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Reasoning on imprecisely defined functions. (English) Zbl 1007.03024

Novák, Vilém (ed.) et al., Discovering the world with fuzzy logic. Heidelberg: Physica-Verlag. Stud. Fuzziness Soft Comput. 57, 331-366 (2000).
Standard functions assign to input elements single output elements. Formally, they can be described by cases (for finite range), i.e., by means of a standard partition of the input space \(X\) induced from the range of the discussed function. The problem of finding an “inverse function” \((y \mapsto x)\) can be transformed into a deduction theorem of classical propositional logic. The reviewed book chapter generalizes the above problem to the case of imprecisely defined functions linked to MV-algebras and to infinite-valued Łukasiewicz logic. The chapter starts with a motivating prologue including an example (recognition of hand-written letters), showing the need to compute the inverse of an imprecisely specified function \(F\), but also showing its links to the infinite-valued calculus of Łukasiewicz. Later it is shown that the infinite-valued consequence relation provides a universal algorithm for analyzing imprecisely defined functions. For example, the existence of a Turing machine is shown that is able to decide in a finite number of steps the satisfiability of the block-recognition problem (observe that the author works with finite MV-partitions). The results of this chapter promise interesting applications in computer-aided deduction and decision in several domains where imprecisely defined functions occur.
For the entire collection see [Zbl 0979.00015].

MSC:

03B52 Fuzzy logic; logic of vagueness
06D35 MV-algebras
68T37 Reasoning under uncertainty in the context of artificial intelligence
68T10 Pattern recognition, speech recognition
03B50 Many-valued logic
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