SAT in Monadic Gödel Logics: A Borderline between Decidability and Undecidability

Abstract

Abstract. We investigate satisfiability in the monadic fragment of firstorder Gödel logics. These are a family of finite-and infinite-valued logics where the sets of truth values V are closed subsets of [0, 1] containing 0 and 1. We identify conditions on the topological type of V that determine the decidability or undecidability of their satisfiability problem.

Syntactic Cut-Elimination for Intuitionistic Fuzzy Logic via Linear Nested Sequents

“…Moreover, the present formalism could offer insight regarding which fragments interpolate (or if all of IF interpolates) by applying the so-called proof-theoretic method of interpolation [18,20]. Additionally, it could be fruitful to adapt linear nested sequents to other first-order Gödel logics and to investigate decidable fragments [2] by providing proof-search algorithms with implementations (e.g. [17] provides an implementation of proof-search in Prolog for a class of modal logics within the linear nested sequent framework).…”

Arithmetical complexity of fuzzy predicate logics — A survey II

“…In [2] the authors prove that the set of monadic formulas in prenex form that are (standard) tautologies of the Gödel logic (over [0, 1]) is undecidable. See also [1,5].…”

“…It is natural to ask what exactly is the complexity of these Π 2 -hard sets; we have given no answer (except the known result on V ↑ itself); but we have shown that adding the Delta connective to the language makes the sets of tautologies of these logics non-arithmetical. See also [5].…”

First-order Nilpotent minimum logics: first steps

“…The main one is the search for a full classification, in analogy with the one done for Gödel logics in [BPZ07, BCF07, Pre03], of the (existence of) first-order logics associated to the various subalgebras of [0, 1] NM : for the subalgebras whose set of first-order tautologies is not recursively axiomatizable, instead, it could be studied its arithmetical complexity (for Gödel logics this has been done in [BPZ07, BCF07, Pre03, Háj10a, Háj10b, Háj05]). Another theme that has not been analysed here concerns the (first-order) satisfiability problem about the subalgebras of [0, 1] NM (for Gödel logics this has been done in [BCP09]).…”