Showing papers on "Modal operator published in 2000"

Input/output logics

Abstract: In a range of contexts, one comes across processes resembling inference, but where input propositions are not in general included among outputs, and the operation is not in any way reversible. Examples arise in contexts of conditional obligations, goals, ideals, preferences, actions, and beliefs. Our purpose is to develop a theory of such input/output operations. Four are singled out: simple-minded, basic (making intelligent use of disjunctive inputs), simple-minded reusable (in which outputs may be recycled as inputs), and basic reusable. They are defined semantically and characterised by derivation rules, as well as in terms of relabeling procedures and modal operators. Their behaviour is studied on both semantic and syntactic levels.

On the Complexity of Explicit Modal Logics

Abstract: Explicit modal logic was introduced by S. Artemov. Whereas the traditional modal logic uses atoms □F with a possible semantics "F is provable", the explicit modal logic deals with atoms of form t:F, where t is a proof polynomial denoting a specific proof of a formula F. Artemov found the explicit modal logic LP in this new format and built an algorithm that recovers explicit proof polynomials corresponding to modalities in every derivation in K. Godel's modal provability calculus S4. In this paper we study the complexity of LP as well as the complexity of explicit counterparts of the modal logics K, D, T, K4, D4 found by V. Brezhnev. The main result: the satisfiability problem for each of these explicit modal logics belongs to the class Σ2p of the polynomial hierarchy. Similar problem for the original modal logics is known to be PSPACE-complete. Therefore, explicit modal logics have much better upper complexity bounds than the original modal logics.

Modal Logics and Philosophy

Abstract: Unlike most modal logic textbooks, which are both forbidding mathematically and short on philosophical discussion, Modal Logics and Philosophy places its emphasis firmly on showing how useful modal logic can be as a tool for formal philosophical analysis. In Part 1 of the book, the reader is introduced to some standard systems of modal logic and encouraged through a series of exercises to become proficient in manipulating these logics. The emphasis is on possible world semantics for modal logics and the semantic emphasis is carried into the formal method, Jeffrey-style truth-trees. Standard truth-trees are extended in a simple and transparent way to take possible worlds into account. Part 2 systematically explores the applications of modal logic to philosophical issues such as truth, time, processes, knowledge and belief, obligation and permission. The second edition sees the addition of two new chapters on conditionals. The first, in Part 1, presents the formalities of a range of conditional logics, and the second, in Part 2, discusses some of the philosophical issues raised by them. Other chapters have been revised and updated, including some reordering of content in Part 1, to strengthen the book as a fully comprehensive introduction to modal logics and their application suitable for course use.

A Simple and Tractable Extension of Situation Calculus to Epistemic Logic

Abstract: The frame problem and the representation of knowledge change have deserved a lot of works. In particular, at the Cognitive Robotics Group, at Toronto, several researchers in the last ten years have produced quite interesting papers in a uni- form logical framework based on Situation Calulus [Rei91, SL93, LR94, LL98]. In [Rei91] Reiter has proposed a simple solution to the frame problem. Scherl and Levesque in [SL93] have defined an extension to Epistemic Logic to represent knowledge dynamics in contexts where some actions may produce knowledge, like, for instance, sensing actions for a robot. This approach has been extended by Lakemeyer and Levesque in [LL98] to modal operators of the kind “I know and only know”. Also, they have given a formal semantics and axiomatics, and they proved soundness and completeness of the axiomatics.

Two-Phase Deontic Logic

Abstract: We show that for the adequate representation of some examples of normative reasoning a combination of different operators is needed, where each operator validates different inference rules. The combination of different modal operators imposes the restriction on the proof theory of the logic that a proof rule can be blocked in a derivation due to the fact that another proof rule has been used earlier in the derivation. In this paper we only use two operators and therefore we call the restriction the two-phase approach in the proof theory, which we formalize in two-phase labeled deontic logic (2ldl) and in two-phase dyadic deontic logic (2dl). The preference-based semantics of 2dl is based on an explicit deontic preference ordering between worlds, representing different degrees of ideality. The two different modal operators represent two different usages of the preference ordering, called minimizing and ordering. 1. Why deontic logic derivations must consist of two phases 1.1. Van Fraassen’s paradox Van Fraassen (1973) presents a logical analysis of dilemmas. In a logic that formalizes reasoning about dilemmas we cannot accept the conjunction rule, because it derives©(p∧¬p) from the dilemma©p∧©¬p, whereas ‘ought implies can’ ¬ © (p ∧ ¬p). On the other hand we do not want to reject the conjunction rule in all cases. For example, we want to derive ©(p ∧ q) from ©p ∧ ©q when p and q are distinct propositional atoms. That is, we have to add a restriction on the conjunction rule such that we only derive ©(α1 ∧α2) from ©α1 and ©α2 if α1 ∧ α2 is consistent. Van Fraassen calls the latter inference pattern Consistent Aggregation, which we write as the restricted conjunction rule (rand). He encounters a problem in the formalization of obliga2 L. VAN DER TORRE AND Y. TAN tions, and wonders if he needs a language in which he can talk directly about the imperatives as well. A variant of this problem is illustrated in the following example. Example 1 (Van Fraassen’s paradox). Assume a monadic deontic logic without nested modal operators1 in which dilemmas like ©p ∧ ©¬p are consistent, but which validates ¬©⊥, where ⊥ stands for any contradiction like p∧¬p. Moreover, assume that it satisfies replacement of logical equivalents and at least the following two inference patterns Restricted Conjunction rule (rand), also called consistent aggregation, and Weakening (w), where ↔ 3φ can loosely be read as φ is possible (or propositionally consistent). rand: ©α1,©α2, ↔ 3(α1 ∧ α2) ©(α1 ∧ α2) w: ©α1 ©(α1 ∨ α2) Moreover, assume the two premises ‘Honor thy father or thy mother!’ ©(f ∨ m) and ‘Honor not thy mother!’ ©¬m. The derivation of Figure 1 illustrates how the desired conclusion ‘thou shalt honor thy father’ ©(f ∨m) ©¬m ©(f ∧ ¬m) rand ©f w Figure 1. Van Fraassen’s paradox (1) ©f can be derived from the premises. Unfortunately, the derivation of Figure 2 illustrates that we cannot accept restricted conjunction and ©p ©(f ∨ p) w ©¬p ©(f ∧ ¬p) rand ©f w Figure 2. Van Fraassen’s paradox (2) weakening in a monadic deontic logic, because we can derive the counterintuitive obligation ©f from a deontic dilemma ©p ∧ ©¬p. The point of this paradox is that every ©(β), of which ©(f) is a special case, would be derivable. Van Fraassen asks himself whether restricted conjunction can be formalized, and he observes interesting technical questions. In this paper we pursue some of these technical questions. 2dl.tex; 15/06/2001; 17:08; no v.; p.2 TWO-PHASE DEONTIC LOGIC 3 ‘But can this happy circumstance be reflected in the logic of the ought-statements alone? Or can it be expressed only in a language in which we can talk directly about the imperatives as well? This is an important question, because it is the question whether the inferential structure of the ‘ought’ language game can be stated in so simple a manner that it can be grasped in and by itself. Intuitively, we want to say: there are simple cases, and in the simple cases the axiologist’s logic is substantially correct even if it is not in general – but can we state precisely when we find ourselves in such a simple case? These are essentially technical questions for deontic logic, and I shall not pursue them here.’ (van Fraassen, 1973) As far as we know, there is no discussion on Van Fraassen’s paradox in the deontic logic literature.2 We analyze Van Fraassen’s paradox in Example 1 by forbidding application of rand after w has been applied. This blocks the counterintuitive derivation in Figure 2 and it does not block the intuitive derivation in Figure 1, as we show below. Our formalization of two-phase reasoning works as follows. In the logic, the two phases are represented by two different types of obligations, written as phase-1 obligations ©1 and phase-2 obligations ©2 . The premises are phase-1 obligations, the conclusions are phase-2 obligations and the two phases are linked to each other with the following inference pattern rel.

A modal logic for epistemic tests

Abstract: We study a modal logic of knowledge and action, focussing on test actions. Such knowledge-gathering actions increase the agents' knowledge. We propose a semantics, and associatean axiomatics and a rewriting-based proof procedure.

Qualitative and Probabilistic Models of Full Belief

Abstract: .Summary. Let L be a language containing the modal operator B –f orfull belief .A n information model is a set E of stable L-theories ([19], [21], [30], [27]). A sentence is valid if it is accepted in all theories of every model. We show that a formula A is valid if and only if it is a theorem of the modal system S5. The second part of the paper considers the possibility of providing probabilistic foundations for full belief. We study a proposal presented by Bas van Fraassen in [31]. Some of the results proved in [31] are generalized (especially theorem 5.2) and the dynamic aspects of the model are discussed. We show that a characterizations of probability kinematics compatible with van Fraassen’s proposal is cumulative - inputs introduced at any stage in a series of changes are rigidly maintained through subsequent changes. Finally we consider the range of applicability of cumulative models of belief change.

A conservative approach to distributed belief fusion

Abstract: In this paper, we develop logics for merging beliefs of agents with different degrees of reliability. The logics are obtained by combining the multiagent epistemic logic and multi-sources reasoning systems. Every ordering for the reliability of the agents is represented by a modal operator, so we can reason with the merging information under different situations The approach is conservative in the sense that if an agent's belief is in conflict with those of higher priorities, then his belief is completely discarded front the merged result. We consider two strategies for the conservative merging of beliefs. In the first one, if inconsistency occurs at some level, then all beliefs at the lower levels are discarded simultaneously, so it is called level cutting strategy. For the second one, only the level at which the inconsistency occurs is skipped, so it is called level skipping strategy. The formal semantics and axiomatic systems for these two strategies are presented.

Modality and Databases

Abstract: Two things are done in this paper. First, a modal logic in which one can quantify over both objects and concepts is presented; a semantics and a tableau system are given. It is a natural modal logic, extending standard versions, and capable of addressing several well-known philosophical difficulties successfully. Second, this modal logic is used to introduce a rather different way of looking at relational databases. The idea is to treat records as possible worlds, record entries as objects, and attributes as concepts, in the modal sense. This makes possible an intuitively satisfactory relational database theory. It can be extended, by the introduction of higher types, to deal with multiple-valued attributes and more complex things, though this is further than we take it here.

A Hierarchy of Modal Event Calculi: Expressiveness and Complexity

Abstract: We consider a hierarchy of modal event calculi to represent and reason about partially ordered events. These calculi are based on the model of time and change of Kowalski and Sergot’s Event Calculus (EC): given a set of event occurrences, EC allows the derivation of the maximal validity intervals (MVIs) over which properties initiated or terminated by those events hold. The formalisms we analyze extend EC with operators from modal logic. They range from the basic Modal Event Calculus (MEC), that computes the set of all current MVIs (MVIs computed by EC) as well as the sets of MVIs that are true in some/every refinement of the current partial ordering of events (◊-/□;-MVIs), to the Generalized Modal Event Calculus (GMEC),that extends MEC by allowing a free mix of boolean connectives and modal operators. We analyze and compare the expressive power and the complexity of the proposed calculi, focusing on intermediate systems between MEC and GMEC. We motivate the discussion by using a fault diagnosis problem as a case study.

On modal logics characterized by models with relative accessibility relations: Part I

Abstract: This work is divided in two papers (Part I and Part II) In Part I, we study a class of polymodal logics (herein called the class of "Rare-logics") for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability are established by faithfully translating the Rare-logics into more standard modal logics The main idea of the translation consists in eliminating the Boolean terms by taking advantage of the components construction and in using various properties of the classes of semilattices involved in the semantics The novelty of our approach allows us to prove new decidability results (presented in Part II), in particular for information logics derived from rough set theory and we open new perspectives to define proof systems for such logics (presented also in Part II)

An Intention Model for Agent

Abstract: Intentions, an integral part of the mental state of an agent, play an important role in determining the behavior of rational agents. There are several models of intention based on normal modal logic. But these theories suffer from the omniscience problem seriously. In this paper, the authors argue that intention is not a normal modal operator, and present another intention model. It doesn't have the logical omniscience problem and other related problems such as side effect problem, etc. Compared with Konolige and Pollack's model of intention, this model not only is simpler and more natural, but also satisfies the K axiom and the Joint Consistency. Actually it gives a new method for semantic representation of non normal modal operators based on normal possible worlds.

Logical systems for reasoning about multi-agent belief, information acquisition and trust

Abstract: In this paper, we consider the influence of trust on the assimilation of acquired information into an agent's belief. By use of modal logic tools, we characterize the relationship among belief, information acquisition and trust both semantically and axiomatically. The belief and information acquisition are respectively represented by KD45 and KD normal modal operators, whereas trust is expressed by a modal operator with minimal semantics. One characteristic axiom of the basic system is if agent i believes that agent j has told him the truth of p and he trusts the judgement of j on p, then he will also believe p. In addition to the basic system, some variants and further axioms for trust and information acquisition are also presented to show the expressive richness of the logic.

Evidence theory in multivalued models of modal logic

Abstract: A modal logic interpretation of Dempster-Shafer theory is developed in the framework of multivalued models of modal logic, i.e. models in which in any possible world an arbitrary number (possibly zero) of atomic propositions can be true. Several approaches to conditioning in multivalued models of modal logic are presented.

Higher-Order Modal Logic - A Sketch

Abstract: First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege's morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the "true" semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in Godel's ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation

Multi-Agent Only Knowing

Abstract: Levesque introduced a notion of ``only knowing'', with the goal of capturing certain types of nonmonotonic reasoning. Levesque's logic dealt with only the case of a single agent. Recently, both Halpern and Lakemeyer independently attempted to extend Levesque's logic to the multi-agent case. Although there are a number of similarities in their approaches, there are some significant differences. In this paper, we reexamine the notion of only knowing, going back to first principles. In the process, we simplify Levesque's completeness proof, and point out some problems with the earlier definitions. This leads us to reconsider what the properties of only knowing ought to be. We provide an axiom system that captures our desiderata, and show that it has a semantics that corresponds to it. The axiom system has an added feature of interest: it includes a modal operator for satisfiability, and thus provides a complete axiomatization for satisfiability in the logic K45.

Labelled Modal Sequents

Abstract: In this paper we present a new labelled sequent calculus for modal logic. The proof method works with a more ``liberal'' modal language which allows inferential steps where different formulas refer to different labels without moving to a particular world and there computing if the consequence holds. World-paths can be composed, decomposed and manipulated through unification algorithms and formulas in different worlds can be compared even if they are sub-formulas which do not depend directly on the main connective. Accordingly, such a sequent system can provide a general definition of modal consequence relation. Finally, we briefly sketch a proof of the soundness and completeness results.

A note on the Declarative reading(s) of Logic Programming

Abstract: This paper analyses the declarative readings of logic programming. Logic programming - and negation as failure - has no unique declarative reading. One common view is that logic programming is a logic for default reasoning, a sub-formalism of default logic or autoepistemic logic. In this view, negation as failure is a modal operator. In an alternative view, a logic program is interpreted as a definition. In this view, negation as failure is classical objective negation. From a commonsense point of view, there is definitely a difference between these views. Surprisingly though, both types of declarative readings lead to grosso modo the same model semantics. This note investigates the causes for this.

Derivability in Locally Quantified Modal Logics via Translation in Set Theory

Abstract: Two of the most active research areas in automated deduction in modal logic are the use of translation methods to reduce its derivability problem to that of classical logic and the extension of existing automated reasoning techniques, developed initially for the propositional case, to first-order modal logics. This paper addresses both issues by extending the translation method for propositional modal logics known as □-as-Pow (read "box-as-powerset") to a widely used class of first-order modal logics, namely, the class of locally quantified modal logics. To do this, we prove a more general result that allows us to separate (classical) first-order from modal (propositional) reasoning. Our translation can be seen as an example application of this result, in both definition and proof of adequateness.

Lukasiewicz inference in Layman's probability theory

Abstract: A prior work introduced Layman's probability theory as a formal system for reasoning with linguistic likelihood, wherein the logical and is interpreted as the arithmetic min, and the or is interpreted as the max. Likelihood modifiers (likely, very likely, somewhat unlikely, etc.) are treated as modal operators. The system is two-leveled, with the lower level being a multivalent logic and the upper level being bivalent. In the previous treatment, the lower level employed only the connectives V, /spl and/, and /spl sim/. The paper adds the /spl rarr/ connective, provides this with the well-known Lukasiewicz interpretation, and explores its properties with respect to the system as a whole.

Non-linear belief revision : Foundations and applications : by John Cantwell

Abstract: Three structures for belief revision: plausability relations on states, relations of epistemic entrenchment on propositions and systems of spheres (hypertheories), are generalised to the non-linear (non-connected) case. The further generalisation to the case of sets of such structures is also investigated. A formal language with dynamic and doxastic (belief) modal operators (DDL) is used to establish interdefinability properties between the structures and complete axiomatisations are given. The problem of iterated belief revision on non-linear structures is considered. The ideas and results of Darwiche & Pearl (1997) for iterated belief revision on linear structures are generalised to the non-linear case and axiomatisations in DDL are given. The structures and results are used to investigate a particular application: the problem when there are multiple, possibly unreliable and possibly contradictory sources of information. A trustworthiness relation on sources of information together with the information supplied by the sources is used to generate a non-linear plausibility relation on states and forms the basis for a decision procedure about what to believe. A DDL-style language with multiple non-prioritised revision operators is used to investigate the resulting structures and complete axiomatisations are given.

A Decidable Temporal Logic for Temporal Prepositions

Abstract: This paper investigates the connection between temporal logic and the semantics of some temporal expressions in English. Specifically, we show how the meanings of English temporal preposition-phrases can be expressed as modal operators in an interval-based temporal logic which we call ETL0. We show how cascaded temporal preposition phrases and temporal preposition phrases with complex complements can be translated into ETL0. Finally, we present a decision-procedure for ETL0 and establish its correctness. The main contribution of the paper is to provide a characterization, in terms familiar to modal logicians, of the expressive resources made available by an important class of temporal expressions in English.

An Algebraic Approach to KBS. Application to Computer Network Selection

Abstract: This article deals with an implementation of a theoretical result that relates tautological consequence in many-valued logics to the ideal membership problem in Algebra. The implementation is applied to automated extraction of knowledge and verification of consistency in Propositional KBS expressed in terms of many-valued logics formulae. As an example, a small KBS expressed in Lukasiewicz’s three-valued logic (with modal operators), describing how to select a computer network, is studied.

A Duplication and Loop Checking Free Proof System for S4

Abstract: Most of the sequent/tableau based proof systems for the modal logic S4 need to duplicate formulas and thus are required to adopt some method of loop checking [7, 13, 10]. In what follows we present a tableau-like proof system for S4, based on D’Agostino and Mondadori’s classical KE [3], which is free of duplication and loop checking. The key feature of this system (let us call it KES4) consists in its use of (i) a label formalism which models the semantics of the modal operators according to the usual conditions for S4; and (ii) a label unification scheme which tells us when two labels “denote” the same world in the S4-model(s) generated in the course of proof search. Moreover, it uses special closure conditions to check models for putative contradictions.