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271
The Family of Stable Models
, 1993
"... The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a socalled knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P it is the well ..."
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Cited by 64 (4 self)
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The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a socalled knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P it is the wellfounded model. There is also a dual largest stable model, S k P , which has not been considered before. There is another ordering based on degree of truth. Taking the meet and the join, in the truth ordering, of the two extreme stable models s k P and S k P just mentioned, yields the alternating fixed points of [29], denoted s t P and S t P here. From s t P and S t P in turn, s k P and S k P can be produced again, using the meet and join of the knowledge ordering. All stable models are bounded by these four valuations. Further, the methods of proof apply not just to logic programs considered classically, but to logic programs over any bilattice meeting certain co...
Spatial Reasoning with Topological Information
 Ph.D. thesis, Institut fur Informatik, AlbertLudwigsUniversitat Freiburg
, 1998
"... . This chapter summarizes our ongoing research on topological spatial reasoning using the Region Connection Calculus. We are addressing different questions and problems that arise when using this calculus. This includes representational issues, e.g., how can regions be represented and what is the re ..."
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Cited by 60 (2 self)
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. This chapter summarizes our ongoing research on topological spatial reasoning using the Region Connection Calculus. We are addressing different questions and problems that arise when using this calculus. This includes representational issues, e.g., how can regions be represented and what is the required dimension of the applied space. Further, it includes computational issues, e.g., how hard is it to reason with the calculus and are there efficient algorithms. Finally, we also address cognitive issues, i.e., is the calculus cognitively adequate. 1 Introduction When describing a spatial configuration or when reasoning about such a configuration, often it is not possible or desirable to obtain precise, quantitative data. In these cases, qualitative reasoning about spatial configurations may be used. Different aspects of space can be treated in a qualitative way. Among others there are approaches considering orientation, distance, shape, topology, and combinations of these. A summary o...
EXPTIME tableaux for ALC
 ARTIFICIAL INTELLIGENCE
, 2000
"... The last years have seen two major advances in Knowledge Representation and Reasoning. First, many interesting problems (ranging from Semistructured Data to Linguistics) were shown to be expressible in logics whose main deductive problems are EXPTIMEcomplete. Second, experiments in automated reaso ..."
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Cited by 58 (4 self)
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The last years have seen two major advances in Knowledge Representation and Reasoning. First, many interesting problems (ranging from Semistructured Data to Linguistics) were shown to be expressible in logics whose main deductive problems are EXPTIMEcomplete. Second, experiments in automated reasoning have substantially broadened the meaning of “practical tractability”. Instances of realistic size for PSPACEcomplete problems are now within reach for implemented systems. Still, there is a gap between the reasoning services needed by the expressive logics mentioned above and those provided by the current systems. Indeed, the algorithms based on treeautomata, which are used to prove EXPTIMEcompleteness, require exponential time and space even in simple cases. On the other hand, current algorithms based on tableau methods can take advantage of such cases, but require double exponential time in the worst case. We propose a tableau calculus for the description logic ALC for checking the satisfiability of a concept with respect to a TBox with general axioms, and transform it into the first simple tableaubased decision procedure working in single exponential time. To guarantee the ease of implementation, we also discuss the effects that optimizations (propositional backjumping, simplification, semantic branching, etc.) might have on our complexity result, and introduce a few optimizations ourselves.
A Canonical Model of the Region Connection Calculus
 Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR98
, 1997
"... Canonical models are very useful for determining simple representation formalism for qualitative relations. Allen's interval relations, e.g., can thereby be represented using the start and the end point of the intervals. Such a simple representation was not possible for regions of higher dim ..."
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Cited by 51 (5 self)
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Canonical models are very useful for determining simple representation formalism for qualitative relations. Allen's interval relations, e.g., can thereby be represented using the start and the end point of the intervals. Such a simple representation was not possible for regions of higher dimension as used by the Region Connection Calculus. In this paper we present a canonical model which allows regions and relations between them to be represented as points of the topological space and information about their neighbourhoods. With this formalism we are able to prove that whenever a set of RCC8 formulas is consistent there exists a realization in any dimension, even when the regions are constrained to be (sets of) polytopes. For three and higher dimensional space this is also true for internally connected regions. Using the canonical model we give algorithms for generating consistent scenarios. 1 Introduction The Region Connection Calculus (RCC) is a topological approach t...
Satisfiability problem in description logics with modal operators
 IN PROCEEDINGS OF THE SIXTH CONFERENCE ON PRINCIPLES OF KNOWLEDGE REPRESENTATION AND REASONING
, 1998
"... The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logi ..."
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Cited by 46 (21 self)
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The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satisfiability checking algorithms for these logics. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. Models with both finite and arbitrary constant domains are investigated. We begin by considering description logics with only one modal operator and then prove a general transfer theorem which makes it possible to lift the obtained results to many systems of polymodal description logic.
Provability logic
 Handbook of Philosophical Logic, 2nd ed
, 2004
"... We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t:F that read t is a justification for F. Justification Logic absorbs basic principles origin ..."
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Cited by 44 (14 self)
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We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t:F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the wellknown Justified True Belief vs. Knowledge problem. As a case study, we formalize Gettier examples in Justification Logic and reveal hidden assumptions and redundancies in Gettier reasoning. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a robust system of justifications. This renders a new, evidencebased foundation for epistemic logic. 1
MetaProgramming with Names and Necessity
, 2002
"... Metaprogramming is a discipline of writing programs in a certain programming language that generate, manipulate or execute programs written in another language. In a typed setting, metaprogramming languages usually contain a modal type constructor to distinguish the level of object programs (which ..."
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Cited by 44 (7 self)
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Metaprogramming is a discipline of writing programs in a certain programming language that generate, manipulate or execute programs written in another language. In a typed setting, metaprogramming languages usually contain a modal type constructor to distinguish the level of object programs (which are the manipulated data) from the meta programs (which perform the computations). In functional programming, modal types of object programs generally come in two flavors: open and closed, depending on whether the expressions they classify may contain any free variables or not. Closed object programs can be executed at runtime by the meta program, but the computations over them are more rigid, and typically produce less e#cient residual code. Open object programs provide better inlining and partial evaluation, but once constructed, expressions of open modal type cannot be evaluated.
The Logic of Justification
 Cornell University
, 2008
"... We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t:F that read t is a justification for F. Justification Logic absorbs basic principles origin ..."
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Cited by 44 (5 self)
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We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t:F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the wellknown Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a robust system of justifications. This renders a new, evidencebased foundation for epistemic logic. As a case study, we offer a resolution of the GoldmanKripke ‘Red Barn ’ paradox and analyze Russell’s ‘prime minister example ’ in Justification Logic. Furthermore, we formalize the wellknown Gettier example and reveal hidden assumptions and redundancies in Gettier’s reasoning. 1
DifferentialAlgebraic Dynamic Logic for DifferentialAlgebraic Programs
"... Abstract. We generalise dynamic logic to a logic for differentialalgebraic programs, i.e., discrete programs augmented with firstorder differentialalgebraic formulas as continuous evolution constraints in addition to firstorder discrete jump formulas. These programs characterise interacting discr ..."
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Cited by 41 (27 self)
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Abstract. We generalise dynamic logic to a logic for differentialalgebraic programs, i.e., discrete programs augmented with firstorder differentialalgebraic formulas as continuous evolution constraints in addition to firstorder discrete jump formulas. These programs characterise interacting discrete and continuous dynamics of hybrid systems elegantly and uniformly. For our logic, we introduce a calculus over real arithmetic with discrete induction and a new differential induction with which differentialalgebraic programs can be verified by exploiting their differential constraints algebraically without having to solve them. We develop the theory of differential induction and differential refinement and analyse their deductive power. As a case study, we present parametric tangential roundabout maneuvers in air traffic control and prove collision avoidance in our calculus.
Efficient LoopCheck for Backward Proof Search in Some NonClassical Propositional Logics
, 1996
"... . We consider the modal logics KT and S4, the tense logic K t , and the fragment IPC (^;!) of intuitionistic logic. For these logics backward proof search in the standard sequent or tableau calculi does not terminate in general. In terms of the respective Kripke semantics, there are several kinds of ..."
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Cited by 39 (1 self)
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. We consider the modal logics KT and S4, the tense logic K t , and the fragment IPC (^;!) of intuitionistic logic. For these logics backward proof search in the standard sequent or tableau calculi does not terminate in general. In terms of the respective Kripke semantics, there are several kinds of nontermination: loops inside a world (KT), innite resp. looping branches (S4, IPC (^;!) ), and innite branching degree (K t ). We give uniform sequentbased calculi that contain specically tailored loopchecks such that the eciency of proof search is not deteriorated. Moreover all these loopchecks are easy to implement and can be combined with optimizations. Note that our calculus for S4 is not a pure contractionfree sequent calculus, but this (theoretical) defect does not hinder its application in practice. 1 Introduction For many nonclassical propositional logics, backward proof search in the usual sequent calculi does not terminate in general. For all the logics we consider in th...