Abstract
The present chapter is devoted to an analysis of computational complexity of fuzzy propositional calculi and to the determination of the degree of undecidability of fuzzy predicate calculi. The results are important but will be only rarely used in other chapters, so the reader unfamiliar with the theory of (polynomial) complexity and/or arithmetical hierarchy may skip the chapter as a whole. On the other hand, in Section 1 we summarize all necessary material so that the non-expert in complexity or arithmetical hierarchy may also read the chapter and understand the results. In Section 2 we study the computational complexity of Ł, G, Π (propositional Łukasiewicz, Gödel and product logic) and show, among other things, that their sets of 1-tautologies T AUT Ł1 , T AUT G1 , T AUT Π1 are all co-NP-complete. In Section 3 we study the corresponding predicate calculi Ł∀, G∀, Π∀. We show that the set T AUT Π1 of Gödel predicate logic is Σ2; it remains open if it is in Π2 or is still more complex). Hence all these predicate calculi are undecidable.
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© 1998 Springer Science+Business Media Dordrecht
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Hájek, P. (1998). Questions of Computational Complexity and Undecidability. In: Metamathematics of Fuzzy Logic. Trends in Logic, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5300-3_6
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DOI: https://doi.org/10.1007/978-94-011-5300-3_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-0370-7
Online ISBN: 978-94-011-5300-3
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