Showing papers on "Modal operator published in 2010"

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.

Gradable epistemic modals, probability, and scale structure

Abstract: The epistemic modals possible, probable, likely, and certain require a semantics which explains their behavior both as modal operators and as gradable adjectives. An analysis of these items in terms of Kennedy & McNally's theory of gradability suggests that they are associated with a single, fully closed scale of possibility. An implementation using the standard theory of modality due to Kratzer is shown to make incorrect predictions in several domains. However, if the scale of possibility is identified with standard numerical probability, the facts about gradability are explained and the undesirable predictions of Kratzer's theory are avoided.

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.

Decidability of the Logics of the Reflexive Sub-interval and Super-interval Relations over Finite Linear Orders

Abstract: An interval temporal logic is a propositional, multi-modal logic interpreted over interval structures of partial orders. The semantics of each modal operator are given in the standard way with respect to one of the natural accessibility relations defined on such interval structures. In this paper, we consider the modal operators based on the (reflexive) sub-interval relation and the (reflexive) super-interval relation. We show that the satisfiability problems for the interval temporal logics featuring either or both of these modalities, interpreted over interval structures of finite linear orders, are all PSPACE-complete. These results fill a gap in the known complexity results for interval temporal logics.

Algebraic Notions of Termination

Abstract: Five algebraic notions of termination are formalised, analysed and compared: wellfoundedness or Noetherity, L\"ob's formula, absence of infinite iteration, absence of divergence and normalisation. The study is based on modal semirings, which are additively idempotent semirings with forward and backward modal operators. To model infinite behaviours, idempotent semirings are extended to divergence semirings, divergence Kleene algebras and omega algebras. The resulting notions and techniques are used in calculational proofs of classical theorems of rewriting theory. These applications show that modal semirings are powerful tools for reasoning algebraically about the finite and infinite dynamics of programs and transition systems.

De re and de dicto modality in islamic traditional logic

Abstract: In Islamic traditional textbooks modal operators sometimes come before propositions, sometimes before the predicates and sometimes at the end of propositions. This makes the interpretation of modality in each case as de re or de dicto difficult. Given Ibn SςnΪ's discussion of and sensitivity to this distinction, in this paper by examining the position of modality in the contradictories and converses of modal categorical propositions as well as their positions in modal syllogisms I will try to find a reasonable answer to this important issue. Key Terms: Ibn SςnΪ, Khūnajς, modality de re, modality de dicto, traditional logic, conversion, syllogism, contradiction.

The Complexity of Reasoning for Fragments of Autoepistemic Logic

Abstract: Autoepistemic logic extends propositional logic by the modal operator L. A formula that is preceded by an L is said to be "believed". The logic was introduced by Moore 1985 for modeling an ideally rational agent's behavior and reasoning about his own beliefs. In this paper we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of counting the number of stable expansions of a given knowledge base. To the best of our knowledge this is the first paper analyzing the counting problem for autoepistemic logic.

Completeness of a Branching-Time Logic with Possible Choices

Abstract: In this paper we present BTC, which is a complete logic for branchingtime whose modal operator quantifies over histories and whose temporal operators involve a restricted quantification over histories in a given possible choice. This is a technical novelty, since the operators of the usual logics for branching-time such as CTL express an unrestricted quantification over histories and moments. The value of the apparatus we introduce is connected to those logics of agency that are interpreted on branching-time, as for instance Stit Logics.

A modal view of the semantics of theoretical sentences

Abstract: Modal logic has been applied in many different areas, as reasoning about time, knowledge and belief, necessity and possibility, to mention only some examples. In the present paper, an attempt is made to use modal logic to account for the semantics of theoretical sentences in scientific language. Theoretical sentences have been studied extensively since the work of Ramsey and Carnap. The present attempt at a modal analysis is motivated by there being several intended interpretations of the theoretical terms once these terms are introduced through the axioms of a theory.

A Dynamic Logic of Agency II: Deterministic VCA, Coalition Logic, and Game Theory

Abstract: We continue the work initiated in Herzig and Lorini (J Logic Lang Inform, in press) whose aim is to provide a minimalistic logical framework combining the expressiveness of dynamic logic in which actions are first-class citizens in the object language, with the expressiveness of logics of agency such as STIT and logics of group capabilities such as CL and ATL. We present a logic called WCA {Deterministic Dynamic logic of Agency) which supports reasoning about actions and joint actions of agents and coalitions, and agentive and coalitional capabilities. In WCA it is sup- posed that, once all agents have selected a joint action, the effect of this joint action is deterministic. In order to assess WCA we prove that it embeds Coalition Logic. We then extend WCA with modal operators for agents' preferences, and show that the resulting logic is sufficiently expressive to capture the game-theoretic concepts of best response and Nash equilibrium.

Cut-Elimination and Proof Search for Bi-Intuitionistic Tense Logic

Abstract: We consider an extension of bi-intuitionistic logic with the traditional modalities from tense logic Kt. Proof theoretically, this extension is obtained simply by extending an existing sequent calculus for bi-intuitionistic logic with typical inference rules for the modalities used in display logics. As it turns out, the resulting calculus, LBiKt, seems to be more basic than most intuitionistic tense or modal logics considered in the literature, in particular, those studied by Ewald and Simpson, as it does not assume any a priori relationship between the diamond and the box modal operators. We recover Ewald's intuitionistic tense logic and Simpson's intuitionistic modal logic by modularly extending LBiKt with additional structural rules. The calculus LBiKt is formulated in a variant of display calculus, using a form of sequents called nested sequents. Cut elimination is proved for LBiKt, using a technique similar to that used in display calculi. As in display calculi, the inference rules of LBiKt are ``shallow'' rules, in the sense that they act on top-level formulae in a nested sequent. The calculus LBiKt is ill-suited for backward proof search due to the presence of certain structural rules called ``display postulates'' and the contraction rules on arbitrary structures. We show that these structural rules can be made redundant in another calculus, DBiKt, which uses deep inference, allowing one to apply inference rules at an arbitrary depth in a nested sequent. We prove the equivalence between LBiKt and DBiKt and outline a proof search strategy for DBiKt. We also give a Kripke semantics and prove that LBiKt is sound with respect to the semantics, but completeness is still an open problem. We then discuss various extensions of LBiKt.

Applying possibility and belief operators to conditional statements

Abstract: In this paper we outline a model for grounding of natural language statements being conditional statements extended with modal operators of possibility and belief. This work extends a detailed theory of modal language grounding proposed elsewhere. At first, some comments and intuitive semantics of modal conditional statements are presented. At second, a general structure of formal model, that covers this semantics is proposed. This model refers to the idea of possible worlds semantics. The grounding of modal conditional statements is defined by means of original concepts of epistemic satisfaction relation. The work sets up theoretical basis for further analytic and experimental evaluations.

A proof system for temporal reasoning with sequential information

Abstract: A new logic, sequence-indexed linear-time temporal logic (SLTL), is obtained semantically from the standard linear-time temporal logic LTL by adding a sequence modal operator which represents a sequence of symbols. By the sequence modal operator of SLTL, we can appropriately express "sequential information" in temporal reasoning. A Gentzen-type sequent calculus for SLTL is introduced, and the completeness and cut-elimination theorems for this calculus are proved. SLTL is also shown to be PSPACE-complete and embeddable into LTL.

Intention change via local assignments

Abstract: We present a logical approach to intention change. Inspired by Bratman's theory, we define intention as the choice to perform a given action at a certain time point in the future. This notion is modeled in a modal logic containing a temporal modality and modal operators for belief and choice. Intention change is then modeled by a specific kind of dynamic operator, that we call 'local assignment'. This is an operation on the model that changes the truth value of atomic formulae at specific time points. Two particular kinds of intention change are considered in some detail: intention generation and intention reconsideration.

A Logical Model of Intention and Plan Dynamics

Abstract: We propose a formal semantics of intention and plan dynamics based on the notion of local assignment. The function of a local assignment is to change the truth value of a given proposition at a specific time point along a history. We combine a static modal logic including a temporal modality and modal operators for mental attitudes belief and choice, with three kinds of dynamic modalities and corresponding three kinds of local assignments operating on agent's beliefs, on agent's choices and on the physical world. An agent's intention is defined in our approach as the agent's choice to perform a given action at a certain time point in the future and two operations called intention generation and intention reconsideration are defined as specific kinds of local assignments on choices. In Section 1 we introduce a static logic of time, action, and mental attitudes. In Section 2 we add the dynamic notion of local assignment to the logic of Section 1. In Section 3, we focus on two specific kinds of local assignment on choice which allow to model the processes of intention and plan generation and reconsideration.

Learning in a changing world, an algebraic modal logical approach

Abstract: We develop an algebraic modal logic that combines epistemic and dynamic modalities with a view to modelling information acquisition (learning) by automated agents in a changing world. Unlike most treatments of dynamic epistemic logic, we have transitions that "change the state" of the underlying system and not just the state of knowledge of the agents. The key novel feature that emerges is the need to have a way of "inverting transitions" and distinguishing between transitions that "really happen" and transitions that are possible. Our approach is algebraic, rather than being based on a Kripke-style semantics. The semantics are given in terms of quantales. We study a class of quantales with the appropriate inverse operations and prove properties of the setting. We illustrate the ideas with toy robot-navigation problems. These illustrate how an agent learns information by taking actions.

A Timed Logic for Modeling and Reasoning about Security Protocols.

Abstract: Many logical methods are usually considered suitable to express the static properties of security protocols while unsuitable to model dynamic processes or properties. However, a security protocol itself is in fact a dynamic process over time, and sometimes it is important to be able to express time-dependent security properties of protocols. In this paper, we present a new timed logic based on predicate modal logic, in which time is explicitly expressed in parameters of predicates or modal operators. This makes it possible to model an agent’s actions, knowledge and beliefs at different and exact time points, which enables us to model both protocols and their properties, especially time-dependent properties. We formalize semantics of the presented logic, and prove its soundness. We also present a modeling scheme for formalizing protocols and security properties of authentication and secrecy under the logic. The scheme provides a flexible and succinct framework to reason about security protocols, and essentially enhances the power of logical methods for protocol analysis. As a case study, we then analyze a timed-release protocol using this framework, and discover a new vulnerability that did not appear previously in the literature. We provide a further example to show additional advantages of the modeling scheme in the new logic.

Inconsistency-adaptive modal logics: on how to cope with modal inconsistency

Abstract: In this paper, I will characterize a new class of inconsistency-adaptive logics, namely inconsistency-adaptive modal logics. These logics cope with inconsistencies in a modal context. More specifically, when faced with inconsistencies, inconsistency-adaptive modal logics avoid explosion, but still allow the derivation of sufficient consequences to adequately explicate the part of human reasoning they are intended for.

A dyadic operator for the gradation of desirability

Abstract: We propose a normal modal deontic logic based on a dyadic operator, similar in structure to the temporal "until". By bringing significant expressiveness to the logic, it allows both the definition of a monadic desirability operator similar to the SDL obligation, and the expression of the relative level of desirability of target formulae. The interpretation of this logic on a linear structure of worlds ordered by desirability makes its semantics more intuitive and concrete than the SDL deontic accessibility relation. We also show that the core modality of the logic permits to represent the Chisholm and Forrester paradoxes of deontic logic in a more precise way, which does not lead to inconsistencies.

Topological Modal Logics with Difference Modality

Abstract: We consider propositional modal logic with two modal operators $\Box$ and $\D$. In topological semantics $\Box$ is interpreted as an interior operator and $\D$ as difference. We show that some important topological properties are expressible in this language. In addition, we present a few logics and proofs of f.m.p. and of completeness theorems.

Tableaux for the Lambek-Grishin calculus

Abstract: Categorial type logics, pioneered by Lambek, seek a proof-theoretic understanding of natural language syntax by identifying categories with formulas and derivations with proofs. We typically observe an intuitionistic bias: a structural configuration of hypotheses (a constituent) derives a single conclusion (the category assigned to it). Acting upon suggestions of Grishin to dualize the logical vocabulary, Moortgat proposed the Lambek-Grishin calculus (LG) with the aim of restoring symmetry between hypotheses and conclusions. We develop a theory of labeled modal tableaux for LG, inspired by the interpretation of its connectives as binary modal operators in the relational semantics of Kurtonina and Moortgat. As a linguistic application of our method, we show that grammars based on LG are context-free through use of an interpolation lemma. This result complements that of Melissen, who proved that LG augmented by mixed associativity and -commutativity was exceeds LTAG in expressive power.

A Multi-Context System Computing Modalities

Abstract: The system description presents the conception and a pro- totypical implementation of a multi-context system, used for computing and implementing temporal modalities within given data without the use of modal operators. Instead, an external constraint based rule sys- tem is used for computing the corresponding temporal relations, making use of the way a multi-context system works for transporting the needed information between contexts and knowledge bases.

Lecture Notes on Judgments and Propositions

Abstract: Logic is the study of reasoning. Since most mathematicians believe they are working on the discovery of objective and absolute truths, mathematical logic has focused on how to justify judgments A is true and A is false, where A denotes propositions about mathematical objects such as integers or real numbers. Every proposition is objectively either true or false, and logical connectives are functions on truth values. Philosophical logic takes a broader view and investigates how to reason about complex judgments such as A is possible, A is necessary, K knows A, K believes A, K affirms A and even A is obligatory or A is permitted, where K denotes agents or principals and A denotes propositions. Modal operators such as “K knows” can not be merely functions on truth values. For example, whetherK knows A does not depend solely on whether A is true or false. Judgments have a subjective quality which is generally denied in mathematical logic. In computer science, both objective and subjective views on logic have numerous applications. The objective or classical approach, most influenced by mathematics, is exemplified by Hoare logic to reason about imperative programs. Classical program logic defines programs as a new kind of mathematical object, their meaning being given explicitly as functions from states to states. The classical approach then develops inference rules for reasoning about properties of programs. The subjective or intuitionistic approach is exemplified by constructive type theory where the very definition of the logical connectives is tied to their computational interpretation.

Completeness Proof by Semantic Diagrams for Transitive Closure of Accessibility Relation.

Abstract: We treat the smallest normal modal propositional logic with two modal operators 2 and 2+ While 2 is interpreted in Kripke models by the accessibility relation R, 2+ is interpreted by the transitive closure of R Intuitively the formula 2+φ means the infinite conjunction 2φ ∧ 22φ ∧ 222φ ∧ · · · There is a Hilbert style axiomatization of this logic (a characteristic axiom is 2φ ∧ 2+(φ → 2φ) → 2+φ, called “induction axiom”), and its completeness with respect to finite models was shown by the canonical model method This paper gives an alternative proof of this completeness We use the method of “semantic diagram”, which is a variant of semantic tableaux, as follows Given an unprovable formula φ, we first make a small model (consisting of one world that forces φ to be false); then we add worlds step by step using the Hilbert system as an oracle, and finally we get a finite countermodel for φ The point is how to handle 2+ in this construction

Multi-attribute preference logic

Abstract: Preferences for objects are commonly derived from ranked sets of properties or multiple attributes associated with these objects. There are several options or strategies to qualitatively derive a preference for one object over another from a property ranking. We introduce a modal logic, called multi-attribute preference logic, that provides a language for expressing such strategies. The logic provides the means to represent and reason about qualitative multi-attribute preferences and to derive object preferences from property rankings. The main result of the paper is a proof that various well-known preference orderings can be defined in multi-attribute preference logic.

The Interval-Valued Truth Degree Theory of the Modal Formulas

Abstract: The modal logic, as a decidable fragment of predict logic, not only solved the paradox of material implication thoroughly, but also have important properties. The present paper defines the standard model and the interval-valued truth degree of modal formulas after analyzing the idea of possible world. Then the harmonious theorem is proved, that is, the interval-value truth degree of formulas without modal operators degenerate into a point and the value is just equal to its Borel truth degree.

Chapter Seven. Modal Logic And Possible Worlds

Abstract: Seventeenth-century discussions on divine knowledge and will thoroughly employ modal categories like contingency and necessity. In fact, the major systematic positions can be distinguished by the way they relate modal aspects to divine agency. Since the early twentieth century, important developments have been achieved in the study of modal logic, enabling us to analyze the involved modal issues more exactly. In this chapter, these tools are introduced to evaluate the seventeenth-century positions and to develop a contemporary analysis of divine agency that takes account of its modal aspects. Modern modal logic with its symbolic language and possible world semantics delivers a framework to evaluate the various seventeenth- century conceptions of divine agency, which all have a clear modal structure.Keywords: divine knowledge; modal logic

A proof system with bounded non-determinism in database transformations

Abstract: Database Abstract State Machines (DB-ASMs) provide a universal computation model for database transformations that encompass queries and updates. In this paper we present a proof system for DB-ASMs. It is shown that the proof system for DB-ASMs is sound. As DB-ASMs are restricted by allowing quantifiers over only the database part of a state which is a finite structure, we can formalise nondeterminism of DB-ASMs by utilising a modal operator [ ] for an update set or multiset generated by a DB-ASM rule. In doing so, we lay down a solid foundation for the completeness proof of the proof system proposed for DB-ASMs in this paper.