Showing papers on "Modal operator published in 2007"

Dynamic Epistemic Logic

Abstract: Dynamic Epistemic Logic is the logic of knowledge change. This is not about one logical system, but about a whole family of logics that allows us to specify static and dynamic aspects of multi-agent systems. This book provides various logics to support such formal specifications, including proof systems. Concrete examples and epistemic puzzles enliven the exposition. The book also contains exercises including answers and is eminently suitable for graduate courses in logic. A sweeping chapter-wise outline of the content of this book is the following. The chapter 'Introduction' informs the reader about the history of the subject, and its relation to other disciplines. 'Epistemic Logic' is an overview of multi-agent epistemic logic - the logic of knowledge - including modal operators for groups, such as general and common knowledge. 'Belief Revision' is an overview on how to model belief revision, both in the 'traditional' way and in a dynamic epistemic setting. 'Public Announcements' is a detailed and comprehensive introduction into the logic of knowledge to which dynamic operators for truthful public announcement are added. Many interesting applications are also presented in this chapter: a form of cryptography for ideal agents also known as 'the russian cards problem', the sum-and-product riddle, etc. 'Epistemic Actions' introduces a generalization of public announcement logic to more complex epistemic actions. A different perspective on that matter is independently presented in 'Action Models'. 'Completeness' gives details on the completeness proof for the logics introduced in 'Epistemic Logic', 'Public Announcements', and 'Action Models'. 'Expressivity' discusses various results on the expressive power of the logics presented.

Free Choice and the Theory of Scalar Implicatures

Abstract: This chapter will be concerned with the conjunctive interpretation of a family of disjunctive constructions. The relevant conjunctive interpretation, sometimes referred to as a ‘free choice effect,’ (FC) is attested when a disjunctive sentence is embedded under an existential modal operator. I will provide evidence that the relevant generalization extends (with some caveats) to all constructions in which a disjunctive sentence appears under the scope of an existential quantifier, as well as to seemingly unrelated constructions in which conjunction appears under the scope of negation and a universal quantifier.

What can we achieve by arbitrary announcements?: A dynamic take on Fitch's knowability

Abstract: Public announcement logic is an extension of multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: sφ expresses that φ is true after an arbitrary announcement ψ. As this includes the trivial announcement ⊤, one might as well say that sφ expresses what remains true after any announcement: it therefore corresponds to truth persistence after (definable) relativisation. The dual operation dφ expresses that there is an announcement after which φ. This gives a perspective on Fitch's knowability issues: for which formulas φ does it hold that φ → dKφ? We give various semantic results, and we show completeness for a Hilbert-style axiomatisation of this logic.

Alternation-free modal mu-calculus for data trees

Abstract: An alternation-free modal mu-calculus over data trees is introduced and studied. A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element ("datum") from an infinite set. For expressing data-sensitive properties, the calculus is equipped with freeze quantification. A freeze quantifier stores in a register the datum labelling the current tree node, which can then be accessed for equality comparisons deeper in the formula. The main results in the paper are that, for the fragment with forward modal operators and one register, satisfiability over finite data trees is decidable but not primitive recursive, and that for the subfragment consisting of safety formulae, satisfiability over countable data trees is decidable but not elementary. The proofs use alternating tree automata which have registers, and establish correspondences with nondeterministic tree automata which have faulty counters. Allowing backward modal operators or two registers causes undecidability. As consequences, decidability is obtained for two data-sensitive fragments of the XPath query language. The paper shows that, for reasoning about data trees, the forward fragment of the calculus with one register is a powerful alternative to a recently proposed first-order logic with two variables.

Alternation-free modal mu-calculus for data trees (Extended abstract)

Abstract: An alternation-free modal µ-calculus over data trees is introduced and studied. A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element (“datum”) from an infinite set. For expressing data-sensitive properties, the calculus is equipped with freeze quantification. A freeze quantifier stores in a register the datum labelling the current tree node, which can then be accessed for equality comparisons deeper in the formula. The main results in the paper are that, for the fragment with forward modal operators and one register, satisfiability over finite data trees is decidable but not primitive recursive, and that for the subfragment consisting of safety formulae, satisfiability over countable data trees is decidable but not elementary. The proofs use alternating tree automata which have registers, and establish correspondences with nondeterministic tree automata which have faultycounters. Allowing backward modaloperatorsor two registers causes undecidability. As consequences, decidability is obtained for two datasensitive fragments of the XPath query language. The paper shows that, for reasoning about data trees, the forward fragment of the calculus with one register is a powerful alternative to a recently proposed first-order logic with two variables.

Future and non-future modal sentences

Abstract: In this paper, I argue for two principles to determine the temporal interpretation of modal sentences in English, given a theory in which modals are interpreted against double conversational backgrounds and an ontology in which possible worlds branch towards the future, The Disparity Principle requires that a modal sentence makes distinctions between worlds in the modal base. The Non- disparity Principle requires that a modal sentence does not make distinctions on the basis of facts settled at speech time. Selection of the modal base will set these principles against each other, or allow for their cooperative interaction. For a root modal base, there is a conflict and disparity wins. The resulting interpretation is future. For a non-root modal base, the principles cooperate. Non-disparity determines a non-future interpretation and disparity requires the sentence to go beyond what is known by the speaker.

15 Combining modal logics

Abstract: Publisher Summary This chapter surveys two key combination methods––namely, fusions and products. It also examines some other combination methods. The properties of combined logics depend on those of their components plus the particular method of combination. The idea of combining modal logics is natural for many applications. The chapter explores the way in which modal logics can be combined and highlights the consequences of combining them. The formation of fusions is the simplest and the most natural way of combining modal logics. The formation of Cartesian products of various structures—vector and topological spaces, algebras—is a standard mathematical way of capturing the multidimensional character of the world. In modal logic, products of Kripke frames are natural constructions allowing it to reflect interactions between modal operators representing time, space, knowledge, or actions. The product construction as a combination method on modal logics is introduced and is used in applications in computer science and artificial intelligence.

A Deep Inference System for the Modal Logic S5

Abstract: We present a cut-admissible system for the modal logic S5 in a formalism that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. The system enjoys systematicity and modularity, two important properties that should be satisfied by modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axioms.

Were, would, might and a compositional account of counterfactuals

Abstract: This paper has two purposes. We first give a new dynamic account of epistemic modal operators that account for both their test-like behaviour with respect to whole information states and their capacity to induce quantificational dependencies across worlds (modal subordination). We then use this theory, together with an analysis of conditionals and irrealis moods, to give a fully compositional semantics of indicative and counterfactual conditionals. In our analysis, the distinction between counterfactual and indicative conditionals follows directly from the interaction between the semantics of the conditional and irrealis operators and the semantics of the particular modals involved in the conditional consequent. We indicate some theoretical and logical consequences of our approach.

Formal Ontology of Action: a Unifying Approach

Abstract: In this dissertation, a formal ontology of action named OntoSTIT+, is developed. It assumes branching time, allows the same action token to unfold in a different way (some of its courses can be successful and some of them not), provides for the classification of action types, incorporates mental attitudes of an agent and divides all action tokens into activities, accomplishments, achievements and states. Formally, OntoSTIT+ is eleven sorted first-order logic, extended with belief, desire and intention modal operators, and has around seventy axioms and twenty definitions. Each axiom and definition is justified and motivated by reference to a relevant research field such as linguistics, philosophy of action and AI. OntoSTIT+ is also based on the works in the modal logic of agency. This interdisciplinary character of OntoSTIT+, on the one hand, complies with the methodology of creating formal ontologies, and on the other hand, unifies the research fields, whose results (concerning action and agency) have been taken into account in OntoSTIT+. In general, OntoSTIT+ is a ground for many theories of action, and hence, allows for a comparison between these theories, and for a possible integration and interdisciplinary research across research areas. Moreover, OntoSTIT+ is a theory which specifies the meaning of the concepts of agency and action. It facilitates meaning negotiation among agents, as long as the concept of action is taken into account. OntoSTIT+ can be also successfully applied for designing practical reasoning and action execution of the rational, artificial agents, and for an ontology based natural language processing.

Intelligent agents and common sense reasoning

Abstract: Publisher Summary This chapter describes the use of modal logic techniques to describe mental attitudes of intelligent systems. Modal logic plays an important role in the field of artificial intelligence (AI). Modal logics are used in AI in a number of different ways. Two of its important roles are discussed in the chapter. The first deals with the use of modal logic for the description of intelligent agents, dealing with the theory and practice of the construction of autonomous software or hardware entities that act intelligently. The second is as a model of common sense reasoning, and covers modal treatments of counterfactual conditionals and non-monotonic reasoning in a variety of guises, including default reasoning. Intelligent agents have become a major field of research in AI. Epistemic logic deals with the mental attitude of knowledge while doxastic logic treats belief. The role of epistemic/doxastic logic in AI, and on the description of intelligent agents in particular is discussed in the chapter. The rise of “dynamic” versions of epistemic and doxastic logic have given new impetus to the study of belief revision in the setting of modal logic.

Modular algorithms for heterogeneous modal logics

Abstract: State-based systems and modal logics for reasoning about them often heterogeneously combine a number of features such as non-determinism and probabilities. Here, we show that the combination of features can be reflected algorithmically and develop modular decision procedures for heterogeneous modal logics. The modularity is achieved by formalising the underlying state-based systems as multi-sorted coalgebras and associating both a logical and an algorithmic description to a number of basic building blocks. Our main result is that logics arising as combinations of these building blocks can be decided in polynomial space provided that this is the case for the components. By instantiating the general framework to concrete cases, we obtain PSPACE decision procedures for a wide variety of structurally different logics, describing e.g. Segala systems and games with uncertain information.

On the expressiveness of MTL variants over dense time

Abstract: The basic modal operator bounded until of Metric Temporal Logic (MTL) comes in several variants. In particular it can be strict (when it does not constrain the current instant) or not, and matching (when it requires its two arguments to eventually hold together) or not. This paper compares the relative expressiveness of the resulting MTL variants over dense time. We prove that the expressiveness is not affected by the variations when considering non-Zeno interpretations and arbitrary nesting of temporal operators. On the contrary, the expressiveness changes for flat (i.e., without nesting) formulas, or when Zeno interpretations are allowed.

A dynamic description logic for representation and reasoning about actions

Abstract: We present a dynamic description logic for representation and reasoning about actions, with an approach that embrace actions into the description logic ALCO@. With this logic, description logic concepts can be used for describing the state of the world, and the preconditions and effects of atomic actions; Complex actions can be modeled with the help of standard action operators, such as the test, sequence, choice, and iteration operators; And both atomic actions and complex actions can be used as modal operators to construct formulas. We develop a terminable and correct algorithm for checking the satisfiability of formulas. Based on the algorithm, many reasoning tasks on actions are effectively carried out, including the realizability, executability, projection and planning problems.

Optimal regression for reasoning about knowledge and actions

Abstract: We show how in the propositional case both Reiter's and Scherl & Levesque's solutions to the frame problem can be modelled in dynamic epistemic logic (DEL), and provide an optimal regression algorithm for the latter. Our method is as follows: we extend Reiter's framework by integrating observation actions and modal operators of knowledge, and encode the resulting formalism in DEL with announcement and assignment operators. By extending Lutz' recent satisfiability-preserving reduction to our logic, we establish optimal decision procedures for both Reiter's and Scherl & Levesque's approaches: satisfiability is NP-complete for one agent, PSPACE-complete for multiple agents and EXPTIME-complete when common knowledge is involved.

3 Complexity of modal logic

Abstract: Publisher Summary This chapter introduces the field of computational complexity in modal logic and provides some fundamental answers. The basic modal language, when interpreted over relational models, can be regarded as a decidable fragment of classical logic. The chapter explores the computational complexity of determining validity, or of performing tasks like model checking. The parameters that affect modal complexity results are discussed. The chapter deals with modal systems whose class of models is definable in first-order logic expanded with a least fixed point operator (FO(LFP)), which implies that membership is decidable in polynomial time. The actual algorithms for deciding the satisfiability problem under constraints and the local satisfiability problem for a number of typical cases are reviewed. Hintikka set elimination is an algorithm, which constructs the model whose idea comes straight from the proof of the truth lemma. A globally satisfiable constraint, which, when satisfied, forces a branch in the model containing an exponential number of different Hintikka set is created.

Modal-Like Operators in Boolean Lattices, Galois Connections and Fixed Points

Abstract: In this work, four modal-like operators on Boolean lattices are introduced and their theory is presented from lattice-theoretical, topological and algebraic point of view. It is also shown how rough set approximation operators, modal operators in temporal logic, and linguistic modifiers determined by L-sets can be interpreted as modal-like operators.

Optimal Regression for Reasoning about Knowledge and Actions.

Abstract: We show how in the propositional case both Reiter's and Scherl & Levesque's solutions to the frame problem can be modelled in dynamic epistemic logic (DEL), and provide an optimal regression algorithm for the latter. Our method is as follows: we extend Reiter's framework by integrating observation actions and modal operators of knowledge, and encode the resulting formalism in DEL with announcement and assignment operators. By extending Lutz' recent satisfiability-preserving reduction to our logic, we establish optimal decision procedures for both Reiter's and Scherl & Levesque's approaches: satisfiability is NP-complete for one agent, PSPACE-complete for multiple agents and EXPTIME-complete when common knowledge is involved.

On the acquisition of modality

Abstract: Epistemic modality concerns what is possible or necessary given what is known and what the available evidence is (von Fintel 2005). Semantically epistemic modal operators encode modal force (necessity or possibility) and get interpreted against a conversational background which includes the speaker's beliefs or the available evidence. Necessity in a given world encodes truth in all alternative possible worlds, whereas possibility encodes truth in at least one alternative possible world (Hintikka 1969). Along similar lines, Kratzer (1981:43) states that "a proposition is a simple necessity in a world w with respect to the conversational background f if, and only if, it follows from f(w)." However, "a proposition is a simple possibility in a world w with respect to the conversational background f if, and only if, it is compatible with f(w)." For instance, sentence (1a) is a necessary proposition in a world where the speaker has definitive evidence that it will rain. Hence, the embedded proposition in (la) follows from her previous knowledge. On the other hand, (l b), is only compatible with the speaker's previous knowledge or the evidence. Pragmatically, epistemic modal verbs typically give rise to conversational implicatures of the following sort:

Understanding the Brandenburger-Keisler Paradox

Abstract: Adam Brandenburger and H. Jerome Keisler have recently discovered a two person Russell-style paradox. They show that the following configurations of beliefs is impossible: Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong. In [7] a modal logic interpretation of this paradox is proposed. The idea is to introduce two modal operators intended to represent the agents’ beliefs and assumptions. The goal of this paper is to take this analysis further and study this paradox from the point of view of a modal logician. In particular, we show that the paradox can be seen as a theorem of an appropriate hybrid logic.

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 axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coalgebraic semantics. As a consequence, recent results on coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, become applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of these results. As an extended example, we apply our framework to recently defined deontic logics.

KE Tableaux for Public Announcement Logic

Abstract: Public announcement logic (PAL) is a simple dynamic epistemic logic extending reasoning about knowledge of agents with a modal operator for simultaneous and transparent knowledge updates This logic is no more expressive than epistemic logic (EL) without updates, but exhibits compact representation of a number of complex epistemic situations A labeled tableau proof system to reason with these updates directly is presented here This system can analyse and present well-known epistemic puzzles like ‘muddy children’ and ‘three wise men’ Using the KE tableau system as a basis, the modal and propositional characteristics of epistemic updates can be separated

Structuralist logic: Implications, inferences, and consequences

Abstract: On a structuralist account of logic, the logical operators, as well as modal operators are defined by the specific ways that they interact with respect to implication. As a consequence, the same logical operator (conjunction, negation etc.) can appear to be very different with a variation in the implication relation of a structure. We illustrate this idea by showing that certain operators that are usually regarded as extra-logical concepts (Tarskian algebraic operations on theories, mereological sum, products and negates of individuals, intuitionistic operations on mathematical problems, epistemic operations on certain belief states) are simply the logical operators that are deployed in different implication structures. That makes certain logical notions more omnipresent than one would think.

Moore problems in full dynamic doxastic logic

Abstract: Dynamic doxastic logic (DDL) is the modal logic of belief change. In basic DDL a modal operator [*?] carries the informal meaning “after the agent has revised his beliefs by ?” or “after the agent has accepted the information that ?”; it is assumed that the arguments of the star operator * are pure Boolean formulae. That assumption is discarded in full DDL where any pure doxastic formula may be an argument. As noted by other authors, a straight-forward extension of the theory from basic DDL to full DDL invites problems of the kind first discussed by G. E. Moore. In this paper it is argued that a way to escape those problems is to redefine revision in a way that seems appropriate for this semantically richer context. The paper deals only with the one-agent case, but the approach can be extended to the case of multiple agents.

Perfective and Imperfective in French Kinds of abilities and Actuality Entailment (And some notes on epistemic readings)

Abstract: In French as in other languages differentiating the perfective and the imperfective morphologically, modal verbs sometimes behave like implicative verbs in perfective sentences. The account of the data presented here is purely semantical and pragmatical in nature, contrary to Hacquard's (2006) one who proposes a structural account in terms of a scopal difference between the aspectual and the modal operator. Our account relies on an ontological distinction that goes back to Aristotle between classical abilities and what we call action dependent abilities.

Modality and Modal Sense Representation in E-HowNet

Abstract: This paper explains how we define and represent modality in E-HowNet. Following Lyons (1977, reviewed in Hsieh 2003, among others), we hold that modals express a speaker's opinion or attitude toward a proposition and hence have a pragmatic dimension and recognize five kinds of modal categories, i.e. epistemic, deontic, ability, volition and expectation modality. We then present a representational formalism that contains the three most basic components of modal meaning: modal category, positive or negative and strength. Such a formula can define not only modal words but also words that contain modal meanings and cope with co-compositions of modals and the negation construction.