\[Not]", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \notherwise, i.e. ", StyleBox["r ", FontSlant->"Italic"], "occurs in ", StyleBox["P", FontSlant->"Italic"], " but is not the head of a rule in ", StyleBox["P", FontSlant->"Italic"], ". Observe that while ", StyleBox["P ", FontSlant->"Italic"], "is a logical consequence of ", StyleBox["compl", FontSlant->"Italic"], "(", StyleBox["P", FontSlant->"Italic"], "), the theories ", StyleBox["compl", FontSlant->"Italic"], "(", StyleBox["P", FontSlant->"Italic"], ") and ", StyleBox["P ", FontSlant->"Italic"], "are not logically equivalent, in general.", StyleBox[" ", FontSlant->"Italic"], "Finally, we associate any definition of a predicate ", StyleBox["r ", FontSlant->"Italic"], "with a definition of a function (we call again ", StyleBox["r", FontSlant->"Italic"], ") with values in {", StyleBox["True", FontSlant->"Italic"], ",", StyleBox["False", FontSlant->"Italic"], "}", ". Namely, if the definition is ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") \[LeftRightArrow] ", Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], "1"]], FontFamily->"Times New Roman"], StyleBox["\[Or]...\[Or]", FontFamily->"Times New Roman"], Cell[BoxData[ SubscriptBox[ StyleBox["body", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["h", FontSlant->"Italic"]]], FontFamily->"Times New Roman"], " then the ", StyleBox["Mathematica", FontSlant->"Italic"], " definition is", StyleBox[" ", FontSlant->"Italic"], "obtained by substituting \n\t- ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], ") with the expression ", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], "_,...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], "_)\n\t- the equivalence \[LeftRightArrow] with the SetDelayed symbol := \n\ \t- the disjunction \[Or] with the Or symbol ||\n\t- the conjunction \[And] \ with the And symbol &&.\n\t- the existential quantifier \[Exists]", StyleBox[" ", FontSlant->"Italic"], "with a suitable function based on a extended disjunction.\nIf the \ definition is \[Not]", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], ",...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], "), then we consider the assignment", StyleBox[":", FontSlant->"Italic"], "\n\t", StyleBox["r", FontSlant->"Italic"], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], "_,...,", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], "_)", StyleBox[" ", FontSlant->"Italic"], ":", StyleBox["= False.\n", FontSlant->"Italic"], " \n", StyleBox["EXAMPLE. ", FontWeight->"Bold"], "To give an example, we start from the following definite program ", StyleBox["P", FontSlant->"Italic"], ":\n\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["x", FontSlant->"Italic"], ") \n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["john", FontSlant->"Italic"], ") \n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["mary", FontSlant->"Italic"], ",", StyleBox["mary", FontSlant->"Italic"], ") \n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")\[And]", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ") \n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["mary", FontSlant->"Italic"], ") \n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ") \n\t", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ") \n\t", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["Italian", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[And]", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ").\n\t\t\nBy using R1 we transform ", StyleBox["P ", FontSlant->"Italic"], "into the following set ", StyleBox["P' ", FontSlant->"Italic"], "of formulas in functional normal form", "\n\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["y = x", FontSlant->"Italic"], "\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] (", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["carl", FontSlant->"Italic"], ")\[And](", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["john", FontSlant->"Italic"], ") \n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] (", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], ")\[And](", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], ") \n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] (", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["carl", FontSlant->"Italic"], ")\[And]", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")\[And]", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ") \n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], "\n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["louise", FontSlant->"Italic"], "\n\t", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["louise", FontSlant->"Italic"], "\n\t", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["louise", FontSlant->"Italic"], "\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftArrow]", StyleBox["Italian", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[And]", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ").\n\t\n\tNotice that in such a program there is no definition for the \ predicate ", StyleBox["Italian ", FontSlant->"Italic"], "and therefore that no fact ", StyleBox["Italian", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ")", StyleBox[" ", FontSlant->"Italic"], "is a consequence of ", StyleBox["P'.", FontSlant->"Italic"], " Then, in the least Herbrand model we have to consider such a predicate \ constantly false. Again, we obtain the following set ", StyleBox["compl", FontSlant->"Italic"], "(", StyleBox["P", FontSlant->"Italic"], ") of definitions of predicates:\n\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftRightArrow] ((", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["x", FontSlant->"Italic"], ")\[Or]((", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["carl", FontSlant->"Italic"], ")\[And](", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["john", FontSlant->"Italic"], "))\n\t \t\t\[Or]((", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], ")\[And](", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], "))\n\t \t\t\[Or]((", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["carl)", FontSlant->"Italic"], "\[And]", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")\[And]", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")))\n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftRightArrow] ((", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], ")\[Or](", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["louise", FontSlant->"Italic"], "))\n\t", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftRightArrow] ", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["louise", FontSlant->"Italic"], "\n\t", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftRightArrow] ", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["louise", FontSlant->"Italic"], "\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftRightArrow] (", StyleBox["Italian", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[Or](", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[And]", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")))\n\t\[Not]", StyleBox["Italian", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ").\n\nThe final step is to translate ", StyleBox["compl", FontSlant->"Italic"], "(", StyleBox["P", FontSlant->"Italic"], ") into the following set of definitions in ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "SectionFirst", TextAlignment->Left, TextJustification->1], Cell[BoxData[{ \(loves[x_, y_] := \((y === x)\) || \((x === carl && y === john)\)\ || \ \((x === mary && y === mary)\) || \((\((x === carl)\) && women[y]\ && \ beautiful[y])\)\), "\n", \(women[x_] := \ \((x === mary)\) || \((\ x === louise)\)\), "\n", \(young[x_] := \(\(x\)\(===\)\(louise\)\(\ \)\)\), "\n", \(tall[x_] := x === louise\), "\n", \(beautiful[ x_] := \((Italian[x])\) || \((tall[x] && young[x])\)\), "\[IndentingNewLine]", \(Italian[x_] := False\)}], "Input", FontSize->12], Cell[TextData[{ "Observe that no loop occurs in these definitions since the program ", StyleBox["P", FontSlant->"Italic"], " is \"stratified\". \n\tNow we can ask whether a fact is true or not in \ the Herbrand model associated with ", StyleBox["P", FontSlant->"Italic"], ", i.e. whether a fact is a theorem of ", StyleBox["P", FontSlant->"Italic"], " or not. Indeed, we simply have to calculate the (Boolean) value of the \ corresponding function." }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(loves[carl, louise]\)\)\)], "Input", FontSize->12], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "In order to illustrate the role of R2, we can add to ", StyleBox["P", FontSlant->"Italic"], " the definite clause \n\t", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["z", FontSlant->"Italic"], ")\[And]", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["z", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\nin which the variable ", StyleBox["z", FontSlant->"Italic"], " occurs in the ", StyleBox["body", FontSlant->"Italic"], " and does not occur in the head. We cannot translate this clause into a ", StyleBox["Mathematica", FontSlant->"Italic"], " equation as" }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12], Cell[BoxData[ \(sympathizes[x_, y_] := loves[x, z] && loves[z, y]\)], "Input", FontSize->12], Cell[TextData[{ "As an example, if we ask for the truth value of the formula ", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ",", StyleBox["carl", FontSlant->"Italic"], ")", ", we obtain the value ", StyleBox["False", FontSlant->"Italic"], " while the correct answer is ", StyleBox["True", FontSlant->"Italic"], "." }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sympathizes", "[", RowBox[{"carl", StyleBox[",", FontSlant->"Plain"], StyleBox["louise", FontSlant->"Plain"]}], "]"}]], "Input", FontSize->12], Cell[BoxData[ \(False\)], "Output"] }, Open ]], Cell[TextData[{ "The reason is that the evaluation of ", StyleBox["loves", FontSlant->"Italic"], "[", StyleBox["louise", FontSlant->"Italic"], ",", StyleBox["z", FontSlant->"Italic"], "] gives ", StyleBox["False", FontSlant->"Italic"], " since ", StyleBox["z", FontSlant->"Italic"], " is not interpreted as a free variable. Instead, by R2 we can transform ", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",z)\[And]", StyleBox["loves", FontSlant->"Italic"], "(z,", StyleBox["y", FontSlant->"Italic"], ") into the equivalent formula ", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] \[Exists]z(", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["z", FontSlant->"Italic"], ")\[And]", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["z", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")). Then, it remains to interpret by the existential quantifier. In \ accordance with logic programming ideas, we can refer to the Herbrand \ universe ", StyleBox["U", FontSlant->"Italic"], " of the program. Since the language of our program is function-free, ", StyleBox["U", FontSlant->"Italic"], " coincides with the (finite) set of constants. Then the valuation of an \ existential formula reduces to the valuation of a finite disjunction of \ formulas. At first we define the Herbrand universe ", StyleBox["U ", FontSlant->"Italic"], "and its Cartesian product." }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12], Cell[BoxData[{ \(\(U := {carl, john, \ louise, mary}\ ;\)\), "\[IndentingNewLine]", \(Cartesian[l_, m_] := Flatten[Outer[List, l, m], 1]\), "\[IndentingNewLine]", \(UxU := Cartesian[U, U]\)}], "Input", FontSize->12], Cell[TextData[{ "Then, we define the predicate ", StyleBox["Thereis", FontSlant->"Italic"], "[x_,U_,A_] whose meaning is that there is an ", StyleBox["x", FontSlant->"Italic"], " in the universe ", StyleBox["U", FontSlant->"Italic"], " satisfying the formula A." }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12], Cell[BoxData[{ \(Vel[{X_, Y_}] := Or[X, Y]; Vel[L_] := Or[First[L], Vel[Rest[L]]];\ \), "\n", \(Thereis[x_, Universe_, A_] := Vel[Map[Function[x, A], Universe]]\)}], "Input", FontSize->12], Cell[TextData[{ "Finally, we are ready to give a correct definition of the predicate ", StyleBox["Sympathizes", FontSlant->"Italic"], "." }], "Text", FontSize->12], Cell[BoxData[ \(\(sympathizes[x_, y_] := Thereis[z, U, Unevaluated[loves[x, z] && loves[z, y]]];\)\)], "Input",\ FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(sympathizes[carl, louise]\)], "Input", FontSize->12], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ We conclude this section by noticing that there are several methods to \ recover information. As an example we can search for all the individual \ satisfying a given predicate. To this aim we use the function \"Select\". \ \>", "Text", TextAlignment->Left, TextJustification->1, FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(Select[U, beautiful]\)], "Input", FontSize->12], Cell[BoxData[ \({louise}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(loves[{X_, Y_}] := loves[X, Y];\)\), "\n", \(Select[UxU, loves]\)}], "Input", FontSize->12], Cell[BoxData[ \({{carl, carl}, {carl, john}, {carl, louise}, {john, john}, {louise, louise}, {mary, mary}}\)], "Output"] }, Open ]], Cell["\<\ More complexes queries are possible by using \"Function\" to transform a \ logic expression into a function with Boolean values. \ \>", "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(Select[U, Function[X, loves[carl, X] && Not[beautiful[X]]]]\)], "Input",\ FontSize->12], Cell[BoxData[ \({carl, john}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2. Fuzzy logic ", "Section", FontFamily->"Times New Roman", FontSize->12], Cell[TextData[{ "In this section we sketch some basic notions on the inferential apparatus \ of a fuzzy logic (see [8], [13], [14] and [15]). The starting point is that \ instead of the Boolean algebra {", StyleBox["False", FontSlant->"Italic"], ",", StyleBox["True", FontSlant->"Italic"], "} we can consider as set of truth values the interval [0,1]. We interpret \ 0 as ", StyleBox["False", FontSlant->"Italic"], " and 1 as ", StyleBox["True.", FontSlant->"Italic"], " This leads to extend the notions of subset and relation in the following \ way. Given a set ", StyleBox["S", FontSlant->"Italic"], ", we define a ", StyleBox["fuzzy subset", FontSlant->"Italic"], " of ", StyleBox["S", FontSlant->"Italic"], " as a map ", StyleBox["s", FontSlant->"Italic"], " : ", StyleBox["S ", FontSlant->"Italic"], "\[RightArrow] [0,1]. For any ", StyleBox["x", FontSlant->"Italic"], "\[Element]", StyleBox["S", FontSlant->"Italic"], ", we interpret the number ", StyleBox["s", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") as ", StyleBox["the membership degree", FontSlant->"Italic"], " of ", StyleBox["x", FontSlant->"Italic"], " in ", StyleBox["s", FontSlant->"Italic"], ". We say that ", StyleBox["s", FontSlant->"Italic"], " is ", StyleBox["crisp", FontSlant->"Italic"], " provided that for any ", StyleBox["x", FontSlant->"Italic"], "\[Element]", StyleBox["S", FontSlant->"Italic"], " either ", StyleBox["s", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") = 0 or ", StyleBox["s", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") = 1. Then we can identify the (characteristic functions of the) subsets \ of ", StyleBox["S", FontSlant->"Italic"], " with the crisp fuzzy subsets of ", StyleBox["S", FontSlant->"Italic"], ". Given two sets ", StyleBox["S", FontSlant->"Italic"], " and ", StyleBox["T,", FontSlant->"Italic"], " a ", StyleBox["fuzzy relation ", FontSlant->"Italic"], "from ", StyleBox["S", FontSlant->"Italic"], " to ", StyleBox["T", FontSlant->"Italic"], " is a fuzzy subset of the Cartesian product ", StyleBox["S", FontSlant->"Italic"], "\[Times]", StyleBox["T", FontSlant->"Italic"], ", i.e. a map ", StyleBox["f", FontSlant->"Italic"], " : ", StyleBox["S", FontSlant->"Italic"], "\[Times]", StyleBox["T", FontSlant->"Italic"], "\[RightArrow][0,1]. A fuzzy relation ", StyleBox["f", FontSlant->"Italic"], " is also called a (non deterministic) ", StyleBox["fuzzy function", FontSlant->"Italic"], ". The intended meaning is that, given an input ", StyleBox["x", FontSlant->"Italic"], "\[Element]", StyleBox["S", FontSlant->"Italic"], ", the output is the fuzzy subset ", StyleBox["s", FontSlant->"Italic"], " : ", StyleBox["T", FontSlant->"Italic"], " \[RightArrow] [0,1] defined by setting ", StyleBox["s", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ") = ", StyleBox["f", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], "). \n\tConsider a first order language in which properties and relations \ which are vague in nature are admitted. As an example properties as ", StyleBox["\"slow\"", FontSlant->"Italic"], ", ", StyleBox["\"big\"", FontSlant->"Italic"], ", relations as ", StyleBox["\"near\"", FontSlant->"Italic"], ", ", StyleBox["\"similar\"", FontSlant->"Italic"], " and so on. Then we interpret these properties and relations by fuzzy \ subsets and fuzzy relations, respectively. We can extend such an \ interpretation to the whole set of formulas ", StyleBox["\[DoubleStruckCapitalF]", FontFamily->"Math5"], " by a suitable interpretation of the logical connectives and of the \ quantifiers. Usually, one interprets the negation \[Not] by the operation \ \[Tilde](", StyleBox["x", FontSlant->"Italic"], ") = 1-", StyleBox["x ", FontSlant->"Italic"], "and the conjunction \[And] by a suitable continuous ", StyleBox["triangular norm ", FontSlant->"Italic"], Cell[BoxData[ \(\[CircleTimes]\)]], " : [0,1]\[Times][0,1]\[RightArrow][0,1], i.e. a commutative, associative, \ order preserving operation ", Cell[BoxData[ \(\[CircleTimes]\)]], " such that ", StyleBox["x", FontSlant->"Italic"], Cell[BoxData[ \(\[CircleTimes]\)]], "1", StyleBox[" = x", FontSlant->"Italic"], " for any ", StyleBox["x", FontSlant->"Italic"], StyleBox["\[Element]", FontFamily->"Times New Roman"], "[0,1]", StyleBox[". ", FontSlant->"Italic"], "Moreover, the disjunction \[Or] is interpreted by the co-norm ", Cell[BoxData[ \(\(\(\[CirclePlus]\)\(\ \)\)\)]], "associated with ", Cell[BoxData[ \(\[CircleTimes]\)]], ", i.e. the operation defined by setting ", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic"], Cell[BoxData[ RowBox[{"\[CirclePlus]", StyleBox["y", FontSlant->"Italic"]}]], FontFamily->"Times New Roman"], StyleBox["= \[Tilde](\[Tilde]", FontFamily->"Times New Roman"], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic"], Cell[BoxData[ \(\[CircleTimes]\)], FontFamily->"Times New Roman"], StyleBox["\[Tilde]", FontFamily->"Times New Roman"], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["), ", FontFamily->"Times New Roman"], "and the implication \[RightArrow] is interpreted by the operation \ \[RightTeeArrow] defined by setting ", StyleBox["x", FontSlant->"Italic"], "\[RightTeeArrow]", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["Sup", FontSlant->"Italic"], "{", StyleBox["z", FontSlant->"Italic"], "\[Element][0,1] : ", StyleBox["x", FontSlant->"Italic"], Cell[BoxData[ \(\[CircleTimes]\)]], StyleBox["z", FontSlant->"Italic"], "\[LessEqual]", StyleBox["y", FontSlant->"Italic"], "}. Finally, the existential and the universal quantifiers are interpreted \ by the greatest lower bound and the least upper bound, respectively. \n\n", StyleBox["Examples. ", FontWeight->"Bold"], "We can set ", StyleBox["x", FontSlant->"Italic"], Cell[BoxData[ \(\[CircleTimes]\)]], StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["Min", FontSlant->"Italic"], "{", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], "} by obtaining a continuous triangular norm whose co-norm is the ", StyleBox["Max", FontSlant->"Italic"], " operation and whose implication is defined by setting ", StyleBox["x", FontSlant->"Italic"], "\[RightTeeArrow]", StyleBox["y", FontSlant->"Italic"], " = 1 if ", StyleBox["x", FontSlant->"Italic"], "\[LessEqual]", StyleBox["y", FontSlant->"Italic"], " and ", StyleBox["x", FontSlant->"Italic"], "\[RightTeeArrow]", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["y", FontSlant->"Italic"], " otherwise. Another example is the usual product whose associated co-norm \ is the operation ", StyleBox["x", FontSlant->"Italic"], "+", StyleBox["y", FontSlant->"Italic"], "-", StyleBox["xy", FontSlant->"Italic"], " an whose implication is defined by setting ", StyleBox["x", FontSlant->"Italic"], "\[RightTeeArrow]", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["Min", FontSlant->"Italic"], "{", StyleBox["y", FontSlant->"Italic"], "/", StyleBox["x,", FontSlant->"Italic"], "1}. Finally, a famous example is furnished by the Lukasiewicz norm \ obtained by setting ", StyleBox["x", FontSlant->"Italic"], Cell[BoxData[ \(\[CircleTimes]\)]], StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["Max", FontSlant->"Italic"], "{", StyleBox["x+y-", FontSlant->"Italic"], "1,0}. In such a case", StyleBox[" ", FontSlant->"Italic"], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic"], Cell[BoxData[ RowBox[{"\[CirclePlus]", StyleBox["y", FontSlant->"Italic"]}]], FontFamily->"Times New Roman"], StyleBox["= ", FontFamily->"Times New Roman"], StyleBox["Min", FontSlant->"Italic"], "{", StyleBox["x+y", FontSlant->"Italic"], ",1} and ", StyleBox["x", FontSlant->"Italic"], "\[RightTeeArrow]", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["Min", FontSlant->"Italic"], "{1", StyleBox["-x+y,", FontSlant->"Italic"], "1} = (\[Tilde]", StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[")", FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{"\[CirclePlus]", StyleBox["y", FontSlant->"Italic"]}]], FontFamily->"Times New Roman"], ". \n\nAny interpretation of the logical connectives gives corresponding \ definitions of the basic set-theoretical operations. Indeed, the ", StyleBox["complement", FontSlant->"Italic"], " of a fuzzy subset ", StyleBox["s", FontSlant->"Italic"], " is the fuzzy subset ", StyleBox["-s", FontSlant->"Italic"], " defined by setting s(", StyleBox["x", FontSlant->"Italic"], ") = \[Tilde]s(", StyleBox["x", FontSlant->"Italic"], "). The ", StyleBox["union ", FontSlant->"Italic"], "of two fuzzy subsets ", StyleBox["s", FontSlant->"Italic"], " : ", StyleBox["S ", FontSlant->"Italic"], "\[RightArrow] [0,1] and ", StyleBox["t ", FontSlant->"Italic"], ": ", StyleBox["S ", FontSlant->"Italic"], "\[RightArrow] [0,1] is the fuzzy subset ", StyleBox["s", FontSlant->"Italic"], "\[Union]", StyleBox["t", FontSlant->"Italic"], " defined by setting (", StyleBox["s", FontSlant->"Italic"], "\[Union]", StyleBox["t", FontSlant->"Italic"], ")(", StyleBox["x", FontSlant->"Italic"], ") = ", StyleBox["s", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")", Cell[BoxData[ \(\[CirclePlus]\)]], StyleBox["t", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "). The ", StyleBox["intersection ", FontSlant->"Italic"], "is the fuzzy subset ", StyleBox["s", FontSlant->"Italic"], "\[Intersection]", StyleBox["t", FontSlant->"Italic"], " defined by setting (", StyleBox["s", FontSlant->"Italic"], "\[Intersection]", StyleBox["t", FontSlant->"Italic"], ")(", StyleBox["x", FontSlant->"Italic"], ") = ", StyleBox["s", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")", Cell[BoxData[ \(\[CircleTimes]\)]], StyleBox["t", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "). We also define ", StyleBox["s", FontSlant->"Italic"], "\[RightArrow]", StyleBox["t", FontSlant->"Italic"], " as the fuzzy subset defined by setting (", StyleBox["s", FontSlant->"Italic"], "\[RightArrow]", StyleBox["t", FontSlant->"Italic"], ")(", StyleBox["x", FontSlant->"Italic"], ") = ", StyleBox["s", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[RightArrow]", StyleBox["t", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "). Finally,", StyleBox[" we call ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Cartesian product ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["of ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["s", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" and ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["t ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["the fuzzy subset ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["s", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Times]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["t ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[": ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["X", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Times]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow] [0,1] defined by setting (", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["s", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Times]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") = ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["s", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ \(TraditionalForm\`\[CircleTimes]\)], FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["). ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], "\n\tThe deduction apparatus of a fuzzy logic is defined in the following \ way. ", StyleBox["We define a ", FontFamily->"Times New Roman"], StyleBox["fuzzy Hilbert system ", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["as a pair ", FontFamily->"Times New Roman"], StyleBox["\[ScriptCapitalS]", FontFamily->"Math5"], StyleBox[" = (", FontFamily->"Times New Roman"], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", ", FontFamily->"Times New Roman"], StyleBox["\[DoubleStruckCapitalR]", FontFamily->"Math5"], StyleBox[") where ", FontFamily->"Times New Roman"], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" is a fuzzy subset of ", FontFamily->"Times New Roman"], StyleBox["\[DoubleStruckCapitalF]", FontFamily->"Math5"], StyleBox[", ", FontFamily->"Times New Roman"], StyleBox["the fuzzy subset of logical axioms", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", and ", FontFamily->"Times New Roman"], StyleBox["\[DoubleStruckCapitalR]", FontFamily->"Math5"], StyleBox[" is a set of fuzzy rules of inference. In turn, a ", FontFamily->"Times New Roman"], StyleBox["fuzzy inference rule", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" is a pair ", FontFamily->"Times New Roman"], StyleBox["r", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" = (", FontFamily->"Times New Roman"], StyleBox["r'", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[",", FontFamily->"Times New Roman"], StyleBox["r\"", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["), where \n - ", FontFamily->"Times New Roman"], StyleBox["r'", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" is a partial ", FontFamily->"Times New Roman"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["-ary operation on ", FontFamily->"Times New Roman"], StyleBox["\[DoubleStruckCapitalF]", FontFamily->"Math5"], StyleBox[" whose domain we denote by ", FontFamily->"Times New Roman"], StyleBox["Dom", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["r", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["),\n - ", FontFamily->"Times New Roman"], StyleBox["r\"", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" is an ", FontFamily->"Times New Roman"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["-ary operation on ", FontFamily->"Times New Roman"], "[0,1]", StyleBox[" preserving the least upper bound in each variable, i.e.\n\t \t\t\ ", FontFamily->"Times New Roman"], StyleBox["r\"", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`x\_1\)]], StyleBox[",...,", FontFamily->"Times New Roman"], StyleBox[" Sup", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[Element]", FontFamily->"Symbol", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["I", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["y", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", ..., ", FontFamily->"Times New Roman"], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") =", FontFamily->"Times New Roman"], StyleBox[" Sup", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[Element]", FontFamily->"Symbol", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["I", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["r\"", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`x\_1\)]], StyleBox[", ..., ", FontFamily->"Times New Roman"], StyleBox["y", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", ..., ", FontFamily->"Times New Roman"], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[").\nIn other words, an inference rule ", FontFamily->"Times New Roman"], StyleBox["r", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" consists \n - of a syntactical component ", FontFamily->"Times New Roman"], StyleBox["r'", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" that operates on formulas (in fact, it is a rule of inference in \ the usual sense),\n - of a valuation component ", FontFamily->"Times New Roman"], StyleBox["r\"", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" to calculate information on the truth-value of the conclusion \ from information on the truth-values of the premises (see [8], [13] and \ [19]). \nNamely, we interpret an application of an ", FontFamily->"Times New Roman"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["-ary inference rule ", FontFamily->"Times New Roman"], StyleBox["r", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" by saying that:\n\tIF \t\tyou know that ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["1", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",\[Ellipsis],", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" are true (at least) to the degree ", FontFamily->"Times New Roman"], StyleBox[" \[Lambda]", FontFamily->"Symbol"], StyleBox["1", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",\[Ellipsis],", FontFamily->"Times New Roman"], StyleBox["\[Lambda]", FontFamily->"Symbol"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\n\tTHEN \t", FontFamily->"Times New Roman"], StyleBox["r'", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["1", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",\[Ellipsis],", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") is true (at least) at level ", FontFamily->"Times New Roman"], StyleBox["r\"", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["\[Lambda]", FontFamily->"Symbol"], StyleBox["1", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",\[Ellipsis],", FontFamily->"Times New Roman"], StyleBox["\[Lambda]", FontFamily->"Symbol"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[").\nGiven a formula ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[", we interpret ", FontFamily->"Times New Roman"], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[")", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["as the ", FontFamily->"Times New Roman"], StyleBox["a prior", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" information on the truth value of ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[", i.e. the information \"the truth value of ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" is at least ", FontFamily->"Times New Roman"], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[")\". ", FontFamily->"Times New Roman"], StyleBox["A proof ", FontFamily->"Times New Roman"], StyleBox["\[Pi]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" of ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" is a sequence ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["1", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",...", FontFamily->"Times New Roman"], StyleBox[",", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["m", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["of formulas such that ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["m", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["= ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" together with the related \"justifications\". This means that, \ for any formula ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", we must specify whether\n (i) ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" is assumed as a logical axiom; or\n (ii) ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" is assumed as an hypothesis; or\n (iii) ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" is obtained by an inference rule (in this case we must indicate \ also the rule and the formulas from ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["1", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",...,", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["-1 ", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["used to obtain ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["). \nObserve that we have only two proofs of whose length is \ equal to 1. The formula ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" with the justification that ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" is assumed as a logical axiom and the formula ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" with the justification that ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" is assumed as an hypothesis. Moreover, as in the classical case, \ for any ", FontFamily->"Times New Roman"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" \[LessEqual] ", FontFamily->"Symbol"], StyleBox["m", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", the initial segment ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["1", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",...,", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Symbol"], StyleBox[" is a proof of ", FontFamily->"Times New Roman"], StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["we denote by ", FontFamily->"Times New Roman"], StyleBox["\[Pi]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["). Differently from the crisp case, the justifications are \ necessary since different justifications of the same formula give rise to \ different valuations. ", FontFamily->"Times New Roman"], "We assume that the available information is a fuzzy set of formulas ", StyleBox["v", FontSlant->"Italic"], " : ", StyleBox["\[DoubleStruckCapitalF]", FontFamily->"Math5"], " \[RightArrow][0,1] we call ", StyleBox["fuzzy subset of hypotheses. ", FontSlant->"Italic"], "Given a formula ", StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], ", we interpret the value ", StyleBox["v", FontSlant->"Italic"], "(", StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], ") as the claim \"the truth value of ", StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], " is at least ", StyleBox["v", FontSlant->"Italic"], "(", StyleBox["\[Alpha]", FontFamily->"Symbol", FontSlant->"Italic"], ")\". ", StyleBox["Let \[Pi]", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox["be a proof. Then the ", FontFamily->"Times New Roman"], StyleBox["valuation", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["Val", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["\[Pi]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[",", FontFamily->"Times New Roman"], StyleBox["v", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[")", FontFamily->"Times New Roman"], StyleBox[" of ", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["\[Pi]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" with respect to v", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" is defined by induction on the length ", FontFamily->"Times New Roman"], StyleBox["m", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" of ", FontFamily->"Times New Roman"], StyleBox["\[Pi]", FontFamily->"Symbol", FontSlant->"Italic"], StyleBox[" as follows. 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FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")\[And]", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")\t\t[0.9]\n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["mary", FontSlant->"Italic"], ")\t\t\t\t\t[1]\n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\t\t\t\t\t[1]\n\t", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\t\t\t\t\t[0.8]\n\t", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\t\t\t\t\t[0.7]\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["Italian", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\t\t\t[0.6]\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[And]", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\t\t[0.9]\n\t", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",z)\[And]", StyleBox["loves", FontSlant->"Italic"], "(z,", StyleBox["y", FontSlant->"Italic"], ")\t[0.8]\n\nThe proofs from this program are very simple. As an example, \ in order to prove ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["louise", FontSlant->"Italic"], "), observe that by the Particularization Rule\t\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["louise", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\[And]", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ") \tat degree 0.9.\nThen, since\n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\t\t\t\t\t\tat degree 1,\nit remain to prove the formula ", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], "). Now this can be done in two way. We can observe directly that\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ") \t\t\t\t\tat degree 0.5,\nand therefore to conclude \n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["louise", FontSlant->"Italic"], ")\t\t\t\t\tat degree 1\[CircleTimes]0.5\[CircleTimes]0.9 = 0.45\nAlso, \ since by Particularization\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\[And]", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\t\tat degree 0.9\nand \n\t", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\t\t\t\t\t\tat degree 0.7\n\t", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ")\t\t\t\t\t\tat degree 0.8\nwe can prove\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["louise", FontSlant->"Italic"], ") \t\t\t\t\tat degree 0.7\[CircleTimes]0.8\[CircleTimes]0.9 = 0.504.\nIn \ such a case we prove\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["louise", FontSlant->"Italic"], ") \t\t\t\t\tat degree 0.9\[CircleTimes]1\[CircleTimes]0.504 = 0.4536.\n\ Thus, in accordance with Definition 2.1, ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["louise", FontSlant->"Italic"], ") was proved at degree ", StyleBox["Max", FontSlant->"Italic"], "{0.45, 0.4536} = 0.4536. \n\tNow, in order to translate such a fuzzy \ program into a set of definitions, we can try to go on by the same \ normalization method as in Section 1. Now, it is easy to see that R1 and R2 \ apply also in this case and therefore that the fuzzy program ", StyleBox["p ", FontSlant->"Italic"], "is equivalent with the", " following fuzzy program (see the note at the end of the section)." }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12], Cell[TextData[{ "\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["x", FontSlant->"Italic"], "\t\t\t\t\t[0.6]\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\[LeftArrow](", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["carl", FontSlant->"Italic"], ")\[And](", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["john", FontSlant->"Italic"], ")\t\t\t[0.7]\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\[LeftArrow](", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], ")\[And](", StyleBox["y", FontSlant->"Italic"], " = ", StyleBox["mary", FontSlant->"Italic"], ")\t\t\t[0.9]\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\[LeftArrow](", StyleBox["x", FontSlant->"Italic"], " = ", StyleBox["carl", FontSlant->"Italic"], ")\[And]", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")\[And]", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")\t\t[0.9]\n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], "\t\t\t\t\t[1]\n\t", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["louise", FontSlant->"Italic"], "\t\t\t\t\t[1]\n\t", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["x", FontSlant->"Italic"], "=", StyleBox["louise", FontSlant->"Italic"], "\t\t\t\t\t[0.8]\n\t", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["louise", FontSlant->"Italic"], "\t\t\t\t\t[0.7]\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["Italian", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\t\t\t\t\t[0.6]\n\t", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[LeftArrow]", StyleBox["tall", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\[And]", StyleBox["young", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")\t\t\t\t[0.9]\n\t", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] \[Exists]", StyleBox["z", FontSlant->"Italic"], "(", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",z)\[And]", StyleBox["loves", FontSlant->"Italic"], "(z,", StyleBox["y", FontSlant->"Italic"], "))\t\t[0.8]\n\nUnfortunately we cannot substitute the set of clauses whose \ head is ", StyleBox["love", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") with an unique formula as\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\[LeftArrow]((", StyleBox["y", FontSlant->"Italic"], "=", StyleBox["x", FontSlant->"Italic"], ")\[Or]((", StyleBox["x", FontSlant->"Italic"], "=", StyleBox["carl", FontSlant->"Italic"], ")\[And](", StyleBox["x", FontSlant->"Italic"], "=", StyleBox["john", FontSlant->"Italic"], "))\n \[Or]((", StyleBox["x", FontSlant->"Italic"], "=", StyleBox["mary", FontSlant->"Italic"], ")\[And](", StyleBox["y", FontSlant->"Italic"], "=", StyleBox["mary", FontSlant->"Italic"], "))\[Or]((", StyleBox["x", FontSlant->"Italic"], "=", StyleBox["carl", FontSlant->"Italic"], ")\[And]", StyleBox["women", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], ")\[And]", StyleBox["beautiful", FontSlant->"Italic"], "(", StyleBox["y", FontSlant->"Italic"], "))) [\[Lambda]].\nIn fact, in such a case we should able to prove both \ ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["mary", FontSlant->"Italic"], ",", StyleBox["mary", FontSlant->"Italic"], ") and ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["john", FontSlant->"Italic"], ") at degree \[Lambda]. Instead, the clause\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\[LeftArrow](", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], ")\[And](", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["mary", FontSlant->"Italic"], ")\t\t\t[0.9]\nenables us to prove ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["mary", FontSlant->"Italic"], ",", StyleBox["mary", FontSlant->"Italic"], ") at degree 0.9 and the clause\n\t", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ")\[LeftArrow](", StyleBox["x ", FontSlant->"Italic"], "= ", StyleBox["carl", FontSlant->"Italic"], ")\[And](", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["john", FontSlant->"Italic"], ") \t\t\t[0.7]\nenables us to prove ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["carl", FontSlant->"Italic"], ",", StyleBox["john", FontSlant->"Italic"], ") at degree 0.7. Then, in accordance with Definition 2.1, to calculate the \ degree at which we can prove a fact as ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ") we have to consider all the proofs arising from the clauses whose head \ is ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") and then to calculate the maximum of the corresponding valuations. This \ leads to the following definitions where we have to define at first the \ interpretation of the existential operator and of the triangular norm (here, \ we assume that ", Cell[BoxData[ \(\[CircleTimes]\)]], " is the usual product). Also, we define the identity as a function \ assuming values in {0,1} and not in {", StyleBox["True", FontSlant->"Italic"], ",", StyleBox[" False", FontSlant->"Italic"], "}." }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12], Cell[BoxData[{ \(Thereis[X_, Universe_, condition_] := Max[Map[Function[X, condition], Universe]]\), "\n", \(x_\[CircleTimes]y_ := \((x*y)\)\ ; \ eq[x_, y_] := If[y === x, 1, 0]\)}], "Input", FontSize->12], Cell[BoxData[{ \(loves[x_, y_] := Max[\[IndentingNewLine]eq[x, y]\[CircleTimes]0.6, \[IndentingNewLine]\((eq[x, carl]\[CircleTimes]eq[y, john])\)\[CircleTimes]0.7, \[IndentingNewLine]\((eq[x, mary]\[CircleTimes]eq[y, mary])\)\[CircleTimes]0.9, \[IndentingNewLine]\((eq[x, carl]\[CircleTimes]women[y])\)\[CircleTimes]\((\ beautiful[y]\[CircleTimes]0.9)\)]\), "\[IndentingNewLine]", \(\(\(loves[{x_, y_}] := loves[x, y]\)\(\[IndentingNewLine]\) \)\), "\n", \(\(\(women[x_] := \ Max[\[IndentingNewLine]eq[x, mary], \[IndentingNewLine]eq[x, louise]]\)\(\[IndentingNewLine]\) \)\), "\n", \(young[x_] := \(\(eq[x, louise]\[CircleTimes]0.8\)\(\ \)\)\), "\n", \(\(\(tall[x_] := eq[x, louise]\[CircleTimes]0.7\)\(\n\) \)\), "\[IndentingNewLine]", \(\(\(beautiful[x_] := Max[\[IndentingNewLine]Italian[ x]\[CircleTimes]0.6, \[IndentingNewLine]\((tall[ x]\[CircleTimes]young[x])\)\[CircleTimes]0.9]\)\(\n\) \)\), "\[IndentingNewLine]", \(\(\(sympathizes[x_, y_] := Thereis[z, U, Unevaluated[\((loves[x, z]\[CircleTimes]loves[z, y])\)\[CircleTimes]0.8]]\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(Italian[x_] := 0\)}], "Input", FontSize->12], Cell["\<\ To recover information, we can ask directly for the truth value of a fact\ \>", "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(loves[carl, louise]\)], "Input", FontSize->12], Cell[BoxData[ \(0.4536`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sympathizes[carl, louise]\)], "Input", FontSize->12], Cell[BoxData[ \(0.21772800000000003`\)], "Output"] }, Open ]], Cell[BoxData[ \(TraditionalForm\`sympathizes[carl, louise]\)], "Input"], Cell["\<\ Also, we can fix a threshold \[Lambda] and to select all the elements (the \ pairs) satisfying a given predicate at least with the threshold \[Lambda] \ \>", "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(Select[UxU, Function[Z, loves[Z]\[CircleTimes]beautiful[Z[\([2]\)]] > 0.3]]\)], "Input", FontSize->12], Cell[BoxData[ \({{louise, louise}}\)], "Output"] }, Open ]], Cell[TextData[{ "Finally, we can list all the answers to a query together with the related \ degrees.Namely, we select the answers whose truth degree is different from \ zero. To this aim we define the predicate ", StyleBox["\"pos\"", FontSlant->"Italic"] }], "Text", FontSize->12], Cell[BoxData[ \(pos[{a_, b_}] := b > 0\)], "Input", FontSize->12], Cell["And we use the Select function", "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(Select[Map[Function[X, {X, loves[carl, X]}], U], pos]\)], "Input", FontSize->12], Cell[BoxData[ \({{carl, 0.6`}, {john, 0.7`}, {louise, 0.4536`}}\)], "Output"] }, Open ]], Cell["\<\ Or, to obtain answers ordered with respect to the truth degree\ \>", "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(Sort[Select[Map[Function[X, {X, loves[carl, X]}], U], pos], Last[#2] < Last[#1] &]\)], "Input", FontSize->12], Cell[BoxData[ \({{john, 0.7`}, {carl, 0.6`}, {louise, 0.4536`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sympathizes[carl, louise]\)], "Input", FontSize->12], Cell[BoxData[ \(0.21772800000000003`\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Note. ", FontWeight->"Bold"], "To justify R2, consider the formula \n", StyleBox["\tsympathizes", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",z)\[And]", StyleBox["loves", FontSlant->"Italic"], "(z,", StyleBox["y", FontSlant->"Italic"], ")\t[0.8]\nwhere ", StyleBox["z", FontSlant->"Italic"], " is a free variable. Then, from the point of view of the proposed \ deduction apparatus such a formula is equivalent to the fuzzy subset of \ formulas\n\t", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["c\"", FontSlant->"Italic"], ")\[And]", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c\"", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \t[0.8]\nwhere ", StyleBox["c\"", FontSlant->"Italic"], " varies in ", StyleBox["U", FontSlant->"Italic"], ". Let ", StyleBox["c ", FontSlant->"Italic"], "and ", StyleBox["c' ", FontSlant->"Italic"], "be two elements in ", StyleBox["U. ", FontSlant->"Italic"], "Then, for any ", StyleBox["c\" ", FontSlant->"Italic"], StyleBox["\[Element]", FontFamily->"Times New Roman"], " ", StyleBox["U, ", FontSlant->"Italic"], "we can prove ", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ") at degree |", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c\"", FontSlant->"Italic"], ")|\[CircleTimes]|", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c\"", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ")|\[CircleTimes]0.8. where |", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c\"", FontSlant->"Italic"], ")| and |", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c\"", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ")| denote the degree at which we can prove ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c\"", FontSlant->"Italic"], ") and ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c\"", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], "), respectively. Consequently, in accordance with Definition 2.1, we can \ prove ", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ") at degree\n\t ", StyleBox["Max", FontSlant->"Italic"], "{|", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c\"", FontSlant->"Italic"], ")|\[CircleTimes]|", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c\"", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ")|\[CircleTimes]0.8 : ", StyleBox["c\"", FontSlant->"Italic"], " ", StyleBox["\[Element]", FontFamily->"Times New Roman"], StyleBox["U", FontFamily->"Times New Roman", FontSlant->"Italic"], "}.\nOn the other hand, if we start form the formula\n\t", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"], ") \[LeftArrow] \[Exists]z(", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",z)\[And]", StyleBox["loves", FontSlant->"Italic"], "(z,", StyleBox["y", FontSlant->"Italic"], "))\t[0.8],\nthen, since the valuation of \[Exists]z(", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ",z)\[And]", StyleBox["loves", FontSlant->"Italic"], "(z,", StyleBox["y", FontSlant->"Italic"], ")) is ", StyleBox["Max", FontSlant->"Italic"], "{|", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c\"", FontSlant->"Italic"], ")|\[CircleTimes]|", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c\"", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ")| : ", StyleBox["c\"", FontSlant->"Italic"], " ", StyleBox["\[Element]", FontFamily->"Times New Roman"], StyleBox["U", FontFamily->"Times New Roman", FontSlant->"Italic"], "}, we can prove ", StyleBox["sympathizes", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ") at degree (", StyleBox["Max", FontSlant->"Italic"], "{|", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ",", StyleBox["c\"", FontSlant->"Italic"], ")|\[CircleTimes]|", StyleBox["loves", FontSlant->"Italic"], "(", StyleBox["c\"", FontSlant->"Italic"], ",", StyleBox["c'", FontSlant->"Italic"], ")| : ", StyleBox["c\"", FontSlant->"Italic"], " ", StyleBox["\[Element]", FontFamily->"Times New Roman"], StyleBox["U", FontFamily->"Times New Roman", FontSlant->"Italic"], "})\[CircleTimes]0.8. Since \[CircleTimes] is distributive with respect to \ ", StyleBox["Max", FontSlant->"Italic"], ", t", "his proves the validity of R2. Observe that such an equivalence holds only \ since the existential quantifier is evaluated by the ", StyleBox["Max", FontSlant->"Italic"], " function in the Herbrand universe", StyleBox["U", FontSlant->"Italic"], ". This explains also way in fuzzy logic the existential quantifier is \ interpreted by the least upper bound ", StyleBox["Sup", FontSlant->"Italic"], StyleBox[" : ", FontFamily->"Times New Roman"], StyleBox["P", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["([0,1])\[RightArrow][0,1]", FontFamily->"Times New Roman"], ". This in spite of the fact that this quantifier looks to be an extension \ of the disjunction and therefore that the natural candidate for its \ interpretation is the operator ", Cell[BoxData[ \(\[CirclePlus]\)], FontFamily->"Times New Roman", FontSize->18], " ", StyleBox[": ", FontFamily->"Times New Roman"], StyleBox["P", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["([0,1])\[RightArrow][0,1] defined by setting ", FontFamily->"Times New Roman"], Cell[BoxData[ \(\[CirclePlus]\)], FontFamily->"Times New Roman", FontSize->18], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["X", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[") = ", FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{ StyleBox["Sup", FontSlant->"Italic"], RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SubscriptBox[ StyleBox["x", FontSlant->"Italic"], "1"], "\[CirclePlus]"}], "..."}], "\[CirclePlus]", SubscriptBox[ StyleBox["x", FontSlant->"Italic"], StyleBox["n", FontSlant->"Italic"]]}]}]}]], FontFamily->"Times New Roman"], StyleBox[" : ", FontFamily->"Times New Roman"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox[ StyleBox["x", FontSlant->"Italic"], "1"], "\[Element]", StyleBox["X", FontSlant->"Italic"]}], StyleBox[",", FontSlant->"Italic"], "...", " ", ",", StyleBox[ SubscriptBox["x", StyleBox["n", FontSlant->"Italic"]], FontSlant->"Italic"]}]], FontFamily->"Times New Roman"], StyleBox["\[Element]", FontFamily->"Times New Roman"], StyleBox["X", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["}", FontFamily->"Times New Roman"], ". " }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["4. Triangular-norm based fuzzy control", "Section", FontFamily->"Times New Roman", FontSize->12], Cell[TextData[{ StyleBox["As an application of the sketched fuzzy logic programming theory, \ we will consider the main success of fuzzy set theory: ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["fuzzy control", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[". Recall that the aim of classical control theory is to \ individuate a function ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["f", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->True}], StyleBox[" : ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["X", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontColor->GrayLevel[0]], StyleBox[" Y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" such that ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["f", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->True}], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") is the correct control given the input ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[". To do this, the starting point is a general theory about the \ phenomenon under consideration. From this theory we obtain some differential \ equations and then ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["f", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->True}], StyleBox[" is obtained as a solution of these equations. The paradigm used \ in fuzzy control theory, as devised by Zadeh in [19] and by Mamdani in [11], \ is totally different. Indeed, in fuzzy control one tries to obtain ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["f", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->True}], StyleBox[" from the verbal information given by an expert on the control \ under consideration (imagine the expert as a cleaver \ \[OpenCurlyDoubleQuote]old hand\[CloseCurlyDoubleQuote] with no theoretical \ knowledge or mathematical competence). Such an information is expressed by a \ system of IF-THEN rules as\n\n", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["IF", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x is Little \t \t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" \ty is Slow,\n\t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["IF", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x is Small\t\t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" \ty is Fast,\n \t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["IF", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x is Medium \t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" \ty is Moderate,\t\n\t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["IF ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x is Big \t \t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\ty is Veryfast,\n\t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["IF ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x is Verybig \t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" \ty is Moderate,\n\n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["where", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" \"Little\", \"Slow\", \"Small\", \"Fast\", \"Medium\", \ \"Moderate\", \"Big\", \"Veryfast\", \"Verybig\" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["are labels for the fuzzy quantities", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" \n\tlittle : X ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1] ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[", \t\tsmall : X ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[", \t\tmedium : X ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\n\tbig : X ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[", \t \tverybig : X ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[",\t\tslow : Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[", \n\tfast : Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[", \tmoderate : Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[",\tveryfast : Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[", \n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["respectively. The whole system of rules can be represented by the \ table:\n", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\n ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], Cell[BoxData[ StyleBox[GridBox[{ {"x", "y"}, {"Little", "Slow"}, {"Small", "Fast"}, {"Medium", "Moderate"}, {"Big", "Veryfast"}, {"Verybig", "Moderate"} }, GridFrame->True, RowLines->True, ColumnLines->True], FontSlant->"Italic", FontColor->GrayLevel[0]]]], StyleBox["\n \n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["Also, any pair defines a fuzzy point ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["obtained as the corresponding Cartesian product and the whole \ system is associated with the fuzzy function obtained as the union of the \ fuzzy points obtained in this way. In our example,\n\t", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["f", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" = (", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["little", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Times]", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["slow)", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Union](", FontFamily->"Times New Roman", FontSize->9, FontColor->GrayLevel[0]], StyleBox["small", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Times]", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["fast)", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Union](", FontFamily->"Times New Roman", FontSize->9, FontColor->GrayLevel[0]], StyleBox["medium", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Times]", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["moderate)", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Union](", FontFamily->"Times New Roman", FontSize->9, FontColor->GrayLevel[0]], StyleBox["big", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Times]", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["veryfast)", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Union](", FontFamily->"Times New Roman", FontSize->9, FontColor->GrayLevel[0]], StyleBox["verybig", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Times]", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["moderate).", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\n\n", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Definition 4.1. (Conjunction-based fuzzy control).", FontFamily->"Times New Roman", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["A system of IF-THEN fuzzy rules is a system of rules like", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\n\t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["IF", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x is A", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" y is B", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1\n\t. . .\n\t", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["IF", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x is A", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" y is B", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\n", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["where the labels ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["A", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" and ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["B", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" are interpreted by the fuzzy quantities ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" : X ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["and ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["b", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" : Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[". ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["We associate any rule", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["IF", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x is A", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" y is B", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" with the fuzzy point ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[Times]b", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" : X \[Times] Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1] and the whole system of rules with the fuzzy function", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" f : X \[Times] Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1] defined by setting", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" f = ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Union]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["i=", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",...,n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" a", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[Times]b", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[".\n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\nUsually the union is defined by the ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Max ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["operator and therefore \n\t", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["f", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") = ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Max", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["{", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\[CircleTimes]", 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", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["X \[Times] Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow][0,1] we can derive a crisp function ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["g", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" : ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["X ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" by a ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\"defuzzification process\"", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[". Usually the defuzzification process is obtained by the centroid \ method where we set, for every ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["r", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Element]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["X", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",\n \t\t\t\t\t", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["g", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["r", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["=", FontFamily->"Times New Roman", FontSize->16, FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSize->16, FontSlant->"Italic", FontColor->GrayLevel[0]], Cell[BoxData[ FractionBox[ RowBox[{ StyleBox[\(\[Integral]\_Y\), ScriptLevel->0], RowBox[{"f", RowBox[{"(", RowBox[{ StyleBox["r", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"]}], ")"}], StyleBox["y", FontSlant->"Italic"], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0]}]}], RowBox[{ StyleBox[\(\[Integral]\_Y\), ScriptLevel->0], RowBox[{"f", RowBox[{"(", RowBox[{ StyleBox["r", FontSlant->"Italic"], ",", StyleBox["y", FontSlant->"Italic"]}], ")"}], StyleBox["\[ThinSpace]", ScriptLevel->0], StyleBox[\(\[DifferentialD]y\), ScriptLevel->0]}]}]]], FontFamily->"Times New Roman", FontSize->16, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\tThe final phase is the learning process in which the rules and \ the fuzzy quantities associated with the labels are changed until we can \ accept ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["g", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" as a good approximation of the ideal function ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["f", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->True}], StyleBox[". \n\tAs an example, we can define the fuzzy quantities by the \ function triangle. 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It is also surprising that to the IF-THEN structure of the system \ corresponds an algorithm based on conjunction. Obviously, the first \ temptation is to interpret the IF-THEN in a rule as a logical implication. As \ an example, we could interpret a rule as \"If", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x ", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["is", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" Little", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" then", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" y", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" is ", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Slow", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\" by the first order formula ", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Little", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]S", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["low", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["y", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["). Indeed, in literature exists also the following definition:", TextAlignment->Left, TextJustification->1, FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\n\n", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Definition 4.2.", FontFamily->"Times New Roman", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" (Implication-based fuzzy control).", FontFamily->"Times New Roman", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" Consider a system of IF-THEN fuzzy rules like", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\n\t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["IF", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x is A", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" y is B", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1\n\t. . .\n\t", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["IF", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" x is A", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["THEN", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" y is B", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\n", FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["where the labels ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["A", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" and ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["B", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" are interpreted by the fuzzy quantities ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" : X ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1] and ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["b", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" : Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[". ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["Then the implication-based fuzzy function ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], Cell[BoxData[ SubscriptBox[ StyleBox["g", FontSlant->"Italic"], "2"]], FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" : X \[Times] Y ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[0,1] associated with such a system is defined by setting ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], Cell[BoxData[ SubscriptBox[ StyleBox["g", FontSlant->"Italic"], "2"]], FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" =", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Intersection]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["i=1,...,n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" a", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["b", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[".\n\n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["Usually, the intersection is defined by the operator ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Min ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["and therefore ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\n\t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], Cell[BoxData[ SubscriptBox[ StyleBox["g", FontSlant->"Italic"], "2"]], FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" = Min", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i=1,...,n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["{", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["a", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")\[RightTeeArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["b", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["i", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["i = ", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1,...,", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["}.", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" \n", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["The following is a program to calculate ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ SubscriptBox[ StyleBox["g", FontSlant->"Italic"], "2"]], FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[". Recall that i", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["n the case the triangular norm is the product, we have that ", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[RightTeeArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" = ", FontColor->GrayLevel[0]], StyleBox["Min", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["{", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["/", FontColor->GrayLevel[0]], StyleBox["x,", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["1}. 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Cell[BoxData[ \(" The implication-based fuzzy control"\)], "Print"] }, Open ]], Cell[TextData[{ "The result of this approach is rather unsatisfactory since the initial \ intuition is not well represented. This is not surprising. As an exampl", StyleBox["e, assume that ", FontFamily->"Times New Roman"], StyleBox["r ", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["is an input such that ", FontFamily->"Times New Roman"], StyleBox["Little", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["r", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[") is false. Then the formula ", FontFamily->"Times New Roman"], StyleBox["Little", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["r", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[")", FontFamily->"Times New Roman"], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Slow", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["(", FontFamily->"Times New Roman"], StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[") is true for any ", FontFamily->"Times New Roman"], StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["\[Element]", FontFamily->"Times New Roman"], StyleBox["Y ", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["and therefore any ", FontFamily->"Times New Roman"], StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox["\[Element]", FontFamily->"Times New Roman"], StyleBox["Y", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" gives a correct control, an absurdity.", FontFamily->"Times New Roman"], "\n\tAs matter of fact, we cannot interpret a system of IF-THEN fuzzy rules \ as a set of logical implications in a direct way. In the next section we will \ show how to translate the system into a set of formulas in fuzzy logic." }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["5. Fuzzy Control as a chapter of programming logic", "Section", FontFamily->"Times New Roman", FontSize->12], Cell[TextData[{ "The main idea in this section is that any fuzzy control system ", StyleBox["S", FontSlant->"Italic"], " (as usually defined) is representable as a fuzzy theory in a fuzzy logic, \ namely, by a fuzzy program. 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FontColor->GrayLevel[0]], StyleBox[")]\n\t", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["fast", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") \t\t \t[", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["fast", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["t", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")]\n", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["\t. . .\nwhere \n ", FontColor->GrayLevel[0]], StyleBox[" - ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["\[Lambda]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ \(\_1\)], FontFamily->"Times New Roman"], StyleBox[", ..., ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["\[Lambda]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ \(\_5\)], FontFamily->"Times New Roman"], StyleBox[" are elements in [0,1],\n -", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[" we write \[Alpha] ", FontColor->GrayLevel[0]], StyleBox[" ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ RowBox[{"[", SubscriptBox["\[Lambda]", StyleBox["i", FontSlant->"Italic"]]}]], FontFamily->"Times New Roman"], StyleBox["]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox[" to denote that ", FontColor->GrayLevel[0]], StyleBox["p", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(\[Alpha]) = ", FontColor->GrayLevel[0]], Cell[BoxData[ SubscriptBox["\[Lambda]", StyleBox["i", FontSlant->"Italic"]]], FontFamily->"Times New Roman"], StyleBox[",", FontColor->GrayLevel[0]], StyleBox["\n", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[" - we list only the clauses \[Alpha] for which ", FontColor->GrayLevel[0]], StyleBox["p", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(\[Alpha]) \[NotEqual] 0,\n - ", FontColor->GrayLevel[0]], StyleBox["X", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" and ", FontColor->GrayLevel[0]], StyleBox["Y ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["are finite approximations of the intervals [0,10] and [0,5], \ respectively,\n - ", FontColor->GrayLevel[0]], StyleBox["r", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[" is any element in ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["X", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[" and ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["t", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[" is any element in ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["Y", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[" we interpret as constants.\nThe idea is that the fuzzy program ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["p ", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["is not a tool to define a function ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["f ", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[":", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["X", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["\[RightArrow]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["Y", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[" representing the correct answer ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["f", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["(", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") given the input ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[". The fuzzy program ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["p", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox[" is able only to define a binary fuzzy predicate \"", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["g", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["ood", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\", expressing, given an input ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[", our graded opinion (degree of preference, tastes) on a possible \ control ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[". We denote by ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["g", FormatType->TextForm, FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["ood", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["r", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["t", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") the degree at which we can prove the fact ", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["g", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["ood", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["r", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["t", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") given the available information. Therefore, the number ", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["good", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["r", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["t", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") represents the information ", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["\"given r the control y is good at degree good", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["r,t", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")\". It is immediate that by applying the procedure of Section 3 \ for fuzzy logic programming in the case ", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["\[Lambda]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ \(\_1\)], FontFamily->"Times New Roman"], "=", StyleBox[" ...= ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["\[Lambda]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ \(\_5\)], FontFamily->"Times New Roman"], StyleBox[" ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["= 1, we obtain the fuzzy function proposed by the classical fuzzy \ control. In general, the fuzzy function ", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox["good", FormatType->TextForm, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" depends on the parameters ", FormatType->TextForm, FontColor->GrayLevel[0]], StyleBox[" ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["\[Lambda]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ \(\_1\)], FontFamily->"Times New Roman"], StyleBox[", ..., ", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["\[Lambda]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ \(\_5\)], FontFamily->"Times New Roman"], "." }], "Text", CellMargins->{{13, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, FontSize->12], Cell[BoxData[ \(good[x_, y_, l1_, l2_, l3_, l4_, l5_] := 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", FormatType->TextForm, FontFamily->"Times New Roman", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["\[Lambda]", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ \(\_5\)], FontFamily->"Times New Roman"], " have to be the result of a learning process to obtain a function ", StyleBox["g ", FontSlant->"Italic"], "sufficiently close to the ideal function ", StyleBox["f", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->True}], " . In this process it is useful to introduce further parameters in the \ fuzzy program. As an example in defining the interpretations of the \ predicates ", StyleBox["l", FontSlant->"Italic"], StyleBox["ittle", FontSlant->"Italic"], ", ", StyleBox["slow", FontSlant->"Italic"], ",... . This is in accordance with the fact that the interpretation of a \ vague predicate is, in turn, vague, i.e. there are infinite many fuzzy \ subsets which are reasonable interpretations of the predicate. Observe that \ our logical approach to fuzzy control makes the learning process in fuzzy \ control very similar to the learning process in inductive logic. In both the \ cases we have to find a logical theory fitting well with the available data. \ Moreover the learning process looks to be more simple in our case. This \ thanks to the presence of real parameters, the fact that we can accept \ approximate solutions and the fact that the whole process is continuous with \ respect to these parameters. " }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 6. Completeness, linguistic modifiers, negative information and others\ \>", "Section", FontFamily->"Times New Roman", FontSize->12], Cell[TextData[{ StyleBox["The logical approach to fuzzy control suggests new tools and \ possible extensions of the usual techniques. As an example, observe that the \ information about the exact controls are ", FontColor->GrayLevel[0]], StyleBox["complete", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" provided that for every possible input ", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" a control ", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" exists such that ", FontColor->GrayLevel[0]], StyleBox["good", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") is satisfied, i.e. provided that the \[ForAll]", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\[Exists]", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["good", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") is true. Then, in account of the fact that the existential \ quantifier is interpreted by the least upper bound and the universal \ quantifier by the greatest lower bound, we can propose the following \ definition.\n\n", FontColor->GrayLevel[0]], StyleBox["Definition 6.1.", FontFamily->"Times New Roman", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" The ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["degree of completeness", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" of a fuzzy function ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["good", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" is the number \n completeness[", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["good", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["] = ", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], Cell[BoxData[ StyleBox[ RowBox[{ SubscriptBox["Inf", RowBox[{ StyleBox["x", FontSlant->"Italic"], "\[Element]", StyleBox["X", FontSlant->"Italic"]}]], \(Sup\_\(y \[Element] Y\)\)}], FontSlant->"Italic"]], FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["good", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["(", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[",", FontFamily->"Times New Roman", FontColor->GrayLevel[0]], StyleBox["y", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[")", FontFamily->"Times New Roman", FontColor->GrayLevel[0]] }], "Text", TextAlignment->Left, TextJustification->1, FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[{ \(completeness[z_, l1_, l2_, l3_, l4_, l5_] := \n\t Min[Table[ Max[Table[ z[x, y, l1, l2, l3, l4, l5], {y, 0, 5, 0.2}]], \n\t\t\t\t\t{x, 0, 10, 0.2}]]\), "\[IndentingNewLine]", \(completeness[z_] := \n\t Min[Table[ Max[Table[z[x, y], {y, 0, 5, 0.2}]], \n\t\t\t\t\t{x, 0, 10, 0.2}]]\)}], "Input", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(completeness[good]\)], "Input", FontSize->12], Cell[BoxData[ \(0.3733333333333335`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(completeness[good, 0.4, 1, 1, 0.4, 1]\)], "Input", FontSize->12], Cell[BoxData[ \(0.26666666666666666`\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Another interesting possibility is the introduction of ", FontColor->GrayLevel[0]], StyleBox["linguistic modifiers ", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["in the language used to describe the correct control. As an \ example, we can define the linguistic modifiers ", FontColor->GrayLevel[0]], StyleBox["clearly", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["] and ", FontColor->GrayLevel[0]], StyleBox["vaguely", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["[", FontColor->GrayLevel[0]], StyleBox["x", FontFamily->"Times New Roman", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["]. 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