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A Sequent Calculus with Implicit Term Representation
[chapter]
2010
Lecture Notes in Computer Science
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We define a cutelimination procedure for this calculus and show that it produces the same cut-free proofs as the standard calculus, but, due to the implicit representation of terms, it provides exponentially ...
We investigate a modification of the sequent calculus which separates a first-order proof into its abstract deductive structure and a unifier which renders this structure a valid proof. ...
On the other hand the implicit representation of terms is used to give a considerably simplified proof of a characterisation of the form of witness terms obtainable by cut-elimination in terms of a regular ...
doi:10.1007/978-3-642-15205-4_28
fatcat:o5ctxuk5rrb6lizbbcau7kdrgq
Intuitionistic Sequent-Style Calculus with Explicit Structural Rules
[chapter]
2011
Lecture Notes in Computer Science
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Sequent calculus -LJ Implicit structural rules (Ax) Γ, A A (→ L ) Γ A Γ, B C Γ, A → B C (→ R ) Γ, A B Γ A → B (Cut) Γ A Γ, A B Γ B
Sequent term calculi Pottinger, Zucker 1970s comparing cut-elimination ...
Natural deduction
λ-calculus;
Hilbert's axiomatic system
combinators;
Sequent calculus
λ Gtz -calculus
Sequent calculus with explicit structural rules
??? ...
doi:10.1007/978-3-642-22303-7_7
fatcat:ysu3u2xzjzdgficcvygpgcl5ou
Compressing Polarized Boxes
2013
2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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The sequential nature of sequent calculus provides a simple definition of cut-elimination rules that duplicate or erase sub-proofs. ...
Moreover, implicit boxes are more parallel than explicit boxes, as they realize a larger quotient. ...
We deal with MELLP sequent calculus only indirectly: we will relate our syntax to Laurent's proof nets, which are themselves related to sequent calculus. Nets. ...
doi:10.1109/lics.2013.49
dblp:conf/lics/Accattoli13
fatcat:vg26sbwapne7fchwqpkd56rh4y
Importing SMT and Connection proofs as expansion trees
2015
Electronic Proceedings in Theoretical Computer Science
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By representing the proofs in the same framework, all the algorithms and tools available for expansion trees (compression, visualization, sequent calculus proof construction, proof checking, etc.) can ...
The expansion proofs can also be used as a validation tool for the proof objects produced. ...
Expansion proofs are a compact representation for first and higher order sequent calculus proofs. ...
doi:10.4204/eptcs.186.3
fatcat:frjp276fq5amfcljaic4z44oii
The λ-Calculus and the Unity of Structural Proof Theory
2009
Theory of Computing Systems
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In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. ...
A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions. ...
Kleene [19] showed how to define sequent calculus with implicit structural rules, built in the logical rules. ...
doi:10.1007/s00224-009-9183-9
fatcat:ihe5exp4ovbrjozo3qgq7fgsla
Completing Herbelin's Programme
[chapter]
2007
Lecture Notes in Computer Science
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Herbelin worked with a fragment of sequent calculus with constraints on left introduction. ...
This requires the introduction of a lambda-like calculus for full sequent calculus and an extension of natural deduction that gives meaning to "applicative contexts" and "applicative terms". ...
Conclusions: Herbelin's seminal suggestion in [11] is that the (computational) difference between sequent calculus and natural deduction may be reduced to a mere question of representation of λ-terms ...
doi:10.1007/978-3-540-73228-0_10
fatcat:pdfhumqqf5cslfeysqp6lbvmd4
Structural Cut Elimination
2000
Information and Computation
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The classical sequent calculus with proof terms satisfies the following properties. Proof. By simple structural inductions. ...
Appendix B gives a formulation of cut elimination as a translation from a sequent calculus with cut to a sequent calculus without cut. ...
Let D :: (1 wÄ d + 2) be a classical sequent derivation possibly containing cut. Thus there exists a cut-free derivation D$ :: (1 wÄ d $ 2) Proof. ...
doi:10.1006/inco.1999.2832
fatcat:olw6f7svunaalkzziyiullzcom
CONSTRUCTIVE CLASSICAL LOGIC AS CPS-CALCULUS
2000
International Journal of Foundations of Computer Science
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We establish the Curry-Howard isomorphism between constructive classical logic and CPS-calculus. CPS-calculus exactly means the target language of Continuation Passing Style(CPS) transforms. ...
i.e. t : A x ) ) with t-cut with implicit contraction inFigureB.1. ...
Our term assignment faithfully reects the structure of sequent calculus. ...
doi:10.1142/s0129054100000065
fatcat:qt4xrvsvsrc5xf62vheeqvavkq
Proving isomorphism of first-order logic proof systems in HOL
[chapter]
1998
Lecture Notes in Computer Science
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Besides, by p r o ving a theorem which states that provability in attened sequent calculus implies provability in standard sequent calculus, we show how some meta-logical results about Hilbertian axiomatization ...
and natural deduction can be translated to sequent calculus. ...
Sequent calculus (SC) represents a special case, being a proof system for sequents with several (more than one) formulae in the succedent. ...
doi:10.1007/bfb0055143
fatcat:l3zojkhjgbckdbjkhiktlwmpoa
Refocusing Generalised Normalisation
[chapter]
2007
Lecture Notes in Computer Science
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In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. ...
A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions. ...
This rules needs a mix of head and tail focus. Maybe a good system for dealing with such rules is a variant of the unified calculus with reduction modulo the equation θ(F N, L) = θ(F, N :: L). 1 ...
doi:10.1007/978-3-540-73001-9_27
fatcat:yike52mbgneqxdfmbnikmo44n4
Cut Elimination for Classical Proofs as Continuation Passing Style Computation
[chapter]
1998
Lecture Notes in Computer Science
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These works are all based on logic in Gentzen-style sequent calculus [7] . We unify the proof theoretical approach (i.e. SN and CR cut-elimination procedure) and the reduction system approach (CPS). ...
In Grin's result, Plotkin's call-by-name (CBN) CPS translation on simply-typed -calculus induces a G odel's double-negation translation on their types. ...
Our term assignment faithfully reects the structure of sequent calculus. Thus inductive denition of substitution on terms agrees with induction on the length of derivation for sequent calculus. ...
doi:10.1007/3-540-49366-2_5
fatcat:hkdngckiqjhrffbrhdnujunnhy
A Sequent Calculus for Type Theory
[chapter]
2006
Lecture Notes in Computer Science
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In order to express proof-search in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proof-search and strongly related to natural ...
PTSC are equipped with a normalisation procedure, adapted from Herbelin's and defined by local rewrite rules as in Cut-elimination, using explicit substitutions. ...
Although encodings from natural deduction to sequent calculus and viceversa have been widely studied [Gen35, Pra65, Zuc74] , the representation in sequent calculus of type theories is relatively undeveloped ...
doi:10.1007/11874683_29
fatcat:252c4vsmevfrzoe2x66m75gdfe
A tutorial on computational classical logic and the sequent calculus
2018
Journal of functional programming
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We begin by reviewing Gentzen's LK sequent calculus and show how the Curry–Howard isomorphism still applies to give us a different basis for expressing computation. ...
We present a model of computation that heavily emphasizes the concept of duality and the interaction between opposites–production interacts with consumption. ...
In terms of provability-the question of which sequents can conclude a valid proof tree-the versions of LK with explicit and implicit structural rules are the same. ...
doi:10.1017/s0956796818000023
fatcat:ff6fprv6vfhrjgqhwnll3vqcwe
On the intuitionistic force of classical search
2000
Theoretical Computer Science
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We develop a system of realizers (proof-objects) for sequents in classical propositional logic (the types) by extending Parigot's -calculus, a system of realizers for classical free deduction (cf. natural ...
As an application, we develop a proof procedure based on the natural extension of the notion of uniform proof to the multiple-conclusioned classical sequent calculus Harrop fragment of intuitionistic logic ...
Acknowledgements We gratefully acknowledge the UK EPSRC for supporting part of this work via Research Grants GR=J46616 and GR=K41687 under the common title "Search Modules I: Representation and Combination ...
doi:10.1016/s0304-3975(99)00178-4
fatcat:bc26d3pcs5fojbmexgaed6ddbu
Deduction versus Computation: The Case of Induction
[chapter]
2002
Lecture Notes in Computer Science
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When mechanizing proof construction, explicit induction is used in proof assistants and implicit induction is used in rewrite based automated theorem provers. ...
We show how this applies to a uniform understanding of the so called induction by rewriting method and how this relates directly to the general use of an induction principle. ...
This presentation of first-order logic relies on the sequent calculus modulo a congruence defined on terms and propositions. ...
doi:10.1007/3-540-45470-5_3
fatcat:4bjblblahbhkrhbmrrmxx2ivsu
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