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A Sequent Calculus with Implicit Term Representation [chapter]

Stefan Hetzl
2010 Lecture Notes in Computer Science  
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]

Silvia Ghilezan, Jelena Ivetić, Pierre Lescanne, Dragisa Žunić
2011 Lecture Notes in Computer Science  
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

Beniamino Accattoli
2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science  
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

Giselle Reis
2015 Electronic Proceedings in Theoretical Computer Science  
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

José Espírito Santo
2009 Theory of Computing Systems  
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]

José Espírito Santo
2007 Lecture Notes in Computer Science  
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

Frank Pfenning
2000 Information and Computation  
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

ICHIRO OGATA
2000 International Journal of Foundations of Computer Science  
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]

Anna Mikhajlova, Joakim von Wright
1998 Lecture Notes in Computer Science  
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]

José Espírito Santo
2007 Lecture Notes in Computer Science  
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]

Ichiro Ogata
1998 Lecture Notes in Computer Science  
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]

Stéphane Lengrand, Roy Dyckhoff, James McKinna
2006 Lecture Notes in Computer Science  
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

PAUL DOWNEN, ZENA M. ARIOLA
2018 Journal of functional programming  
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

Eike Ritter, David Pym, Lincoln Wallen
2000 Theoretical Computer Science  
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]

Eric Deplagne, Claude Kirchner
2002 Lecture Notes in Computer Science  
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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