Showing papers on "Normal modal logic published in 2004"

Expressive Logics for Coalgebras via Terminal Sequence Induction

Abstract: This paper presents a logical characterization of coalgebraic behavioral equivalence. The characterization is given in terms of coalgebraic modal logic, an abstract framework for reasoning about, and specifying properties of, coalgebras, for an endofunctor on the category of sets. Its main feature is the use of predicate liftings which give rise to the interpretation of modal operators on coalgebras. We show that coalgebraic modal logic is adequate for reasoning about coalgebras, that is, behaviorally equivalent states cannot be distinguished by formulas of the logic. Subsequently, we isolate properties which also ensure expressiveness of the logic, that is, logical and behavioral equivalence coincide.

Connexive Modal Logic.

Abstract: Connexive logic is a neglected direction in non-classical logic. In the present paper, first an axiomatic system of connexive propositional logic is presented. This logic, C, is shown to be sound and complete with respect to a class of relational models. It seems that this semantics is, in fact, the first known intuitively plausible interpretation of a system of connexive logic. The presentation of C suggests that connexive logic is constructive. It is a variant of David Nelson’s constructive logics with strong negation. In Nelson’s logics the verification conditions of implications are dynamic, whereas all falsification conditions are static conditions of falsification on the spot. In C, both the verification and the falsification conditions of implications are dynamic. This is enough to ensure that C is connexive and can be given a comprehensible and clear interpretation in terms of information states. In a second step, the language of the system C is extended by the modal operators and ♦ to obtain a connexive analogue of the smallest normal modal propositional logic K. Aiming at a connexive analogue of K that can be faithfully embedded by a modal translation into QC, quantified C, we arrive at a system that will be called CK, connexive K. The system CK is a connexive version of the constructive modal logic FSK characterized in [12]. CK is shown to be sound and complete with respect to relational models and to be decidable. We shall also critically discuss the evaluation clauses for the modal operators that are induced by the standard translation from modal propositional logic into first-order logic. In the context of the connexive base logic, the falsification clauses of formulas A induced by the standard translation appear to be intuitively implausible. In any case, both syntactic duality axioms ∼ A ↔ ♦ ∼ A and ∼ ♦ ↔ ∼ A fail to hold. It seems that CK is the first system of connexive modal logic considered in the modal logic literature. This paper may therefore be seen as a contribution to establishing connexive modal logic as a respectable branch of modal logic. Advances in Modal Logic, Volume 5. c © 2005, Heinrich Wansing. 368 Heinrich Wansing 1 Aristotle’s Theses and Boethius’ Theses The following principle is well-known as “Aristotle’s Thesis”:

A Systematic Proof Theory for Several Modal Logics.

Abstract: The family of normal propositional modal logic systems is given a very systematic organisation by their model theory This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert-Lewis style and conditions on the accessibility relation on frames By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is ad-hoc While we can formulate several modal logics in the sequent calculus that enjoy cut-elimination, their formalisation arises through system-bysystem fine tuning to ensure that the cut-elimination holds, and the correspondence to the axioms of the Hilbert-Lewis systems becomes opaque This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the Hilbert-Lewis axiomatisation We show that the calculus possesses a cut-elimination property directly analogous to cut-elimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics

Utilitarian Deontic Logic.

Abstract: This paper aims to examine Horty’s proposal of utilitarian deontic logic [7]. It will focus on his dominance operators by way of simplified semantics. An axiomatization of the logic of the simplified semantics will be shown by the construction of a finite countermodel. Analysis of the proof suggests possibilities of further investigations. The presented version of Horty’s proposal does not make essential differences from standard deontic logic: the deontic operators are normal, and thus paradoxes of deontic logic occur. Non-monotonic agency operators should be considered.

On the Complexity of Fragments of Modal Logics

Abstract: We study and give a summary of the complexity of 15 basic normal monomodal logics under the restriction to the Horn fragment and/or bounded modal depth. As new results, we show that: a) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by k 2 in the modal logics K4 and KD4 is PSPACE-complete, in K is NP-complete; b) the satisfiability problem of modal formulas with modal depth bounded by 1 in K4, KD4, and S4 is NP-complete; c) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by 1 in K, K4, KD4, and S4 is PTIME-complete. In this work, we also study the complexity of the multimodal logics Ln under the mentioned restrictions, where L is one of the 15 basic monomodal logics. We show that, for n 2 : a) the satisfiability problem of sets of Horn modal clauses in K5n, KD5n, K45n, and KD45n is PSPACE-complete; b) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by k 2 in Kn, KBn, K5n, K45n, KB5n is NP-complete, and in KDn, Tn, KDBn, Bn, KD5n, KD45n, S5n is PTIME-complete.

Inflationary fixed points in modal logic

Abstract: We consider an extension of modal logic with an operator for constructing inflationary fixed points, just as the modal μ-calculus extends basic modal logic with an operator for least fixed points. Least and inflationary fixed-point operators have been studied and compared in other contexts, particularly in finite model theory, where it is known that the logics IFP and LFP that result from adding such fixed-point operators to first-order logic have equal expressive power. As we show, the situation in modal logic is quite different, as the modal iteration calculus (MIC), we introduce has much greater expressive power than the μ-calculus. Greater expressive power comes at a cost: the calculus is algorithmically much less manageable.

A logic for reasoning about coherent conditional probability: A modal fuzzy logic approach

Abstract: In this paper we define a logic to reason about coherent conditional probability, in the sense of de Finetti. Under this view, a conditional probability μ(· | ·) is a primitive notion that applies over conditional events of the form “ϕgiven ψ”, where ψ is not the impossible event. Our approach exploits an idea already used by Hajek and colleagues to define a logic for (unconditional) probability in the frame of fuzzy logics. Namely, in our logic for each pair of classical propositions ϕ and ψ, we take the probability of the conditional event “ϕgiven ψ”, ϕ∣ψ for short, as the truth-value of the (fuzzy) modal proposition P(ϕ | ψ), read as “ϕ∣ψ is probable”. Based on this idea we define a fuzzy modal logic FCP(ŁΠ), built up over the many-valued logic Ł\(\Pi\frac{1}{2}\) (a logic which combines the well-known Lukasiewicz and Product fuzzy logics), which is shown to be complete with respect to the class of probabilistic Kripke structures induced by coherent conditional probabilities. Finally, we show that checking coherence of a probability assessment to an arbitrary family of conditional events is tantamount to checking consistency of a suitable defined theory over the logic FCP(ŁΠ).

Filtering unification and most general unifiers in modal logic

Abstract: We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2+ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

Termination in Modal Kleene Algebra

Abstract: Modal Kleene algebras (MKAs) are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. The paper formalizes and compares different notions of termination, including Lob’s formula, in MKA. It studies exhaustive iteration and gives calculational proofs of two fundamental termination-dependent statements from rewriting theory: the well-founded union theorem by Bachmair and Dershowitz and Newman’s lemma. These results are also of general interest for the termination analysis of programs and state transition systems.

A temporalised belief logic for specifying the dynamics of trust for multi-agent systems

Abstract: Temporalisation is a methodology for combining logics whereby a given logic system can be enriched with temporal features to create a new logic system. TML (Typed Modal Logic) extends classical first-order logic with typed variables and multiple belief modal operators; it can be applied to the description of, and reasoning about, trust for multi-agent systems. Without the introduction of a temporal dimension, this logic may not be able to express the dynamics of trust. In this paper, adopting the temporalisation method, we combine TML with a temporal logic to obtain a new logic, so that the users can specify the dynamics of trust and model evolving theories of trust for multi-agent systems.

Independence-Friendly Modal Logic. Studies in its Expressive Power and Theoretical Relevance.

Abstract: The doctoral dissertation introduces independence-friendly (IF) modal logic as an extension of standard modal logic Making use of the notion of uniform strategy, a game-theoretical interpretation of IF modal logic is formulated It is shown that under this interpretation, IF modal logic has greater expressive power than standard modal logic, and can be translated into first-order logic However, when restricted to a simple tense-logical setting (evaluation over strict linear orders), its expressive power coincides with standard tense logic The syntax of IF modal logic can be modified to allow independence of modal operators from conjunctions and disjunctions It is shown that the resulting modal logic can no longer be translated into first-order logic Two further interpretations of the language of IF modal logic are given, one in terms of backwards-looking operators, the other algebraic The dissertation contains an extensive discussion of tenses in linguis- tics, and explains how the 'backwards-looking operators' interpretation of IF tense logic makes it possible to formally distinguish between two types of independence appearing in connection with tense evaluation It is argued that the linguistic critique against scope theories of tense becomes less appealing when this distinction is made

A Spatial Logic for the Hybrid π -Calculus

Abstract: We present a formal logic for stating properties of systems expressed in the hybrid π-calculus, or Φ-calculus. It is a very expressive logic, subsuming many standard logics like CTL and the modal μ-calculus. Because the π-calculus and the Φ-calculus allow passing of names to achieve reconfigurability of hybrid systems, and because we must abstract over these names, the logic (a hybrid extension of a logic by Caires and Cardelli for the π-calculus) uses a new method of defining abstractions – FM set theory – for expressing syntax and semantics of Φ-calculus models, and for expressing the semantics of spatial logic. We provide several new constructions for this logic, including an assume-guarantee principle, and illustrate many of the semantic features using an extended example of a robotic parts feeder and parts carrier in a minifactory.

The Knower Paradox in the light of provability interpretations of modal logic

Abstract: This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368--387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338-- 355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287--304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy.

A New Combination Procedure for the Word Problem That Generalizes Fusion Decidability Results in Modal Logics

Abstract: Previous results for combining decision procedures for the word problem in the non-disjoint case do not apply to equational theories induced by modal logics—whose combination is not disjoint since they share the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other equational theories.

Modal matters in interpretability logics

Abstract: In this paper we expose a method for building models for interpretability logics. The method can be compared to the method of taking unions of chains in classical model theory. Many applications of the method share a common part. We isolate this common part in a main lemma. Doing so, many of our results become applications of this main lemma. We also briefly describe how our method can be generalized to modal logics with a different signature. With the general method, we prove completeness for the interpretability logics IL, ILM, ILM0 and ILW*. We also apply our method to obtain a classification of the essential Σ1-sentences of essentially reflexive theories. We briefly comment on such a classification for finitely axiomatizable theories. As a digression we proof some results on self-provers. Towards the end of the paper we concentrate on modal matters con- cerning IL(All), the interpretability logic of all reasonable arithmetical theories. We prove the modal incompleteness of the logic ILW*P0. We put forward a new principle R, and show it to be arithmetically sound in any reasonable arithmetical theory. Finally we make some general remarks on the logics ILRW and IL(All).

Modal Logics for Products of Topologies

Abstract: We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 ⊕ S4. We axiomatize the modal logic of products of topological spaces with horizontal, vertical, and standard product topologies. We prove that both of these logics are complete for the product of rational numbers Q× Q with the appropriate topologies.

Definability in Normal Extensions of S4

Abstract: A projective Beth property, PB2, in normal modal logics extending S4 is studied. A convenient criterion is furnished for PB2 to be valid in a larger family of extensions of K4. All locally tabular extensions of the Grzegorczyk logic with PB2 are described. Superintuitionistic logics with the projective Beth property that have no modal companions with this property are found.

All normal extensions of S5-squared are finitely axiomatizable

Abstract: We prove that every normal extension of the bi-modal system S52 is finitely axiomatizable and that every proper normal extension has NP-complete satisfiability problem.

Fuzzy Reasoning Based on Propositional Modal Logic

Abstract: In order to deal with some vague assertions more efficiently, fuzzy modal logics have been discussed by many researchers. This paper introduces the notation of fuzzy assertion based on propositional modal logic. As an extension of the traditional semantics about the modal logics, the fuzzy Kripke semantics are considered and the formal system of the fuzzy reasoning based on propositional modal logic is established and the properties about the satisfiability of the reasoning system are discussed.

A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC{Part II

Abstract: We expand classical first order logic by formalizing a fragment of its metatheory, namely adding a predicate “is a theorem” (`) in its modes of expression. We do this by embedding the classical logic into a very basic version of modal logic, letting the latter’s modal operator play the role of the predicate “is a theorem”. We conclude with a number of illustrations of use and a proof of the conservatism of the extended logic: If it proves A for a classical formula A, then A is indeed a classical theorem.

Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity

Abstract: Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural two-sorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result is established between the category of sober algebras and the category of general Kripke structures. A simple enrichment of the proposed sequent calculi is proved to be complete over standard Kripke structures. The calculi are shown to be analytic in a useful sense.