An Introduction to Proof Theory

Buss, Samuel R.

doi:10.1016/s0049-237x(98)80016-5

Cited by 144 publications

(121 citation statements)

References 24 publications

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“…Necessary and sufficient conditions have been defined for these calculi to admit reductive cut-elimination -a naturally strengthened version of cut-elimination in presence of axioms (see e.g. (Buss 1998)) which in addition aims to shift upward non-eliminable cuts as much as possible. The defined conditions (reductivity and weak substitutivity) are recalled below and applied to knotted commutative calculi.…”

Section: Conditions For (Reductive) Cut-eliminationmentioning

confidence: 99%

“…Reductive cut-elimination is a naturally strengthened version of cut-elimination in presence of axioms (see e.g. (Buss 1998)) which encompasses the "standard" cut-elimination methods working by 1. shifting up cuts and 2. replace them with smaller cuts, when the cut formula is introduced by logical rules in both premisses. The syntactic conditions defined there (reductivity and weak substitutivity) formalize the steps 1. and 2. above.…”

Section: Introductionmentioning

confidence: 99%

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Towards an algorithmic construction of cut-elimination procedures

Ciabattoni

Leitsch

2008

Math. Struct. Comp. Sci.

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We investigate cut-elimination in propositional substructural logics. The problem is to decide whether a given calculus admits (reductive) cut-elimination. We show that, for commutative single-conclusion sequent calculi containing generalized knotted structural rules and arbitrary logical rules, the problem can be decided by resolution-based methods. A general cut-elimination proof for these calculi is also provided. IntroductionGentzen sequent calculi have been the central tool in many proof-theoretical investigations and applications of logic in algebra and computer science. A key property of these calculi is cut-elimination (Gentzen's Hauptsatz), first established by Gentzen (1935) for the sequent calculi LK and LJ for classical and intuitionistic first-order logic. The removal of cuts corresponds to the elimination of intermediate statements (lemmas) from proofs resulting in calculi in which proofs are analytic in the sense that all statements in the proofs are subformulae of the result. Analytic proof calculi for logics are not only an important theoretical tool, useful for understanding relationships between logics and proving metalogical properties like consistency, decidability, admissibility of rules and interpolation, but also the key to develop automated reasoning methods. These calculi also provide an alternative representation of varieties of algebras (see e.g. (Galatos and Ono 2006)) which can then be used to give syntactic proofs of algebraic properties, e.g. amalgamation, for which (in particular cases) semantic methods are not known. Cutelimination is also a powerful tool to prove the completeness of a given analytic sequent calculus with respect to a logic formalized using Hilbert style systems, as the cut rule simulates modus ponens. Cut-elimination proofs have been provided for very many sequent calculi, mainly on a case by case basis (even when the arguments for a given calculus are similar to that of another) and using heavy syntactic arguments usually written without filling in the details. This renders the proof checking difficult and the whole process of eliminating cuts rather opaque. † This work was supported by FWF Projects P18731 and P17995.A. Ciabattoni and A. Leitsch 2 In this paper we perform a resolution-based analysis of cut-elimination in knotted commutative calculi. These are propositional single-conclusion sequent calculi containing arbitrary logical rules (satisfying suitable conditions), the permutation rule and (possibly) unary structural rules generalizing both the weakening and contraction rules in Gentzen's LJ. The considered structural rules are a generalization of the knotted structural rules in (Hori, Ono and Schellinx 1994), whose (n, k) type is of the form: from Γ, A, . . . , A (n times) → C infer Γ, A, . . . , A (k times)→ C, for all n ≥ 0 and k ≥ 1. The (n, k) rule is a restricted form of weakening when n < k, and of contraction when k < n. In (Hori, Ono and Schellinx 1994) extensions of intuitionistic linear logic without the exponential connectives ILL and of i...

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Section: Conditions For (Reductive) Cut-eliminationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Towards an algorithmic construction of cut-elimination procedures

Ciabattoni

Leitsch

2008

Math. Struct. Comp. Sci.

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“…The proof system P K is the Gentzen-style sequent calculus for propositional logic [5,6]. The initial sequents are ⊥ →, → , and A → A, for any propositional formula A.…”

Section: The Proof Systemmentioning

confidence: 99%

“…The normal form we want is cut variable normal form (CVNF) and is defined as follows. It is known how to find a proof with the first two properties [6,5], but, to our knowledge, no property similar to the third has ever been considered.…”

Section: Cut Variable Normal Formmentioning

confidence: 99%

A Propositional Proof System for Log Space

Perron¹

2005

Computer Science Logic

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Abstract. The proof system G * 0 of the quantified propositional calculus corresponds to N C 1 , and G * 1 corresponds to P , but no formula-based proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL * . We begin by defining a class ΣCN F (2) of quantified formulas that can be evaluated in log space. Then GL * is defined as G * 1 with cuts restricted to ΣCN F (2) formulas and no cut formula that is not quantifier free contains a non-parameter free variable. To show that GL * is strong enough to capture log space reasoning, we translate theorems of Σ B 0 -rec into a family of tautologies that have polynomial size GL * proofs. Σ B 0 -rec is a theory of bounded arithmetic that is known to correspond to log space. To do the translation, we find an appropriate axiomatization of Σ B 0 -rec, and put Σ B 0 -rec proofs into a new normal form. To show that GL * is not too strong, we prove the soundness of GL * in such a way that it can be formalized in Σ B 0 -rec. This is done by giving a log space algorithm that witnesses GL * proofs.

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“…Often the existence of a cut-free system for a logic implies an interpolation property for that logic, see any introduction to proof theory, e.g. [7,13,24]. However, if interpolation is a consequence of cut-elimination, then by contraposition we get that the failure of interpolation 'implies' the non-existence of a 'nice' cut-free system.…”

Section: Introductionmentioning

confidence: 99%

Common knowledge does not have the Beth property

Studer

2009

Information Processing Letters

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Common knowledge is an essential notion for coordination among agents. We show that the logic of common knowledge does not have the Beth property and thus it also lacks interpolation. The proof we present is a variant of Maksimova's proof that temporal logics with 'the next' do not have the Beth property. Our result also provides an explanation why it is so difficult to find 'nice' deductive systems for common knowledge.

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An Introduction to Proof Theory

Cited by 144 publications

References 24 publications

Towards an algorithmic construction of cut-elimination procedures

Towards an algorithmic construction of cut-elimination procedures

A Propositional Proof System for Log Space

Common knowledge does not have the Beth property

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