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72
Complete Proof Systems for First Order Interval Temporal Logic
 In Tenth Annual IEEE Symp. on Logic in Computer Science
, 1995
"... Different interval modal logics have been proposed for reasoning about the temporal behaviour of digital systems. Some of them are purely propositional and only enable the specification of qualitative time requirements. Others, such as ITL and the duration calculus, are first order logics which supp ..."
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Cited by 26 (0 self)
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Different interval modal logics have been proposed for reasoning about the temporal behaviour of digital systems. Some of them are purely propositional and only enable the specification of qualitative time requirements. Others, such as ITL and the duration calculus, are first order logics which support the expression of quantitative, realtime requirements. These two logics have in common the presence of a binary modal operator `chop' interpreted as the action of splitting an interval into two parts. Proof systems for ITL or the duration calculus have been proposed but little is known about their power. This paper present completeness results for a variant of ITL where `chop' is the only modal operator. We consider several classes of models for ITL which make different assumptions about time and we construct a complete and sound proof system for each class. 1 Introduction Digital systems are increasingly used in applications where they interact with physical processes. In these appli...
Halforder Modal Logic: How To Prove Realtime Properties
 IN PROCEEDINGS OF THE NINTH ANNUAL SYMPOSIUM ON PRINCIPLES OF DISTRIBUTED COMPUTING
, 1990
"... We introduce a novel extension of propositional modal logic that is interpreted over Kripke structures in which a value is associated with every possible world. These values are, however, not treated as full firstorder objects; they can be accessed only by a very restricted form of quantificati ..."
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Cited by 26 (6 self)
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We introduce a novel extension of propositional modal logic that is interpreted over Kripke structures in which a value is associated with every possible world. These values are, however, not treated as full firstorder objects; they can be accessed only by a very restricted form of quantification: the "freeze" quantifier binds a variable to the value of the current world. We present a complete proof system for this ("halforder") modal logic. As a special case, we obtain the realtime temporal logic TPTL of [AH89]: the models are restricted to infinite sequences of states, whose values are monotonically increasing natural numbers. The ordering relation between states is interpreted as temporal precedence, while the value associated with a state is interpreted as its "real" time. We extend our proof system to be complete for TPTL, and demonstrate how it can be used to derive realtime properties.
A systematic proof theory for several modal logics
 Advances in Modal Logic, volume 5 of King’s College Publications
, 2005
"... abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence betw ..."
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Cited by 24 (1 self)
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abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a HilbertLewis style and conditions on the accessibility relation on frames. By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is adhoc. While we can formulate several modal logics in the sequent calculus that enjoy cutelimination, their formalisation arises through systembysystem fine tuning to ensure that the cutelimination holds, and the correspondence to the axioms of the HilbertLewis systems becomes opaque. This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference. Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the HilbertLewis axiomatisation. We show that the calculus possesses a cutelimination property directly analogous to cutelimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics. 1
Modeling an Agent's Incomplete Knowledge during Planning and Execution
 In Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning
, 1998
"... In many domains agents must be able to generate plans even when faced with incomplete knowledge of their environment. We provide a model to capture the evolution of the agent's knowledge as it engages in the activities of planning (where the agent must attempt to infer the effects of hypothesized ac ..."
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Cited by 19 (5 self)
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In many domains agents must be able to generate plans even when faced with incomplete knowledge of their environment. We provide a model to capture the evolution of the agent's knowledge as it engages in the activities of planning (where the agent must attempt to infer the effects of hypothesized actions) and execution (where the agent must update its knowledge to reflect the actual effects of actions). The effects (on the agent's knowledge) of a planned sequence of actions are very different from the effects of an executed sequence of actions, and one of the aims of this work is to clarify this distinction. The work is also aimed at providing a model that is not only rigorous but can also be of use in developing planning systems.
A Framework for Modal Logic Programming
 In Joint International Conference and Symposium on Logic Programming
, 1996
"... In this paper we present a framework for developing modal extensions of logic programming, which are parametric with respect to the properties chosen for the modalities and which allow sequences of modalities of the form [t], where t is a term of the language, to occur in front of clauses, goals and ..."
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Cited by 18 (3 self)
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In this paper we present a framework for developing modal extensions of logic programming, which are parametric with respect to the properties chosen for the modalities and which allow sequences of modalities of the form [t], where t is a term of the language, to occur in front of clauses, goals and clause heads. The properties of modalities are specified by a set A of inclusion axioms of the form [t 1 ] : : : [t n ]ff oe [s 1 ] : : : [s m ]ff. The language can deal with many of the wellknown modal systems and several examples are provided. Due to its features, it is particularly suitable for performing epistemic reasoning, defining parametric and nested modules, describing inheritance in a hierarchy of classes and reasoning about actions. A goal directed proof procedure of the language is presented, which is modular with respect to the properties of modalities. Moreover, we define a fixpoint semantics, by generalizing the standard construction for Horn clauses, which is used to prov...
Representing Multiple Theories
 In Proceedings of the Twelfth National Conference on Artificial Intelligence
, 1994
"... Most Artificial Intelligence programs lack generality because they reason with a single domain theory that is tailored for a specific task and embodies a host of implicit assumptions. Contexts have been proposed as an effective solution to this problem by providing a mechanism for explicitly stating ..."
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Cited by 17 (0 self)
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Most Artificial Intelligence programs lack generality because they reason with a single domain theory that is tailored for a specific task and embodies a host of implicit assumptions. Contexts have been proposed as an effective solution to this problem by providing a mechanism for explicitly stating the assumptions underlying a domain theory. In addition, contexts can be used to focus reasoning, allow the representation of mutually incoherent domain theories, lift axioms from one context into another, and transcend a context. In this paper we develop a simple propositional logic of context suitable for representing and reasoning with multiple domain theories. We introduce contexts as modal operators, and allow different contexts to have different vocabularies. We analyze the computational properties of the logic, providing the central computational justification for the use of contexts. We show how the logic effectively handles the common uses of contexts. We also discuss the extension...
A Fixpoint Semantics and an SLDResolution Calculus for Modal Logic Programs
, 2001
"... We propose a modal logic programming language called MProlog, which is as expressive as the general modal Horn fragment. We give a fixpoint semantics and an SLDresolution calculus for MProlog in all of the basic serial modal logics KD, T, KDB, B, KD4, S4, KD5, KD45, and S5. For an MProlog program P ..."
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Cited by 15 (13 self)
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We propose a modal logic programming language called MProlog, which is as expressive as the general modal Horn fragment. We give a fixpoint semantics and an SLDresolution calculus for MProlog in all of the basic serial modal logics KD, T, KDB, B, KD4, S4, KD5, KD45, and S5. For an MProlog program P and for L being one of the mentioned logics, we define an operator TL,P, which has the least fixpoint IL,P. This fixpoint is a set of formulae, which may contain labeled forms of the modal operator ✸, and is called the least Lmodel generator of P. The standard model of IL,P is shown to be a least Lmodel of P. The SLDresolution calculus for MProlog is designed with a similar style as for classical logic programming. It is sound and complete. We also extend the calculus for MProlog in the almost serial modal logics KB, K 5, K 45, and KB5. 1
Labelled Modal Logics: Quantifiers
, 1998
"... . In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logic ..."
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Cited by 15 (2 self)
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. In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logics with varying, increasing, decreasing, or constant domains. The result is modular with respect to both properties of the accessibility relation in the Kripke frame and the way domains of individuals change between worlds. Our approach has a modular metatheory too; soundness, completeness and normalization are proved uniformly for every logic in our class. Finally, our work leads to a simple implementation of a modal logic theorem prover in a standard logical framework. 1 Introduction Motivation Modal logic is an active area of research in computer science and artificial intelligence: a large number of modal logics have been studied and new ones are frequently proposed. Each new log...
Multimodal Logic Programming and Its Applications to Modal Deductive Databases
, 2003
"... We give a general framework for developing the least model semantics, xpoint semantics, and SLDresolution calculi for logic programs in multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. 8x 9y R i (x; y)) and some classical rstorder Horn formulas. Our appr ..."
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Cited by 13 (9 self)
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We give a general framework for developing the least model semantics, xpoint semantics, and SLDresolution calculi for logic programs in multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. 8x 9y R i (x; y)) and some classical rstorder Horn formulas. Our approach is direct and no restriction on occurrences of 2 i and 3 i is required. We apply the framework for a large class of basic serial multimodal logics, which are parameterized by an arbitrary combination of generalized versions of axioms T , B, 4, 5 (in the form, e.g., 4 : 2 i ! 2 j 2k) and I : 2 i ! 2 j . Another part of the work is devoted to programming in multimodal logics intended for reasoning about multidegree belief, for use in distributed systems of belief, or for reasoning about epistemic states of agents in multiagent systems.
Multimodal logic programming
 Theoretical Computer Science
, 2006
"... We give a framework for developing the least model semantics, fixpoint semantics, and SLDresolution calculi for logic programs in multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. ∀x ∃y Ri(x, y)) and some classical firstorder Horn formulas. Our approach is dir ..."
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Cited by 12 (7 self)
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We give a framework for developing the least model semantics, fixpoint semantics, and SLDresolution calculi for logic programs in multimodal logics whose frame restrictions consist of the conditions of seriality (i.e. ∀x ∃y Ri(x, y)) and some classical firstorder Horn formulas. Our approach is direct and no special restriction on occurrences of ✷i and ✸i is required. We apply our framework for a large class of basic serial multimodal logics, which are parameterized by an arbitrary combination of generalized versions of axioms T, B, 4, 5 (in the form, e.g., 4: ✷iϕ → ✷j✷kϕ) and I: ✷iϕ → ✷jϕ. Another part of the work is devoted to programming in multimodal logics intended for reasoning about multidegree belief, for use in distributed systems of belief, or for reasoning about epistemic states of agents in multiagent systems. For that we also use the framework, and although these latter logics belong to the mentioned class of basic serial multimodal logics, the special SLDresolution calculi proposed for them are more efficient.