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The Taming of the Cut. Classical Refutations with Analytic Cut
 JOURNAL OF LOGIC AND COMPUTATION
, 1994
"... The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cutfree systems are anomalous from the prooftheoretical, the semantical and the computational point of view. F ..."
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Cited by 52 (1 self)
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The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cutfree systems are anomalous from the prooftheoretical, the semantical and the computational point of view. Firstly, they cannot represent the use of auxiliary lemmas in proofs. Secondly, they cannot express the bivalence of classical logic. Thirdly, they are extremely inefficient, as is emphasized by the "computational scandal" that such systems cannot polynomially simulate the truthtables. None of these anomalies occurs if the cut rule is allowed. This raises the problem of formulating a proof system which incorporates a cut rule and yet can provide a suitable model of classical analytic deduction. For this purpose we present an alternative refutation system for classical logic, that we call KE. This system, though being "close" to Smullyan's tableau method, is not cutfree but includes a class...
Labelled Tableaux for MultiModal Logics
 Theorem Proving with Analytic
, 1995
"... this paper we present a tableaulike proof system for multimodal logics based on D'Agostino and Mondadori's classical refutation system KE [DM94]. The proposed system, that we call KEM , works for the logics S5A and S5P(n) which have been devised by Mayer and van der Hoek [MvH92] for formalizing th ..."
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Cited by 17 (9 self)
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this paper we present a tableaulike proof system for multimodal logics based on D'Agostino and Mondadori's classical refutation system KE [DM94]. The proposed system, that we call KEM , works for the logics S5A and S5P(n) which have been devised by Mayer and van der Hoek [MvH92] for formalizing the notions of actuality and preference. We shall also show how KEM works with the normal modal logics K45, D45, and S5 which are frequently used as bases for epistemic operators  knowledge, belief (see, for example [Hoe93, Wan90]), and we shall briefly sketch how to combine knowledge and belief in a multiagent setting through KEM modularity
Sequent Calculi for Nominal Tense Logics: A Step Towards Mechanization?
, 1999
"... . We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule ..."
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Cited by 15 (4 self)
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. We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule that is not eliminable. A nice computational property of the restriction is, for instance, that at any stage of the proof, only a finite number of potential cutformulae needs to be taken under consideration. Although restrictions on the proof search (preserving completeness) are given in the paper and most of them are theoretically appealing, the use of those calculi for mechanization is however doubtful. Indeed, we present sequent calculi for fragments of classical logic that are syntactic variants of the sequent calculi for the nominal tense logics. 1 Introduction Background. The nominal tense logics are extensions of Prior tense logics (see e.g. [Pri57, RU71]) by adding nomina...
Shakespearian Modal Logic: A Labelled Treatment Of Modal Identity
, 1998
"... this paper we describe a modal proof system arising from the combination of a tableaulike classical system, which incorporates a restricted ("analytical") version of the cut rule, with a label formalism which allows for a speicialised, logic dependant unification algorithm. The system provides a un ..."
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Cited by 14 (12 self)
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this paper we describe a modal proof system arising from the combination of a tableaulike classical system, which incorporates a restricted ("analytical") version of the cut rule, with a label formalism which allows for a speicialised, logic dependant unification algorithm. The system provides a uniform prooftheoretical treatment of firstorder (normal) modal logics with identity, with and without Barcan formula and/or its converse 1
ProofTerms for Classical and Intuitionistic Resolution (Extended Abstract)
, 1996
"... We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resol ..."
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Cited by 12 (3 self)
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We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resolution rules. The techniques use the fflcalculus, a development of Parigot's calculus.
Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
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Cited by 8 (4 self)
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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.
Turning Cycles into Spirals
, 1999
"... Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14 ..."
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Cited by 6 (3 self)
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Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14]. There is a link between the presence of cut formulas with nested quantifiers and the nonelementary expansion needed to prove a theorem without the help of such formulas. If one considers the graph defined by tracing the flow of occurrences of formulas (in the sense of [2]) for proofs allowing a nonelementary compression, one Preprint submitted to Elsevier Preprint 7 November 1997 finds that such graphs contain cycles [5] or almost cyclic structures[6]. These cycles codify in a small space (i.e. a proof with a small number of lines) all the information which is present in the proof once cuts on formulas wit
Cut Formulae and Logic Programming
"... . In this paper we present a mechanism to define names for proofwitnesses of formulae and thus to use Gentzen's cutrule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proofsearch by i ..."
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Cited by 3 (1 self)
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. In this paper we present a mechanism to define names for proofwitnesses of formulae and thus to use Gentzen's cutrule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proofsearch by indicating when an instance of the cutrule should be attempted. By using the cutrule there are proofs that can be made dramatically shorter. We explain how this idea of using the cutrule can be applied to the logic of hereditary Harrop formulae. 1 Introduction The computation mechanisms both for logic and for functional programming are searches for cutfree proofs. First, in pure logic programming the achievement of a goal G w.r.t. a program P can be seen 1 as the search for a proof in Gentzen's intuitionistic sequent calculus LJ [Gen69], of the sequent P ) G, that by Gentzen's cutelimination theorem can be cutfree [Bee89], [Mil90]; a term found as a witness to a proof contains among...
Simplifying proofs in Fitchstyle natural deduction systems
, 2004
"... We present an algorithm for simplifying Fitchstyle natural deduction proofs in classical firstorder logic. We formalize Fitchstyle natural deduction as a denotational proof language, N DL, with a rigorous syntax and semantics. Based on that formalization, we define an array of simplifying transfo ..."
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Cited by 3 (2 self)
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We present an algorithm for simplifying Fitchstyle natural deduction proofs in classical firstorder logic. We formalize Fitchstyle natural deduction as a denotational proof language, N DL, with a rigorous syntax and semantics. Based on that formalization, we define an array of simplifying transformations and show them to be terminating and to respect the formal semantics of the language. We also show that the transformations never increase the size or complexity of a deduction—in the worst case, they produce deductions of the same size and complexity as the original. We present several examples of proofs containing various types of superfluous “detours, ” and explain how our procedure eliminates them, resulting in smaller and cleaner deductions. All of the transformations are fully implemented in SMLNJ, and the complete code listing is available. 1.1