Preservation of Craig interpolation by the product of matrix logics

The product of matrix logics, possibly with additional interaction axioms, is shown to preserve a slightly relaxed notion of Craig interpolation. The result is established symbolically, capitalizing on the complete axiomatization of the product of matrix logics provided by their meet-combination. Along the way preservation of the metatheorem of deduction is also proved. The computation of the interpolant in the resulting logic is proved to be polynomially reducible to the computation of the interpolants in the two given logics. Illustrations are provided for classical, intuitionistic and modal propositional logics.     

The Monodic Fragment of Propositional Term Modal Logic

We study term modal logics, where modalities can be indexed by variables that can be quantified over. We suggest that these logics are appropriate for reasoning about systems of unboundedly many reasoners and define a notion of bisimulation which preserves propositional fragment of term modal logics. Also we show that the propositional fragment is already undecidable but that its monodic fragment (formulas using only one free variable in the scope of a modality) is decidable, and expressive enough to include interesting assertions.

     

Free-Variable Tableaux for Propositional Modal Logics

Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.     

基于广义谢弗竖的分析性模态公理系统

沿着安德森等人开创的方向,我们将分析性公理系统从经典逻辑推向模态逻辑,所定义的广义谢弗竖混合了模态词和广义析舍。在这篇论文中,我们给出常见的正规模态逻辑的分析性公理系统及其强完全性定理和插值定理,并讨论演绎关系的性质：单调性和切割性。     

Normal Modal Substructural Logics with Strong Negation

We introduce modal propositional substructural logics with strong negation, and prove the completeness theorems (with respect to Kripke models) for these logics.     

Solving the $100 modal logic challenge

We present the theoretical foundation, design, and implementation, of a system that automatically determines the subset relation between two given axiomatizations of propositional modal logics. This is an open problem for automated theorem proving. Our system solves all but six out of 121 instances formed from 11 common axiomatizations of seven modal logics. Thus, although the problem is undecidable in general, our approach is empirically successful in practically relevant situations.     

A Loop-Free Decision Procedure for Modal Propositional Logics K4, S4 and S5

The aim of this paper is to present a loop-free decision procedure for modal propositional logics K4, S4 and S5. We prove that the procedure terminates and that it is sound and complete. The procedure is based on the method of Socratic proofs for modal logics, which is grounded in the logic of questions IEL.     

Kripke bundles for intermediate predicate logics and Kripke frames for intuitionistic modal logics

Shehtman and Skvortsov introduced Kripke bundles as semantics of non-classical first-order predicate logics. We show the structural equivalence between Kripke bundles for intermediate predicate logics and Kripke-type frames for intuitionistic modal prepositional logics. This equivalence enables us to develop the semantical study of relations between intermediate predicate logics and intuitionistic modal propositional logics. New examples of modal counterparts of intermediate predicate logics are given.The author would like to express his gratitude to Professor Hiroakira Ono for his comments, and to Professor Tadashi Kuroda for his encouragement.The author wishes to express his gratitude to Professors V. B. Shehtman, D. P. Skvortsov and M. Takano for their comments.     

Every Finitely Reducible Logic has the Finite Model Property with Respect to the Class of ♦-Formulae

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from these classes have the finite model property with respect to the class of -formulae, i.e. each -formula has a -model iff it has a finite -model. Roughly speaking, a -formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators.     

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.     

Natural Deduction for Non-Classical Logics

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of soundness, completeness and proof normalization. We have implemented our work in the Isabelle Logical Framework.     

Arrow Logic and Infinite Counting

We consider arrow logics (i.e., propositional multi-modal logics having three -- a dyadic, a monadic, and a constant -- modal operators) augmented with various kinds of infinite counting modalities, such as 'much more', 'of good quantity', 'many times'. It is shown that the addition of these modal operators to weakly associative arrow logic results in finitely axiomatizable and decidable logics, which fail to have the finite base property.     

Coinductive models and normal forms for modal logics (or how we learned to stop worrying and love coinduction)

We present a coinductive definition of models for modal logics and show that it provides a homogeneous framework in which it is possible to include different modal languages ranging from classical modalities to operators from hybrid and memory logics. Moreover, results that had to be proved separately for each different language—but whose proofs were known to be mere routine—now can be proved in a general way. We show, for example, that we can have a unique definition of bisimulation for all these languages, and prove a single invariance-under-bisimulation theorem.We then use the new framework to investigate normal forms for modal logics. The normal form we introduce may have a smaller modal depth than the original formula, and it is inspired by global modalities like the universal modality and the satisfiability operator from hybrid logics. These modalities can be extracted from under the scope of other operators. We provide a general definition of extractable modalities and show how to compute extracted normal forms. As it is the case with other classical normal forms—e.g., the conjunctive normal form of propositional logic—the extracted normal form of a formula can be exponentially bigger than the original formula, if we require the two formulas to be equivalent. If we only require equi-satisfiability, then every modal formula has an extracted normal form which is only polynomially bigger than the original formula, and it can be computed in polynomial time.     

Term-Modal Logics

Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators.This gives us a flexible and uniform language for reasoning about the agents themselves and their knowledge. This article gives examples of the expressiveness of the languages and provides sequent-style and tableau-based proof systems for the logics. Furthermore we give proofs of soundness and completeness with respect to the possible world semantics.     

Indexed systems of sequents and cut-elimination

Cut reductions are defined for a Kripke-style formulation of modal logic in terms of indexed systems of sequents. A detailed proof of the normalization (cut-elimination) theorem is given. The proof is uniform for the propositional modal systems with all combinations of reflexivity, symmetry and transitivity for the accessibility relation. Some new transformations of derivations (compared to standard sequent formulations) are needed, and some additional properties are to be checked. The display formulations of the systems considered can be presented as encodings of Kripke-style formulations.     

On Modal Logics Characterized by Models with Relative Accessibility Relations: Part II

This work is divided in two papers (Part I and Part II). In Part I, we introduced the class of Rare-logics for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability were established by faithfully translating the Rare-logics into more standard modal logics (some of them contain the universal modal operator).In Part II, we push forward the results from Part I. For Rare-logics with nominals (present at the level of formulae and at the level of modal expressions), we show that the constructions from Part I can be extended although it is technically more involved. We also characterize a class of standard modal logics for which the universal modal operator can be eliminated as far as satifiability is concerned. Although the previous results have a semantic flavour, we are also able to define proof systems for Rare-logics from existing proof systems for the corresponding standard modal logics. Last, but not least, decidability results for Rare-logics are established uniformly, in particular for information logics derived from rough set theory.Since this paper is the continuation of Part I, we do not recall here the definitions of Part I although we refer to them.     

On Modal Logics of Partial Recursive Functions

The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established. Presented by Melvin Fitting     

Amalgamation and interpolation in normal modal logics

This is a survey of results on interpolation in propositional normal modal logics. Interpolation properties of these logics are closely connected with amalgamation properties of varieties of modal algebras. Therefore, the results on interpolation are also reformulated in terms of amalgamation.     

Sequent-systems and groupoid models. I

The purpose of this paper is to connect the proof theory and the model theory of a family of propositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related toBCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hubert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.The first part of the paper, which follows, contains the first two sections, which deal with sequent-systems and Hubert-formulations. The second part, due to appear in the next issue of this journal, will contain the third section, which deals with groupoid models.