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Register a new userAction Logic is Undecidable
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Added by
Stepan Kuznetsov
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Stepan Kuznetsov
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Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. This logic involves Kleene star, axiomatized by an induction scheme. For a stronger system which uses an $omega$-rule instead (infinitary action logic) Buszkowski and Palka (2007) have proved $Pi_1^0$-completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by D. Kozen in 1994. In this article, we show that it is undecidable, more precisely, $Sigma_1^0$-complete. We also prove the same complexity results for all recursively enumerable logics between action logic and infinitary action logic; for fragments of those only one of the two lattice (additive) connectives; for action logic extended with the law of distributivity.
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This paper is concerned with the first-order paraconsistent logic LPQ$^{supset,mathsf{F}}$. A sequent-style natural deduction proof system for this logic is presented and, for this proof system, both a model-theoretic justification and a logical justification by means of an embedding into first-order classical logic is given. For no logic that is essentially the same as LPQ$^{supset,mathsf{F}}$, a natural deduction proof system is currently available in the literature. The given embedding provides both a classical-logic explanation of this logic and a logical justification of its proof system. The major properties of LPQ$^{supset,mathsf{F}}$ are also treated.
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The finite models of a universal sentence $Phi$ in a finite relational signature are the age of a homogeneous structure if and only if $Phi$ has the amalgamation property. We prove that the computational problem whether a given universal sentence $Phi$ has the amalgamation property is PSPACE-hard, even if $Phi$ is additionally Horn and the signature of $Phi$ only contains relation symbols of arity at most three. The decidability of the problem remains open.
In this paper we provide two new semantics for proofs in the constructive modal logics CK and CD. The first semantics is given by extending the syntax of combinatorial proofs for propositional intuitionistic logic, in which proofs are factorised in a linear fragment (arena net) and a parallel weakening-contraction fragment (skew fibration). In particular we provide an encoding of modal formulas by means of directed graphs (modal arenas), and an encoding of linear proofs as modal arenas equipped with vertex partitions satisfying topological criteria. The second semantics is given by means of winning innocent strategies of a two-player game over modal arenas. This is given by extending the Heijltjes-Hughes-Stra{ss}burger correspondence between intuitionistic combinatorial proofs and winning innocent strategies in a Hyland-Ong arena. Using our first result, we provide a characterisation of winning strategies for games on a modal arena corresponding to proofs with modalities.
We solve some decision problems for timed automata which were recently raised by S. Tripakis in [ Folk Theorems on the Determinization and Minimization of Timed Automata, in the Proceedings of the International Workshop FORMATS2003, LNCS, Volume 2791, p. 182-188, 2004 ] and by E. Asarin in [ Challenges in Timed Languages, From Applied Theory to Basic Theory, Bulletin of the EATCS, Volume 83, p. 106-120, 2004 ]. In particular, we show that one cannot decide whether a given timed automaton is determinizable or whether the complement of a timed regular language is timed regular. We show that the problem of the minimization of the number of clocks of a timed automaton is undecidable. It is also undecidable whether the shuffle of two timed regular languages is timed regular. We show that in the case of timed Buchi automata accepting infinite timed words some of these problems are Pi^1_1-hard, hence highly undecidable (located beyond the arithmetical hierarchy).
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