Showing papers on "Normal modal logic published in 2010"

Gentzen Calculi for Modal Propositional Logic

Abstract: The book is about Gentzen calculi for (the main systems of) modal logic. It is divided into three parts. In the first part we introduce and discuss the main philosophical ideas related to proof theory, and we try to identify criteria for distinguishing good sequent calculi. In the second part we present the several attempts made from the 50's until today to provide modal logic with Gentzen calculi. In the third and and final part we analyse new calculi for modal logics, called tree-hypersequent calculi, which were recently introduced by the author. We show in a precise and clear way the main results that can be proved with and about them.

Standard gödel modal logics

Abstract: We prove strong completeness of the □-version and the ◊-version of a Godel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Godel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.

Maximal decidable fragments of Halpern and Shoham's modal logic of intervals

Abstract: In this paper, we focus our attention on the fragment of Halpern and Shoham's modal logic of intervals (HS) that features four modal operators corresponding to the relations "meets", "met by", "begun by", and "begins" of Allen's interval algebra (AABB logic). AABB properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BB of Allen's "begun by/begins" relations and propositional neighborhood logic AA, in its many variants (including metric ones). We prove that the satisfiability problem for AABB, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AABB turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AABB is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, Q, and R.

Modal logics with Belnapian truth values

Abstract: Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification (support of truth) and the other for falsification (support of falsity). Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong ...

On the logic of argumentation theory

Abstract: The paper applies modal logic to formalize fragments of argumentation theory. Such formalization allows to import, for free, a wealth of new notions (e.g., argument equivalence), new techniques (e.g., calculi, model-checking games, bisimulation games), and results (e.g., completeness of calculi, adequacy of games, complexity of model-checking) from logic to argumentation.

A logical framework to deal with variability

Abstract: We present a logical framework that is able to deal with variability in product family descriptions. The temporal logic MHML is based on the classical Hennessy-Milner logic with Until and we interpret it over Modal Transition Systems (MTSs). MTSs extend the classical notion of Labelled Transition Systems by distinguishing possible (may) and required (must) transitions: these two types of transitions are useful to describe variability in behavioural descriptions of product families. This leads to a novel deontic interpretation of the classical modal and temporal operators, which allows the expression of both constraints over the products of a family and constraints over their behaviour in a single logical framework. Finally, we sketch model-checking algorithms to verify MHML formulae as well as a way to derive correct products from a product family description.

Explicit Evidence Systems with Common Knowledge

Abstract: Justification logics are epistemic logics that explicitly include justifications for the agents' knowledge. We develop a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge. We define a Kripke-style semantics that is similar to Fitting's semantics for the Logic of Proofs LP. We show the soundness, completeness, and finite model property of our multi-agent justification logic with respect to this Kripke-style semantics. We demonstrate that our logic is a conservative extension of Yavorskaya's minimal bimodal explicit evidence logic, which is a two-agent version of LP. We discuss the relationship of our logic to the multi-agent modal logic S4 with common knowledge. Finally, we give a brief analysis of the coordinated attack problem in the newly developed language of our logic.

Model Checking Games for the Quantitative μ -Calculus

Abstract: We investigate quantitative extensions of modal logic and the modal μ-calculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It turns out that, if the quantitative μ-calculus is defined in an appropriate way respecting the duality properties between the logical operators, then its model checking problem can indeed be characterised by a quantitative variant of parity games. However, these quantitative games have quite different properties than their classical counterparts, in particular they are, in general, not positionally determined. The correspondence between the logic and the games goes both ways: the value of a formula on a quantitative transition system coincides with the value of the associated quantitative game, and conversely, the values of quantitative parity games are definable in the quantitative μ-calculus.

Rank-1 Modal Logics are Coalgebraic

Abstract: Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatized in rank 1. Here we establish the converse, i.e. every rank-1 modal logic has a sound and strongly complete coalgebraic semantics. This is achieved by constructing for a given modal logic a canonical coalgebraic semantics, consisting of a signature functor and interpretations of modal operators, which turns out to be final among all such structures. The canonical semantics may be seen as a coalgebraic reconstruction of neighbourhood semantics, broadly construed. A finitary restriction of the canonical semantics yields a canonical weakly complete semantics which moreover enjoys the Hennessy–Milner property. As a consequence, the machinery of coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, becomes applicable to arbitrary rank-1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of such results. As an extended example, we apply our framework to recently defined deontic logics. In particular, our methods lead to the new result that these logics are strongly complete.

Decomposing Modal Quantification

Abstract: Providing a compositional interpretation procedure for discourses in which descriptions of complex dependencies between interrelated objects are incrementally built is a key challenge for natural language semantics. This article focuses on the interactions between the entailment particle therefore, modalized conditionals and modal subordination. It shows that the dependencies between individuals and possibilities that emerge out of such interactions can receive a unified compositional account in a system couched in classical type logic that integrates and simplifies van den Berg's dynamic plural logic and the classical Lewis-Kratzer analysis of modal quantification. The main proposal is that modal quantification is a composite notion, to be decomposed in terms of discourse reference to quantificational dependencies that is multiply constrained by the various components that make up a modal quantifier. The system captures the truth-conditional and anaphoric components of modal quantification in an even-handed way and, unlike previous accounts, makes the propositional contents contributed by modal constructions available for subsequent discourse reference.

Uniform interpolation for monotone modal logic

Abstract: We reconstruct the syntax and semantics of monotone modal logic, in the style of Moss' coalgebraic logic. To that aim, we replace the box and diamond with a modality nabla which takes a finite collection of finite sets of formulas as its argument. The semantics of this modality in monotone neighborhood models is defined in terms of a version of relation lifting that is appropriate for this setting. We prove that the standard modal language and our r -based one are effectively equi-expressive, meaning that there are effective translations in both directions. We prove and discuss some algebraic laws that govern the interaction of nabla with the Boolean operations. These laws enable us to rewrite each formula into a special kind of disjunctive normal form that we call transparent. For such transparent formulas it is relatively easy to define the bisimulation quantifiers that one may associate with our notion of relation lifting. This allows us to prove the main result of the paper, viz., that monotone modal logic enjoys the property of uniform interpolation.

A Modal Logic of Epistemic Games

Abstract: We propose some variants of a multi-modal of joint action, preference and knowledge that support reasoning about epistemic games in strategic form. The first part of the paper deals with games with complete information. We first provide syntactic proofs of some well-known theorems in the area of interactive epistemology that specify some sufficient epistemic conditions of equilibrium notions such as Nash equilibrium and Iterated Deletion of Strictly Dominated Strategies (IDSDS). Then, we present a variant of the logic extended with dynamic operators of Dynamic Epistemic Logic (DEL). We show that it allows to express the notion IDSDS in a more compact way. The second part of the paper deals with games with weaker forms of complete information. We first discuss several assumptions on different aspects of perfect information about the game structure (e.g., the assumption that a player has perfect knowledge about the players’ strategy sets or about the preference orderings over strategy profiles), and show that every assumption is expressed by a corresponding logical axiom of our logic. Then we provide a proof of Harsanyi’s claim that all uncertainty about the structure of a game can be reduced to uncertainty about payoffs. Sound and complete axiomatizations of the logics are given, as well as some complexity results for the satisfiability problem.

Proof search specifications of bisimulation and modal logics for the π-calculus

Abstract: We specify the operational semantics and bisimulation relations for the finite φ-calculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The ∇ quantifier helps with the delicate issues surrounding the scope of variables within φ-calculus expressions and their executions (proofs). We illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no side-conditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arise from familar logic distinctions; the interplay between the three quantifiers (∀, ∃, and ∇) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for one-step transitions, bisimulation, and satisfaction in modal logic. We also illustrate how one can encode the φ-calculus with replications, in an extended logic with induction and co-induction.

Complexity results for modal dependence logic

Abstract: Modal dependence logic was introduced very recently by Vaananen. It enhances the basic modal language by an operator dep. For propositional variables p1, ..., pn, dep(p1, ..., pn-1; pn) intuitively states that the value of pn only depends on those of p1, ..., pn-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man's dependence logic, the language consisting of formulas built from literals and dependence atoms using ∧, □, ⋄ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Vaananen's language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy. In this way we completely classifiy the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Vaananen and Sevenster.

Avoiding deontic explosion by contextually restricting aggregation

Abstract: In this paper, we present an adaptive logic for deontic conflicts, called P2.1r, that is based on Goble's logic SDLaPe--a bimodal extension of Goble's logic P that invalidates aggregation for all prima facie obligations. The logic P2.1r has several advantages with respect to SDLaPe. For consistent sets of obligations it yields the same results as Standard Deontic Logic and for inconsistent sets of obligations, it validates aggregation "as much as possible". It thus leads to a richer consequence set than SDLaPe. The logic P2.1r avoids Goble's criticisms against other non-adjunctive systems of deontic logic. Moreover, it can handle all the 'toy examples' from the literature as well as more complex ones.

On modal logics of linear inequalities

Abstract: We consider probabilistic modal logic, graded modal logic and stochastic modal logic, where linear inequalities may be used to express numerical constraints between quantities For each of the logics, we construct a cut-free sequent calculus and show soundness with respect to a natural class of models The completeness of the associated sequent calculi is then established with the help of coalgebraic semantics which gives completeness over a (typically much smaller) class of models With respect to either semantics, it follows that the satisfiability problem of each of these logics is decidable in polynomial space

Modal logics with counting

Abstract: We present a modal language that includes explicit operators to count the number of elements that a model might include in the extension of a formula, and we discuss how this logic has been previously investigated under different guises. We show that the language is related to graded modalities and to hybrid logics. We illustrate a possible application of the language to the treatment of plural objects and queries in natural language. We investigate the expressive power of this logic via bisimulations, discuss the complexity of its satisfiability problem, define a new reasoning task that retrieves the cardinality bound of the extension of a given input formula, and provide an algorithm to solve it.

Label-free natural deduction systems for intuitionistic and classical modal logics

Abstract: In this paper we study natural deduction for the intuitionistic and classical (normal) modal logics obtained from the combinations of the axioms T, B, 4 and 5. In this context we introduce a new multi-contextual structure, called T-sequent, that allows to design simple labelfree natural deduction systems for these logics. After proving that they are sound and complete we show that they satisfy the normalization property and consequently the subformula property in the intuitionistic case.

B and D are enough to make the Halpern-Shoham logic undecidable

Abstract: The Halpern-Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We contribute to this effort by showing (among other results), that the BD fragment (where only the operators "begins" and "during" are allowed) is undecidable over discrete structures.

Knowledge compilation in the modal logic S5

Abstract: In this paper, we study the knowledge compilation task for propositional epistemic logic S5. We first extend many of the queries and transformations considered in the classical knowledge compilation map to S5. We then show that the notion of disjunctive normal form (DNF) can be profitably extended to the epistemic case; we prove that the DNF fragment of S5, when appropriately defined, satisfies essentially the same queries and transformations as its classical counterpart.

Optimal tableau algorithms for coalgebraic logics

Abstract: 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

Modal fixpoint logic: some model theoretic questions

Abstract: This thesis is a study into some model-theoretic aspects of the modal µ-calculus, the extension of modal logic with least and greatest fixpoint operators. We explore these aspects through a fine-structure approach to the µ-calculus. That is, we concentrate on special classes of structures and particular fragments of the language. The methods we use also illustrate the fruitful interaction between the µ-calculus and other methods from automata theory, game theory and model theory.

Frame correspondences in modal predicate logic

Abstract: Understanding modal predicate logic is a continuing challenge, both philosophical and mathematical. In this paper, I study this system in terms of frame correspondences, finding a number of definability results using substitution methods, including new analyses of axioms in intermediate intuitionistic predicate logics. The semantic arguments often have a different flavour from those in propositional modal logic. But eventually, I hit boundaries to first-order definability of frame conditions. I then relate these findings to the known incompleteness theorems for modal predicate logic, and point out some new directions for further research, including the use of strengthened higher-order proof systems for the basic modal language.