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Metalogical Frameworks
, 1992
"... In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the me ..."
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Cited by 60 (18 self)
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In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...
Reasoning About Set Constraints Applied to Tractable Inference in Intuitionistic Logic
 Journal of Logic and Computation
, 1998
"... Automated reasoning about sets has received a considerable amount of interest in the literature. Techniques for such reasoning have been used in, for instance, analyses of programming languages, terminological logics and spatial reasoning. In this paper, we identify a new class of set constraints wh ..."
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Cited by 7 (2 self)
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Automated reasoning about sets has received a considerable amount of interest in the literature. Techniques for such reasoning have been used in, for instance, analyses of programming languages, terminological logics and spatial reasoning. In this paper, we identify a new class of set constraints where checking satisfiability is tractable (i.e. polynomialtime). We show how to use this tractability result for constructing a new tractable fragment of intuitionistic logic. Furthermore, we prove NPcompleteness of several other cases of reasoning about sets. 1 Introduction There has been considerable interest in formalisms for describing and reasoning about sets. We begin by describing some of these. The most wellstudied class of set constraints is, probably, Herbrand set constraints. Such have been suggested as a formalism for describing relationships between sets of terms of a free algebra. A positive set constraint has the form X ` Y , where X and Y are set expressions. Examples of ...
Fast decision procedure for propositional dummett logic based on a multiple premise tableau calculus
, 2008
"... We present a procedure to decide propositional Dummett logic. This procedure relies on a tableau calculus with a multiple premise rule and optimizations. The resulting implementation outperforms the state of the art graphbased procedure. 1 ..."
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Cited by 5 (3 self)
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We present a procedure to decide propositional Dummett logic. This procedure relies on a tableau calculus with a multiple premise rule and optimizations. The resulting implementation outperforms the state of the art graphbased procedure. 1
Algorithms and Complexity for Temporal and Spatial Formalisms
, 1997
"... The problem of computing with temporal information was early recognised within the area of artificial intelligence, most notably the temporal interval algebra by Allen has become a widely used formalism for representing and computing with qualitative knowledge about relations between temporal interv ..."
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Cited by 3 (2 self)
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The problem of computing with temporal information was early recognised within the area of artificial intelligence, most notably the temporal interval algebra by Allen has become a widely used formalism for representing and computing with qualitative knowledge about relations between temporal intervals. However, the computational properties of the algebra and related formalisms are known to be bad: most problems (like satisfiability) are NPhard. This thesis contributes to finding restrictions (as weak as possible) on Allen's algebra and related temporal formalisms (the pointinterval algebra and extensions of Allen's algebra for metric time) for which the satisfiability problem can be computed in polynomial time. Another research area utilising temporal information is that of reasoning about action, which treats the problem of drawing conclusions based on the knowledge about actions having been performed at certain time points (this amounts to solving the infamous frame problem). One ...
A Sequent Calculus for Intuitionistic Default Logic
, 1997
"... Current research on nonmonotonic reasoning shows growing interest on implementation details, so the need for concrete calculi formalizing nonmonotonic logics is clearly recognized. On the other hand, there is also an increased number of works combining intuitionistic logic with various kinds of no ..."
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Cited by 1 (0 self)
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Current research on nonmonotonic reasoning shows growing interest on implementation details, so the need for concrete calculi formalizing nonmonotonic logics is clearly recognized. On the other hand, there is also an increased number of works combining intuitionistic logic with various kinds of nonmonotonic formalisms. As a case in point, intuitionistic versions of both default and autoepistemic logics have been proposed, and tight connections between intuitionistic logic and logic programs (or deductive databases) using hypothetical inferences have been established. In this paper, we present a sequent calculus for default reasoning in the style of Bonatti with intuitionistic logic as the underlying logical structure. In contrast to other proposals, Bonatti's technique allows a very simple and intuitive specification of the calculus, making it an ideal tool for implementation purposes. 1 Introduction Nonmonotonic reasoning techniques are an important cornerstone towards formalizin...
Tableau Calculus for Dummett Logic Based on Present and Next State of Knowledge
"... In this paper we use the Kripke semantics characterization of Dummett logic to introduce a new way of handling nonforced formulas in tableau proof systems. We pursue the aim of reducing the search space by strictly increasing the number of forced propositional variables after the application of non ..."
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In this paper we use the Kripke semantics characterization of Dummett logic to introduce a new way of handling nonforced formulas in tableau proof systems. We pursue the aim of reducing the search space by strictly increasing the number of forced propositional variables after the application of noninvertible rules. The focus of the paper is on a new tableau system for Dummett logic, for which we have an implementation. The ideas presented can be extended to intuitionistic logic as well. 1