Another Generalization of Connexive Logic C

The method used in Chapter 4 to show that every displayable logic enjoys strong cut-elimination was derived from the proof of strong normalizability of typed λ-terms. It does not only apply to display calculi. The present chapter is devoted to a proof of strong cut-elimination in a labelled tableau calculus for the (constant domain) modal predicate logic QS5. Modal tableau calculi which build in the accessibility relation of possible worlds models were first introduced by Kripke [93] and were later ‘linearized’ by various authors, notably Fitting [62], [63], [64] and Mints [114]. As in Gabbay’s [68] theory of labelled deductive systems, the basic declarative unit of these tableau calculi is not just a formula A, but rather a formula plus label (σ, A). In the case of the modal logic S5 the label σ may just be a single positive integer, whereas in general it is a non-empty finite sequence of positive integers. Moreover, for S5 the accessibility relation between labels may be universal and hence neglected. In contrast to labelled tableaux, the modal tableau systems of, for example, Rautenberg [137] and Gore [74] do not use labelled formulas. For a general survey on tableau methods for modal and tense logics, see [79]. The use of labels allows to formulate tableau calculi for certain extensions of the minimal normal modal logic K by imposing constraints on accessibility and on occurrences and the shape of labels on tableau branches. These constraints may be regarded as structural in the sense of not referring to any connectives. In order to emphasize the relation to sequent calculi, we shall work with a tableau calculus TQS5 based on the ordinary notion of a sequent. By defining suitable mappings on cut-free closed tableaux it can easily be shown that the result of dropping cut from TQS5 is equivalent to Fitting’s tableau calculus for first-order S5 with respect to provable formulas (see Section 7.5). Usually modal tableau calculi are formulated without a cut rule. The admissibility of cut is, however, of interest for constructive proofs of equivalence with Hilbert-type systems; compare [93, p. 82]. Moreover, non-constructive proofs of cut-elimination are of little appeal when it comes to extending the notion of formuals-as-types to modal logic (see, for instance, [28], [106]). It is this respect in which the present chapter may be seen to have significance.

We give sound and complete Hilbert-style axiomatizations for propositional dependence logic (PD), modal dependence logic (MDL), and extended modal dependence logic (EMDL) by extending existing axiomatizations for propositional logic and modal logic. In addition, we give novel labeled tableau calculi for PD, MDL, and EMDL. We prove soundness, completeness and termination for each of the labeled calculi.

The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic $$\mathsf{BK}^\Box $$ , which lacks a primitive possibility operator $$\Diamond $$ , is definitionally equivalent with the logic $$\mathsf{BK}$$ , which has both $$\Box $$ and $$\Diamond $$ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with $$\mathsf{BK}^\Box $$ without the absurdity constant. Moreover, a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a four-valued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic $$\mathsf{BK}^\Box \times \mathsf{BK}^\Box $$ over the non-modal vocabulary of MBL. On the way from $$\mathsf{BK}^\Box $$ to MBL, the Fischer Servi-style modal logic $$\mathsf{BK}^\mathsf{FS}$$ is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and $$\mathsf{BK}^\mathsf{FS}$$ is shown to be characterized by the class of all models for $$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$ . Moreover, $$\mathsf{BK}^\mathsf{FS}$$ is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for $$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$ . Moreover, the notion of definitional equivalence is suitably weakened, so as to show that $$\mathsf{BK}^\mathsf{FS}$$ and $$\mathsf{BK}^{\Box }\times \mathsf{BK}^{\Box }$$ are weakly definitionally equivalent.

Our interest in this paper are semantic tableau approaches closely related to bottom-up model generation methods. Using equality-based blocking techniques these can be used to decide logics representable in first-order logic that have the finite model property. Many common modal and description logics have these properties and can therefore be decided in this way. This paper integrates congruence closure, which is probably the most powerful and efficient way to realise reasoning with ground equations, into a modal tableau system with equality-based blocking. The system is described for an extension of modal logic K characterised by frames in which the accessibility relation is transitive and every world has a distinct immediate predecessor. We show the system is sound and complete, and discuss how various forms of blocking such as ancestor blocking can be realised in this setting. Though the investigation is focussed on a particular modal logic, the modal logic was chosen to show the most salient ideas and techniques for the results to be generalised to other tableau calculi and other logics.

The paper presents a tableau calculus for a linear time temporal logic for reasoning about processes and events in concurrent systems. The logic is based on temporal connectives in the style of Transaction Logic [BK94] and explicit quantification over states. The language extends first-order logic with sequential and parallel conjunction, parallel disjunction, and temporal implication. Explicit quantification over states via state variables allows to express temporal properties which cannot be formulated in modal logics. Using the tableau representation of temporal Kripke structures presented for CTL in [MS96] which represents states by prefix terms, explicit quantification over states is integrated into the tableau calculus by an adaptation of the δ-rule from first-order tableau calculi to the linear ordering of the universe of states. Complementing the CTL calculus, the paper shows that this tableau representation is both suitable for modal temporal logics and for logics using temporal connectives.

Tableaux for constructive concurrent dynamic logic

HTab: a Terminating Tableaux System for Hybrid Logic

The long-standing problem of the complete axiomatization of the propositional mu -calculus introduced by D. Kozen (1983) is addressed. The approach can be roughly described as a modified tableau method in the sense that infinite trees labeled with sets of formulas are investigated. The tableau method has already been used in the original paper by Kozen. The reexamination of the general tableau method presented is due to advances in automata theory, especially S. Safra's determinization procedure (1988), connections between automata on infinite trees and games, and experience with the model checking. A finitary complete axiom system for the mu -calculus is obtained. It can be roughly described as a system for propositional modal logic with the addition of a induction rule to reason about least fixpoints. >

Logics for time intervals provide a natural framework for representing and reasoning about timing properties in various 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 temporal logics. In this paper, we introduce a new, very expressive propositional interval temporal logic, called (Non-Strict) Branching CDT (BCDT + ) which extends most of the propositional interval temporal logics proposed in the literature. Then, we provide BCDT + with a generic tableau method which combines features of explicit tableau methods for modal logics with constraint label management and the classical tableau method for first-order logic, and we prove its soundness and completeness.

In this paper we investigate the two-variable fragment of modal logics of relations interpreted on local squares, LC2. A labelled tableau calculus is presented and its soundness and completeness are proven. Further, a termination proof enables us to use the calculus as a theorem prover. The prover has been implemented in Prolog, and we give a short system description. The paper also contains examples for how the system works including translations from other modal logics into LC2. Key words: modal logic, tableau system, theorem proving, guarded fragment

Deciding whether a modal formula is satisfiable with respect to a given set of (global) assumptions is a question of fundamental importance in applications of logic in computer science. Tableau methods have proved extremely versatile for solving this problem for many different individual logics but they typically do not meet the known complexity bounds for the logics in question. Recently, it has been shown that optimality can be obtained for some logics while retaining practicality by using a technique called “global caching”. Here, we show that global caching is applicable to all logics that can be equipped with coalgebraic semantics, for example, classical modal logic, graded modal logic, probabilistic modal logic and coalition logic. In particular, the coalgebraic approach also covers logics that combine these various features. We thus show that global caching is a widely applicable technique and also provide foundations for optimal tableau algorithms that uniformly apply to a large class of modal logics.

An automatic theorem prover for a proof system in the style of dual tableaux for the relational logic associated with modal logic has been introduced. Although there are many well-known implementations of provers for modal logic, as far as we know, it is the first implementation of a specific relational prover for a standard modal logic. There are two main contributions in this paper. First, the implementation of new rules, called () and (), which substitute the classical relational rules for composition and negation of composition in order to guarantee not only that every proof tree is finite but also to decrease the number of applied rules in dual tableaux. Second, the implementation of an order of application of the rules which ensures that the proof tree obtained is unique. As a consequence, we have implemented a decision procedure for modal logic . Moreover, this work would be the basis for successive extensions of this logic, such as , and .

In Automated Deduction for non classical logics and specially for modal logics, efficient (and not only complete) strategies are needed. Ordering strategies are presented for the Fitting's tableaux calculi. Besides orderings of tableaux rules, different variants for backtracking are used. The strategies apply to most usual propositional modal logics: K, T, K4, S4, D, D4, C, CT, C4, CS4, CD, CD4, G. More precisely, they apply to logics for which there exists a tableaux calculus such that the number of sets of formulas introduced in a tree is finite -the analytic tableaux systems satisfy this requirement. The strategies are proved to be complete for most of these logics. The results are presented for S4. The strategies have been implemented and extensively experimented in the tableaux theorem prover running on our Inference Laboratory ATINF. Experiments have shown the efficiency of some of the proposed strategies.

Relational dual tableau decision procedures and their applications to modal and intuitionistic logics

Memory logics are modal logics whose semantics is specified in terms of relational models enriched with additional data structure to represent memory. The logical language is then extended with a collection of operations to access and modify the data structure. In this paper we study their satisfiability and the model checking problems.We first give sound and complete tableaux calculi for the memory logic \(\mathcal {ML}\)(ⓚ, ⓡ, ⓔ) (the basic modal language extended with the operator ⓡ used to memorize a state, the operator ⓔ used to wipe out the memory, and the operator ⓚ used to check if the current point of evaluation is memorized) and some of its sublanguages. As the satisfiability problem of \(\mathcal {ML}\)(ⓚ, ⓡ, ⓔ) is undecidable, the tableau calculus we present is non terminating. Hence, we furthermore study a variation that ensures termination, at the expense of completeness, and we use model checking to ensure soundness. Secondly, we show that the model checking problem is PSpace-complete.KeywordsModel CheckModal LogicHybrid LogicModel Check ProblemModel Check AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.