Belief merging within fragments of propositional logic

Recently, belief change within the framework of fragments of propositional logic has gained increasing attention. Previous works focused on belief contraction and belief revision on the Horn fragment. However, the problem of belief merging within fragments of propositional logic has been neglected so far. This paper presents a general approach to define new merging operators derived from existing ones such that the result of merging remains in the fragment under consideration. Our approach is not limited to the case of Horn fragment but applicable to any fragment of propositional logic characterized by a closure property on the sets of models of its formulae. We study the logical properties of the proposed operators in terms of satisfaction of merging postulates, considering in particular distance-based merging operators for Horn and Krom fragments.Comment: To appear in the Proceedings of the 15th International Workshop on Non-Monotonic Reasoning (NMR 2014

Undecidability of Multiplicative Subexponential Logic

Subexponential logic is a variant of linear logic with a family of exponential connectives--called subexponentials--that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that classical propositional multiplicative linear logic extended with one unrestricted and two incomparable linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

Expressing Additives Using Multiplicatives and Subexponentials

International audienceSubexponential logic is a variant of linear logic with a family of exponential connectives—called subex-ponentials—that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that a classical propositional multiplicative subexponential logic (MSEL) with one unrestricted and two linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable. We then show how the additive con-nectives can be directly simulated by giving an encoding of propositional multiplicative additive linear logic (MALL) in an MSEL with one unrestricted and four linear subexponentials

A minimal classical sequent calculus free of structural rules

Gentzen's classical sequent calculus LK has explicit structural rules for contraction and weakening. They can be absorbed (in a right-sided formulation) by replacing the axiom P,(not P) by Gamma,P,(not P) for any context Gamma, and replacing the original disjunction rule with Gamma,A,B implies Gamma,(A or B). This paper presents a classical sequent calculus which is also free of contraction and weakening, but more symmetrically: both contraction and weakening are absorbed into conjunction, leaving the axiom rule intact. It uses a blended conjunction rule, combining the standard context-sharing and context-splitting rules: Gamma,Delta,A and Gamma,Sigma,B implies Gamma,Delta,Sigma,(A and B). We refer to this system M as minimal sequent calculus. We prove a minimality theorem for the propositional fragment Mp: any propositional sequent calculus S (within a standard class of right-sided calculi) is complete if and only if S contains Mp (that is, each rule of Mp is derivable in S). Thus one can view M as a minimal complete core of Gentzen's LK.Comment: To appear in Annals of Pure and Applied Logic. 15 page

Generic Modal Cut Elimination Applied to Conditional Logics

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature