An analytic tableau calculus for a temporalised belief logic

A tableau is a refutation-based decision procedure for a related logic, and is among the most popular proof procedures for modal logics. In this paper, we present a labelled tableau calculus for a temporalised belief logic called TML⁺, which is obtained by adding a linear-time temporal logic onto a belief logic by the temporalisation method of Finger and Gabbay. We first establish the soundness and the completeness of the labelled tableau calculus based on the soundness and completeness results of its constituent logics. We then sketch a resolution-type proof procedure that complements the tableau calculus and also propose a model checking algorithm for TML⁺ based on the recent results for model checking procedures for temporalised logics. TML⁺ is suitable for formalising trust and agent beliefs and reasoning about their evolution for agent-based systems. Based on the logic TML⁺, the proposed labelled tableau calculus could be used for analysis, design and verification of agent-based systems operating in dynamic environments.     

A Sound and Complete Tableau Calculus for Reasoning about only Knowing and Knowing at Most

We define a tableau calculus for the logic of only knowing and knowing at most ON, which is an extension of Levesque's logic of only knowing O. The method is based on the possible-world semantics of the logic ON, and can be considered as an extension of known tableau calculi for modal logic K45. From the technical viewpoint, the main features of such an extension are the explicit representation of "unreachable" worlds in the tableau, and an additional branch closure condition implementing the property that each world must be either reachable or unreachable. The calculus allows for establishing the computational complexity of reasoning about only knowing and knowing at most. Moreover, we prove that the method matches the worst-case complexity lower bound of the satisfiability problem for both ON and O. With respect to [22], in which the tableau calculus was originally presented, in this paper we both provide a formal proof of soundness and completeness of the calculus, and prove the complexity results for the logic ON.     

A reasoning method for a paraconsistent logic

A proof method for automation of reasoning in a paraconsistent logic, the calculus C1* of da Costa, is presented. The method is analytical, using a specially designed tableau system. Actually two tableau systems were created. A first one, with a small number of rules in order to be mathematically convenient, is used to prove the soundness and the completeness of the method. The other one, which is equivalent to the former, is a system of derived rules designed to enhance computational efficiency. A prototype based on this second system was effectively implemented.     

Analytic Tableaux for all of <Emphasis Type="Italic">SIXTEEN</Emphasis><Subscript>3</Subscript>

In this paper we give an analytic tableau calculus P L _{1 6} for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ? _t , ? _f , ? _i , and ? under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment relations will in general require developing four tableaux, while proving that they are in the ? relation may require six.     

An Axiomatic System and a Tableau Calculus for STIT Imagination Logic

We formulate a Hilbert-style axiomatic system and a tableau calculus for the STIT-based logic of imagination recently proposed in Wansing (2015). Completeness of the axiom system is shown by the method of canonical models; completeness of the tableau system is also shown by using standard methods.     

Tableaus for many-valued modal logic

We continue a series of papers on a family of many-valued modal logics, a family whose Kripke semantics involves many-valued accessibility relations. Earlier papers in the series presented a motivation in terms of a multiple-expert semantics. They also proved completeness of sequent calculus formulations for the logics, formulations using a cut rule in an essential way. In this paper a novel cut-free tableau formulation is presented, and its completeness is proved.Research partly supported by NSF Grant CCR-9104015.     

A general tableau method for propositional interval temporal logics: Theory and implementation

In this paper, we focus our attention on tableau methods for propositional interval temporal logics. These logics provide a natural framework for representing and reasoning about temporal properties in several areas of computer science. However, while various tableau methods have been developed for linear and branching time point-based temporal logics, not much work has been done on tableau methods for interval-based ones. We develop a general tableau method for Venema's CDT logic interpreted over partial orders (BCDT⁺ for short). It combines features of the classical tableau method for first-order logic with those of explicit tableau methods for modal logics with constraint label management, and it can be easily tailored to most propositional interval temporal logics proposed in the literature. We prove its soundness and completeness, and we show how it has been implemented.     

Temporalising Tableaux

As a remedy for the bad computational behaviour of first-order temporal logic (FOTL), it has recently been proposed to restrict the application of temporal operators to formulas with at most one free variable thereby obtaining so-called monodic fragments of FOTL. In this paper, we are concerned with constructing tableau algorithms for monodic fragments based on decidable fragments of first-order logic like the two-variable fragment or the guarded fragment. We present a general framework that shows how existing decision procedures for first-order fragments can be used for constructing a tableau algorithm for the corresponding monodic fragment of FOTL. Some example instantiations of the framework are presented.     

Theory matrices (for modal logics) using alphabetical monotonicity

In this paper I give conditions under which a matrix characterisation of validity is correct for first order logics where quantifications are restricted by statements from a theory. Unfortunately the usual definition of path closure in a matrix is unsuitable and a less pleasant definition must be used. I derive the matrix theorem from syntactic analysis of a suitable tableau system, but by choosing a tableau system for restricted quantification I generalise Wallen's earlier work on modal logics. The tableau system is only correct if a new condition I call alphabetical monotonicity holds. I sketch how the result can be applied to a wide range of logics such as first order variants of many standard modal logics, including non-serial modal logics.     

概称句词项逻辑的树图判定算法

概称句的形式刻画研究始于人工智能。从条件蕴涵引入开始,到建立概称句词项逻辑的形式系统GAG和Gaa,关于概称句这一系列的研究主要是围绕概称句自身性质的探讨,以试图对于概称句推理给出更合理的形式刻画,而没有同时兼顾计算机应用方面的考虑。回归问题的初始,关于概称句的概念理论是否还可以用于计算机科学领域,是这一研究路线所面临的问题。首先要解决的问题是,根据GAG和Gaa模型,公式的可满足性是否有能行的判定方法。对此本文给出了基于GAG语义的树图判定算法,包括相应的可靠性,完备性等证明。     

Gts and interrogative tableaux

A variant of the standard deductive tableau system is introduced, and interrogative rules are added, resulting in a so-called interrogative tableau system. A game-theoretical account of entailment is sketched, and the deductive tableau system is interpreted in these terms. Finally, it is shown how to extend this account of entailment into an account of interrogative entailment, thereby providing a semantics for the interrogative tableau system.     

The relative complexity of analytic tableaux and SL-resolution

In this paper we describe an improvement of Smullyan's analytic tableau method for the propositional calculus-Improved Parent Clash Restricted (IPCR) tableau-and show that it is equivalent to SL-resolution in complexity.     

A Gentzen Calculus for Nothing but the Truth

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof.     

SLAP: Specification logic of actions with probability

A logic for specifying probabilistic transition systems is presented. Our perspective is that of agents performing actions. A procedure for deciding whether sentences in this logic are valid is provided. One of the main contributions of the paper is the formulation of the decision procedure: a tableau system which appeals to solving systems of linear equations. The tableau rules eliminate propositional connectives, then, for all open branches of the tableau tree, systems of linear equations are generated and checked for feasibility. Proofs of soundness, completeness and termination of the decision procedure are provided.     

Completeness Theorems via the Double Dual Functor

The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particular, we obtain completeness theorems for Moisil calculus, n-valued Łukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods. This revised version was published online in June 2006 with corrections to the Cover Date.     

Cut Elimination inside a Deep Inference System for Classical Predicate Logic

Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic. As a consequence we derive Herbrand's Theorem, which we express as a factorisation of derivations.     

A Contraction-free and Cut-free Sequent Calculus for Propositional Dynamic Logic

In this paper we present a sequent calculus for propositional dynamic logic built using an enriched version of the tree-hypersequent method and including an infinitary rule for the iteration operator. We prove that this sequent calculus is theoremwise equivalent to the corresponding Hilbert-style system, and that it is contraction-free and cut-free. All results are proved in a purely syntactic way.     

Completeness of quantum logic

This paper is based on a semantic foundation of quantum logic which makes use of dialog-games. In the first part of the paper the dialogic method is introduced and under the conditions of quantum mechanical measurements the rules of a dialog-game about quantum mechanical propositions are established. In the second part of the paper the quantum mechanical dialog-game is replaced by a calculus of quantum logic. As the main part of the paper we show that the calculus of quantum logic is complete and consistent with respect to the dialogic semantics. Since the dialog-game does not involve the excluded middle the calculus represents a calculus of effective (intuitionistic) quantum logic.In a forthcoming paper it is shown that this calculus is equivalent to a calculus of sequents and more interestingly to a calculus of propositions. With the addition of the excluded middle the latter calculus is a model for the lattice of subspaces of a Hilbert space.On leave of absence from the Institut für Theoretische Physik der Universität zu Köln, W.-Germany.