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A Gentzen-style sequent calculus of constructions with expansion rules

Jonathan P. Seldin
2000 Theoretical Computer Science  
A Gentzen-style L-formulation of the calculus of constructions is presented and proved equivalent to a natural deduction formulation based on that of Seldin (1997) .  ...  In [5, 6] , I presented a natural deduction formulation, TOC0, for Coquand's calculus of constructions.  ...  backwards through the rules involves reductions but not expansions of terms, and this seems to be more in the spirit of Gentzen L-formulations than systems with conversion rules.  ... 
doi:10.1016/s0304-3975(98)00195-9 fatcat:vml2vytutfbp7lp62a5sxlf7wi

Index

2000 Theoretical Computer Science  
Meier (1}2) 339}361 Seldin, J.P., A Gentzen-style sequent calculus of constructions with expansion rules (1}2) 199}215 Snir, S., see S. Moran (1}2) 217}241 Veanes, M., see A.  ...  (The issue number is given in front of the page numbers) Arai, N.H., Tractability of Cut-free Gentzen-type propositional calculus with permutation inference II (1}2) 185}197 Aura, T. and J.  ... 
doi:10.1016/s0304-3975(00)00267-x fatcat:ando4cth6nfvzas7td6gkbxmru

Cut as Consequence

Curtis Franks
2010 History and Philosophy of Logic  
', and (2) the structure of the sequent calculi li and lk allowed Gentzen to conceptualize completeness as a question about the relationships among a system's individual rules (as opposed to the relationship  ...  Gentzen's conception of logic is compelling in its own right. It is also of historical interest because it allows for a better understanding of the invention of natural deduction and sequent calculi.  ...  Acknowledgments The participants in my spring 2009 proof theory seminar at the University of Notre Dame, with their comments and questions, helped me clarify my reading of Gentzen.  ... 
doi:10.1080/01445340.2010.522365 fatcat:dnibzkfrlfapjkdfxfpaj4d2bm

Unified correspondence and proof theory for strict implication

Minghui Ma, Zhiguang Zhao
2016 Journal of Logic and Computation  
Gentzen-style cut-free sequent calculi for BDFNL and its extensions with analytic rules which are transformed from strict implication sequents, are developed.  ...  Many strict implication sequents can be transformed into analytic rules employing one of the main tools of unified correspondence theory, namely (a suitably modified version of) the Ackermann lemma based  ...  The work of the second author has been made possible by the NWO Vidi grant 016.138.314, by the NWO Aspasia grant 015.008.054, and by a Delft Technology Fellowship awarded in 2013.  ... 
doi:10.1093/logcom/exw012 fatcat:b46elixh4nelfjxt6whkicpjdq

Explanation in Natural Language of λ̄μμ͂-Terms [chapter]

Claudio Sacerdoti Coen
2006 Lecture Notes in Computer Science  
Theλµμ-calculus, introduced by Curien and Herbelin, is a calculus isomorphic to (a variant of) the classical sequent calculus LK of Gentzen.  ...  In this paper we embed it with a rendering semantics that provides explanations in pseudo-natural language of its proof terms, in the spirit of the work of Yann Coscoy [3] for the λ-calculus.  ...  Theλµμ-calculus is isomorphic to (a variant of) the classical sequent calculus LK of Gentzen.  ... 
doi:10.1007/11618027_16 fatcat:xsxx3aa5ifbqjnzwnd5nzdrxyq

Page 595 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
Summary: “The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels.  ...  Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1.  ... 

Page 5445 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
Martin Hofmann (D-MNCH-I; Munich) 2002h:03025 03B40 03F05 Seldin, Jonathan P. (3-CONC; Montreal, QC) A Gentzen-style sequent calculus of constructions with expansion rules. (English summary) Theoret.  ...  Summary: “A Gentzen-style L-formulation of the calculus of con- structions is presented and proved equivalent to a natural deduc- tion formulation based on that of [J. P. Seldin, Ann. Pure Appl.  ... 

Intuitionistic Decision Procedures Since Gentzen [chapter]

Roy Dyckhoff
2016 Advances in Proof Theory  
Gentzen [31] solved the decision problem for Int with a calculus LJ, in which the antecedent of each sequent is a list of formulae and the succedent is either empty or a single formula.  ...  Initial sequents are of the form A =⇒ A, where A is a formula. Let C indicate either a formula or an empty succedent.  ...  us, for helpful comments on historical matters-regrettably not all incorporated (thanks to a failure of technology).  ... 
doi:10.1007/978-3-319-29198-7_6 fatcat:siqb5ieuunfijm7kec4uf5ygqu

Sequoia: A Playground for Logicians [chapter]

Giselle Reis, Zan Naeem, Mohammed Hashim
2020 Lecture Notes in Computer Science  
Sequent calculus is a pervasive technique for studying logics and their properties due to the regularity of rules, proofs, and metaproperty proofs across logics.  ...  For that purpose, we provide an intuitive interface where inference rules can be input in L A T E X and are immediately rendered with the corresponding symbols.  ...  We have recently finished the implementation of two new features: checking cut admissibility (using Gentzen-style proofs), and supporting rules with context restrictions (such as the one mentioned in Sect  ... 
doi:10.1007/978-3-030-51054-1_32 fatcat:jjpuhsits5d6vgluspitladqrm

First order abduction via tableau and sequent calculi

MARTA CIALDEA MAYER, FIORA PIRRI
1993 Logic Journal of the IGPL  
The tableau system is sound and complete, i.e. ' is a tautology i ' has a tableau proof. 2.1.2. A Gentzen-type calculus is a proof system given by a set of validity preserving rules.  ...  In the case of the sequent calculus, for example, the rules can be modi ed so that, in the sequents of a deduction, formulae never change The leaves ?  ... 
doi:10.1093/jigpal/1.1.99 fatcat:uvtwdb2gqvfy3i76eey2ud3sbi

Page 3414 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
Harold Schellinx (NL-UTRE-MI; Utrecht) 97f:03083 03F50 Prijatelj, Andreja (SV-LJUBM; Ljubljana) Bounded contraction and Gentzen-style formulation of Lukasiewicz logics.  ...  The author takes care to note which of his model-theoretic constructions makes sense from a constructive point of view, but his arguments are mainly classical, that is, non- intuitionistic.  ... 

Sequent Systems for Modal Logics [chapter]

Heinrich Wansing
2002 Handbook of Philosophical Logic  
Higher-arity sequent systems In search of generalizations of the standard Gentzen-style sequent format, it is a natural move to consider consequence relations with an arity greater than 2.  ...  Došen [1985] has developed certain non-standard sequent systems for S4 and S5. In these Gentzen-style systems one is dealing with sequents of arbitrary finite level.  ...  If A is atomic, Clauses 2-5 in the definition of C hold trivially. Case 3.4 ([F ]) :  ... 
doi:10.1007/978-94-010-0387-2_2 fatcat:woycjp6ygjfb5ihl6urfnns36y

Call-by-value is dual to call-by-name

Philip Wadler
2003 Proceedings of the eighth ACM SIGPLAN international conference on Functional programming - ICFP '03  
The rules of classical logic may be formulated in pairs corresponding to De Morgan duals: rules about & are dual to rules about ∨. A line of work, including that of Filinski  ...  As with sequent calculus, there are logical rules for each connective. Right rules always end with a right sequent and left rules always end with a left sequent.  ...  Gentzen proved sequent calculus satisfies a Cut elimination property: any proof of a sequent can be transformed to a proof of the same sequent that does not contain Cut.  ... 
doi:10.1145/944705.944723 dblp:conf/icfp/Wadler03 fatcat:vvyk2cgsufavjp52svrcls6uxu

Call-by-value is dual to call-by-name

Philip Wadler
2003 SIGPLAN notices  
The rules of classical logic may be formulated in pairs corresponding to De Morgan duals: rules about & are dual to rules about ∨. A line of work, including that of Filinski  ...  As with sequent calculus, there are logical rules for each connective. Right rules always end with a right sequent and left rules always end with a left sequent.  ...  Gentzen proved sequent calculus satisfies a Cut elimination property: any proof of a sequent can be transformed to a proof of the same sequent that does not contain Cut.  ... 
doi:10.1145/944746.944723 fatcat:7iqy2ioydjevdpda2ljw6guuhe

Page 753 of Mathematical Reviews Vol. , Issue 99b [page]

1991 Mathematical Reviews  
0A, OF + OB, and a monotonic modal logic is one with the inference rule RM by which from At B we can infer the sequent 0A+ OB.  ...  A normal modal logic is a modal logic with the necessitation rule and the axiom K: O(A — B) — (OA — OB), a regular modal logic is one with the inference rule RR by which from A, + B we can infer the sequent  ... 
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