Towards a Semantic Characterization of Cut-Elimination

Ciabattoni, Agata; Terui, Kazushige

doi:10.1007/s11225-006-6607-2

Cited by 35 publications

(46 citation statements)

References 17 publications

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“…The range of systems dealt with there is in fact broader than ours, since it deals with various types of structural rules, while in this paper we assume the standard structural rules of minimal logic. On the other hand the results and characterization given in [9] are less satisfactory than those given here. First, in the framework of [9] any connective has essentially infinitely many introduction (and elimination) rules, while our framework makes it possible to convert these infinite sets of rules into a finite set.…”

Section: Related Work and Further Researchcontrasting

confidence: 78%

“…On the other hand the results and characterization given in [9] are less satisfactory than those given here. First, in the framework of [9] any connective has essentially infinitely many introduction (and elimination) rules, while our framework makes it possible to convert these infinite sets of rules into a finite set. Second, our coherence criterion (for non-triviality and cut-elimination)…”

Section: Related Work and Further Researchcontrasting

confidence: 78%

“…Except for [6], the closest to the present one is [9]. The range of systems dealt with there is in fact broader than ours, since it deals with various types of structural rules, while in this paper we assume the standard structural rules of minimal logic.…”

Section: Related Work and Further Researchmentioning

confidence: 97%

“…In contrast, its counterpart in [9] (called reductivity) is not constructive. Third, our notion of strong cut-elimination simply limits the set of possible cut formulas used in a derivation of a sequent from other sequents to those that occur in the premises of that derivation.…”

Section: Related Work and Further Researchmentioning

confidence: 99%

See 3 more Smart Citations

Strict Canonical Constructive Systems

Avron

Lahav

2010

Fields of Logic and Computation

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Abstract. We define the notions of a canonical inference rule and a canonical constructive system in the framework of strict single-conclusion Gentzen-type systems (or, equivalently, natural deduction systems), and develop a corresponding general non-deterministic Kripke-style semantics. We show that every strict constructive canonical system induces a class of non-deterministic Kripke-style frames, for which it is strongly sound and complete. This non-deterministic semantics is used for proving a strong form of the cut-elimination theorem for such systems, and for providing a decision procedure for them. These results identify a large family of basic constructive connectives, including the standard intuitionistic connectives, together with many other independent connectives.

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Section: Related Work and Further Researchcontrasting

confidence: 78%

Section: Related Work and Further Researchcontrasting

confidence: 78%

Section: Related Work and Further Researchmentioning

confidence: 97%

Section: Related Work and Further Researchmentioning

confidence: 99%

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Strict Canonical Constructive Systems

Avron

Lahav

2010

Fields of Logic and Computation

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“…Remark 1. [6] investigates single-conclusion systems with non-standard sets of structural rules. An algebraic semantics using phase spaces is provided for left and right introduction rules for connectives; a result similar to Theorem 4 is shown, namely a connective has a "deterministic semantics"(i.e., the interpretations for the left and right rules coincide) iff admits axiom expansion.…”

Section: Axiom-expansionmentioning

confidence: 99%

Basic Constructive Connectives, Determinism and Matrix-Based Semantics

Ciabattoni

Lahav

Zamansky

2011

Lecture Notes in Computer Science

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Canonical Gentzen-Type Calculi with (n,k)-ary Quantifiers

Zamansky

Avron

2006

Automated Reasoning

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Abstract. Propositional canonical Gentzen-type systems, introduced in [1], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned.[1] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided by two-valued nondeterministic matrices (2Nmatrices).[3] extends these results to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n, k)-ary quantifiers, which bind k distinct variables and connect n formulas. We show that the coherence criterion remains constructive for such systems, and that for the case of k ∈ {0, 1}: (i) a canonical system is coherent iff it has a strongly characteristic 2Nmatrix, and (ii) if a canonical system is coherent, then it admits cut-elimination.

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Towards a Semantic Characterization of Cut-Elimination

Abstract: We introduce necessary and sufficient conditions for a (single-conclusion) sequent calculus to admit (reductive) cut-elimination. Our conditions are formulated both syntactically and semantically.

Cited by 35 publications

References 17 publications

Strict Canonical Constructive Systems

Strict Canonical Constructive Systems

Basic Constructive Connectives, Determinism and Matrix-Based Semantics

Canonical Gentzen-Type Calculi with (n,k)-ary Quantifiers

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