Multi-modal Logic Research Articles

On a theorem of Vaught for first order logic with finitely many variables

We prove that the existence of atomic models for countable atomic theories does not hold for Ln the first order logic restricted to n variables for finite n > 2. Our proof is algebraic, via polyadic algebras. We note that Lnhas been studied in recent times as a multi-modal logic with applications in computer science. 2000 MATHEMATICS SUBJECT CLASSIFICATION. 03C07, 03G15.

CTL AgentSpeak(L): A specification language for agent programs

This work introduces CTL AgentSpeak(L), a logic to specify and verify expected properties of rational agents implemented in the well-known agent oriented programming language AgentSpeak(L). Our approach is closely related to the BDI CTL multi-modal logic, used to reason about agents in terms of their beliefs ( B), desires ( D), intentions ( I), and the temporal logic CTL. A new interpretation for the temporal operators, grounded in the transition system induced by the operational semantics of AgentSpeak(L), is proposed. The main contribution of the approach is a better understanding of the relation between the programming language and its logical specification, enabling us to prove expected or desired properties for any agent programmed in the language, e.g., commitment strategies. The results, as well as the specification language proposed, are very useful to reconcile computational and philosophical aspects of practical reasoning, e.g., approaching single-minded commitment as a policy-based reconsideration case.

Cut-elimination for knowledge logics with interaction

In the article, multimodal logics K4n and S4n with the central agent axiom are analysed. The Hilbert type calculi are presented, then the Gentzen type calculi with cut are derived, and the proofs of thecut-eliminationtheorems are outlined. The work shows that it is possible to construct an analytical Gentzen type calculi for these logics.

Loop-check elimination for non-transitive distributed knowledge logic

A non-transitive distributed knowledge logic TnD, obtained from multi-modal logic Tn by adding distributed knowledge operator, is considered. Sound and complete loop-check-free sequent calculus for this logic is proposed. Termination of derivations in proposed calculus is justified.

Multimodal Separation Logic for Reasoning About Operational Semantics

We show how to reason, in the proof assistant Coq, about realistic programming languages using a combination of separation logic and heterogeneous multimodal logic. A heterogeneous multimodal logic is a logic with several modal operators that are not required to satisfy the same frame conditions. The result is a powerful and elegant system for reasoning about programming languages and their semantics. The techniques are quite general and can be adopted to a wide variety of settings.

PSPACE complexity of modal logic K D45n

We study sequent calculus for multi-modal logic K D45n and its complexity. We introduce a loop-check free sequent calculus. Loop-check is eliminated by using the marked modal operator □i•, which is used as an alternative to sequents with histories ([8], [3], [5]). All inference rules are invertible or semi-invertible. To get this, we use or branches beside common and branches. We prove the equivalence between known sequent calculus and our newly introduced efficient sequent calculus.

Termination of Derivations in a Fragment of Transitive Distributed Knowledge Logic

A transitive distributed knowledge logic is considered. The considered logic S4 nD is obtained from multi-modal logic S4 n by adding transitive distributed knowledge operator. For a fragment of this logic loop-check-free sequent calculus is proposed. The considered fragment is such that it can be applied for specification and verification of safety properties of knowledge-based distributed systems. By relying on the constructed loop-check-free sequent calculus a PSPACE procedure to determine a termination of backward derivation in considered fragment of the logic S4 nD is presented.

Multi-modal and Temporal Logics with Universal Formula--Reduction of Admissibility to Validity and Unification

The article studies multi-modal (in particular temporal, tense, logics) possessing universal formulas. We prove (Theorems 11 and 12) that the admissibility problem for inference rules (inference rules with parameters) is decidable for all decidable multi-modal (in particular, temporal) logics possessing an universal formula and decidable w.r.t. unification (unification with parameters). These theorems use characterizations of admissible rules in terms of valid rules and unifiable formulas. Results are applied to wide range of multi-modal logics, including all linear transitive temporal logics, all temporal logics with bounded zigzag, all multi-modal logics with explicit universal modality. As consequence, we show that the admissibility problem for all such logics is decidable (e.g. for all logics of the series S4nU, n∈ N).

Constructing Finite Least Kripke Models for Positive Logic Programs in Serial Regular Grammar Logics

A serial context-free grammar logic is a normal multimodal logic L characterized by the seriality axioms and a set of inclusion axioms of the form 2tφ→ 2s1 . . .2skφ. Such an inclusion axiom corresponds to the grammar rule t→ s1 . . . sk. Thus the inclusion axioms of L capture a context-free grammar G(L). If for every modal index t, the set of words derivable from t using G(L) is a regular language, then L is a serial regular grammar logic. In this paper, we present an algorithm that, given a positive multimodal logic program P and a set of finite automata specifying a serial regular grammar logic L, constructs a finite least L-model of P . (A model M is less than or equal to model M ′ if for every positive formula φ, if M |= φ then M ′ |= φ.) A least L-model M of P has the property that for every positive formula φ, P |= φ iff M |= φ. The algorithm runs in exponential time and returns a model with size 2 3). We give examples of P and L, for both of the case when L is fixed or P is fixed, such that every finite least L-model of P must have size 2. We also prove that if G is a context-free grammar and L is the serial grammar logic corresponding to G then there exists a finite least L-model of 2sp iff the set of words derivable from s using G is a regular language.

Multi-modal nonmonotonic logics of minimal knowledge

In this paper we introduce multi-modal logics of minimal knowledge. Such a family of logics constitutes the first proposal in the field of epistemic nonmonotonic logic in which the three following aspects are simultaneously addressed: (1) the possibility of formalizing multiple agents through multiple modal operators; (2) the possibility of using first-order quantification in the modal language; (3) the possibility of formalizing nonmonotonic reasoning abilities for the agents modeled, based on the principle of minimal knowledge. We illustrate the expressive capabilities of multi-modal logics of minimal knowledge to provide a formal semantics to peer-to-peer data integration systems, which constitute one of the most recent and complex architectures for distributed information systems.

An Axiomatisation for the Multi-modal Logic of Knowledge and Linear Time LTK

The paper aims at providing the multi-modal propositional logic LTK with a sound and complete axiomatisation. This logic combines temporal and epistemic operators and focuses on m odeling the behaviour of a set of agents operating in a system on the background of a temporal framework. Time is represented as linear and discrete, whereas knowledge is modeled as an S5-like modality. A further modal operator intended to represent environment knowledge is added to the system in order to achieve the expressive power sufficient to describe the piece of information available to the agents at each moment in the flow of time.

Undecidability of Multi-modal Hybrid Logics

This paper establishes undecidability of satisfiability for multi-modal logic equipped with the hybrid binder ↓, with respect to frame classes over which the same language with only one modality is decidable. This is in contrast to the usual behaviour of many modal and hybrid logics, whose uni-modal and multi-modal versions do not differ in terms of decidability and, quite often, complexity. The results from this paper apply to a wide range of frame classes including temporally and epistemically relevant ones.

Logics with the universal modality and admissible consecutions

In this paper1 we study admissible consecutions (inference rules) in multi-modal logics with the universal modality. We consider extensions of multi-modal logic S4n augmented with the universal modality. Admissible consecutions form the largest class of rules, under which a logic (as a set of theorems) is closed. We propose an approach based on the context effective finite model property. Theorem 7, the main result of the paper, gives sufficient conditions for decidability of admissible consecutions in our logics. This theorem also provides an explicit algorithm for recognizing such consecutions. Some applications to particular logics with the universal modality are given.

Multimodal logic programming

We give a framework for developing the least model semantics, fixpoint semantics, and SLD-resolution calculi for logic programs in multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. ∀ x ∃ y R i ( x , y ) ) and some classical first-order Horn clauses. Our approach is direct and no special restriction on occurrences of □ i and ⋄ i is required. We apply our framework for a large class of basic serial multimodal logics, which are parameterized by an arbitrary combination of generalized versions of axioms T, B, 4, 5 (in the form, e.g. 4 : □ i ϕ → □ j □ k ϕ ) and I : □ i ϕ → □ j ϕ . Another part of the work is devoted to programming in multimodal logics intended for reasoning about multidegree belief, for use in distributed systems of belief, or for reasoning about epistemic states of agents in multiagent systems. For that we also use the framework, and although these latter logics belong to the mentioned class of basic serial multimodal logics, the special SLD-resolution calculi proposed for them are more efficient.

Disjunction and modular goal-directed proof search

This article explores goal-directed proof search in first-order multimodal logic. I focus on a family of modal logics which offer the expressive power to specify modular goals and local assumptions. A modular goal must be proved from designated assumptions; conversely, a local assumption can only be used to prove a designated goal. Indefinite modal specifications can avoid combinatorial interactions among independent ambiguities by making separate goals modular and corresponding disjunctive alternatives local. Such specifications can effectively guarantee that provable goals have short proofs. The key result of this article is to establish a sound and complete goal-directed proof system that actively uses the modularity and locality of modal logic to constrain proof search. In particular, logically independent, local ambiguities will not interact in proof search. The challenge is that, in goal-directed proof, a modal prover cannot simply reason locally, in a module, because modularity is a property of formulas rather than proof problems. The result therefore requires prior proof-theoretic justifications of logic programming to be extended, strengthened, and combined with new proof-theoretic analyses of modal deduction.

Logic of Visibility, Perception, and Knowledge and Admissible Inference Rules

We investigate admissible inference rules for the multi-modal logic VSK+ extending the logic VSK – the logic of Visibility, Perception and Knowledge. The logic VSK has been introduced by M.Wooldridge and A. Lomuscio [21]. VSK was intended for reasoning about properties of computational agents situated in some environment. Admissible rules are important for modelling of logical consequence. We consider these rules for VSK+, the logic of a wise agent (one which knows anything visible). The main result of our paper is the construction of an algorithm which determines admissible inference rules in VSK+. The algorithm is based on the proof of existence of computable bounds on the size of special Kripke 3-frames refuting inadmissible rules.

A Logic For Ignorance

We introduce and motivate a non-standard multi-modal logic to represent and reason about ignorance in Multi-Agent Systems. We argue that in Multi-agent systems being able to reason about what agents ignore is just as important as being able to reason about what agents know. We show a sound and complete axiomatisation for the logic. We investigate its applicability by restating the feasibility condition for the FIPA communication primitive of inform.

Minimal structures for modal tableaux: Some examples

In this paper we present some examples of decision procedures based on tableau calculus for some mono- and multimodal logics having a semantics involving properties that are not easily representable in tree-like structures (like e.g. density, confluence and persistence). We show how to handle them in our framework by generalizing usual tableaux (which are trees) to richer structures: rooted directed acyclic graphs.

A first step towardsmodeling semistructured data in hybrid multimodal logic

XML documents and, more generally, semistructured data, can be seen as labelled graphs. In this paper we set a correspondence between such graphs and the models of a language of hybrid multimodal logic. This allows us to characterize a schema for semistructured data as a formula of hybrid multimodal logic, and instances of the schema (data graphs) as models of this formula. We also investigate how to express in such a logic integrity constraints on semistructured data, in particular some classes of constraints widely considered in the literature. The contribution of this work is twofold: 1) We generalize the notion of schema, by proposing a definition of schema were references are “well typed” (contrary to what happens with DTDs). 2) We formalize semistructured data, schema for semistructured data and integrity constraints in a unique framework, namely hybrid multimodal logic.

Residuation, Structural Rules and Context Freeness

The article presents proofs of the context freeness of a family of type logical grammars, namely all grammars that are based on a uni- or multimodal logic of pure residuation, possibly enriched with the structural rules of Permutation and Expansion for binary modes.