Atomic Cut Elimination for Classical Logic

Brünnler, Kai

doi:10.1007/978-3-540-45220-1_9

Cited by 21 publications

(35 citation statements)

References 9 publications

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“…Three approaches to cut elimination in the calculus of structures have been explored in previous papers: in [GS01,Str03b] we relied on permutations of rules, in [BT01] Brünnler and Tiu relied on semantics, and in [Brü03] Brünnler presents a simple syntactic method that employs the atomicity of cut together with certain proof theoretical properties of classical logic. Neither method can be applied in our case: We know a counterexample (found by Alwen Tiu) that shows that the coseq rule cannot be permuted up by the same technique that has been used in [GS01,Str03b].…”

Section: Cut Eliminationmentioning

confidence: 99%

A system of interaction and structure V: the exponentials and splitting

Guglielmi¹,

Straβburger²

2011

Math. Struct. Comp. Sci.

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System NEL is the mixed commutative/non-commutative linear logic BV augmented with linear logic's exponentials, or, equivalently, it is MELL augmented with the non-commutative self-dual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential composition and it faithfully models causal quantum evolution. In this paper, we show cut elimination for NEL, based on a property that we call splitting. NEL is presented in the calculus of structures, which is a deep-inference formalism, because no Gentzen formalism can express it analytically. The splitting theorem shows how and to what extent we can recover a sequent-like structure in NEL proofs. Together with the decomposition theorem, proved in the previous paper of the series, this immediately leads to a cut-elimination theorem for NEL.

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Section: Cut Eliminationmentioning

confidence: 99%

A system of interaction and structure V: the exponentials and splitting

Guglielmi¹,

Straβburger²

2011

Math. Struct. Comp. Sci.

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“…In an instance of i↓ or i↑, the structure A introduced in the conclusion or premise respectively, is called principal structure. This system has a typical shape for deep inference, and it is related to the classical system KS [Brü03a]. Apart from the switch rule s↓, rules are similar to those of the sequent calculus, with a variation on the identity rule.…”

Section: An Intuitionistic Calculus Of Structuresmentioning

confidence: 99%

“…This logically leads us to the completion of JS ∪ {i↑} into the symmetric system SJS, following the usual scheme in deep inference [Brü03a,Str03b]. In the intuitionistic setting, where DeMorgan duality does not exist, the dual of a rule is obtained by inverting premise and conclusion, and then using the result in the opposite context.…”

Section: Symmetric Normalisationmentioning

confidence: 99%

“…On one side, this provides a greater freedom when designing new deductive systems such as completely local proof systems, as done in classical [Brü03a] and linear logic [Str03b,Str03a], or for proving new results as decomposition theorems [Str03b,SG11]. On the other side, this means that the traditional techniques applied in normalisation proofs cannot directly be used.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric normalisation for intuitionistic logic

Guenot

Straßburger

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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We present two proof systems for implication-only intuitionistic logic in the calculus of structures. The first is a direct adaptation of the standard sequent calculus to the deep inference setting, and we describe a procedure for cut elimination, similar to the one from the sequent calculus, but using a non-local rewriting. The second system is the symmetric completion of the first, as normally given in deep inference for logics with a DeMorgan duality: all inference rules have duals, as cut is dual to the identity axiom. We prove a generalisation of cut elimination, that we call symmetric normalisation, where all rules dual to standard ones are permuted up in the derivation. The result is a decomposition theorem having cut elimination and interpolation as corollaries.

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“…As shown in [Bruscoli and Guglielmi 2009], Frege systems and calculus of structures (with cut) p-simulate each other and are therefore equally powerful with respect to proof complexity. However, unlike Frege systems, the calculus of structures is a proof formalism that comes with methods for proof search [Kahramanogulları 2006;Chaudhuri et al 2011;Chaudhuri 2013b] and proof normalization [Brünnler 2003a;Brünnler 2006;. This means that we can now study cut-free proof systems with extension and substitution [Straßburger 2012].…”

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confidence: 99%

On the Power of Substitution in the Calculus of Structures

Novaković¹,

Straßburger

2015

ACM Trans. Comput. Logic

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There are two contributions in this paper. First, we give a direct proof of the known fact that Frege systems with substitution can be p-simulated by the calculus of structures extended with the substitution rule. This is done without referring to the p-equivalence of extended Frege systems and Frege systems with substitution. Second, we then show that the cut-free calculus of structures with substitution is p-equivalent to the cut-free calculus of structures with extension.

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Atomic Cut Elimination for Classical Logic

Cited by 21 publications

References 9 publications

A system of interaction and structure V: the exponentials and splitting

A system of interaction and structure V: the exponentials and splitting

Symmetric normalisation for intuitionistic logic

On the Power of Substitution in the Calculus of Structures

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