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From X to Pi; Representing the Classical Sequent Calculus in the Pi-calculus [article]

Steffen van Bakel, Luca Cardelli, Maria Grazia Vigliotti
2011 arXiv   pre-print
Since X enjoys the Curry-Howard isomorphism for Gentzen's calculus LK, this implies that all proofs in LK have a representation in Pi.  ...  We encode the circuits of the calculus X into this variant of Pi, and show that all reduction (cut-elimination) and assignable types are preserved.  ...  Classical sequents, X , and π The sequent calculus LK, introduced by Gentzen in [15] , is a logical system in which the rules only introduce connectives (but on either side of a sequent), in contrast  ... 
arXiv:1109.4817v1 fatcat:u3fcljuubbfmff5feimp22qska

The Complexity of Theorem Proving in Circumscription and Minimal Entailment [chapter]

Olaf Beyersdorff, Leroy Chew
2014 Lecture Notes in Computer Science  
In particular, we investigate two sequent-style calculi: MLK defined by Olivetti [28] and CIRC introduced by Bonatti and Olivetti [8] , and the tableaux calculus NTAB suggested by Niemelä [26] .  ...  In our analysis we obtain exponential lower bounds for the proof size in NTAB and CIRC and show a polynomial simulation of CIRC by MLK .  ...  When considering LK in proof complexity we treat sequents as strings in binary, built from binary strings representing atoms and connectives.  ... 
doi:10.1007/978-3-319-08587-6_32 fatcat:uqhibcgl35clfozhc7ambszgwi

Principles of Superdeduction

Paul Brauner, Clement Houtmann, Claude Kirchner
2007 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)  
In predicate logic, the proof that a theorem P holds in a theory Th is typically conducted in natural deduction or in the sequent calculus using all the information contained in the theory in a uniform  ...  We introduce a proof-term language and a cutelimination procedure both based on Christian Urban's work on classical sequent calculus.  ...  for helpful discussions, to the Modulo meetings and the Protheo team for many interactions.  ... 
doi:10.1109/lics.2007.37 dblp:conf/lics/BraunerHK07 fatcat:uez7vdqlbvh3jg5272onixx3rm

nanoCoP: A Non-clausal Connection Prover [chapter]

Jens Otten
2016 Lecture Notes in Computer Science  
This paper presents a non-clausal theorem prover for classical first-order logic. It is based on a non-clausal connection calculus and implemented with a few lines of Prolog code.  ...  By working entirely on the original structure of the input formula, the resulting non-clausal proofs are not only shorter, but can also be more easily translated into, e.g., sequent proofs.  ...  The author would like to thank Wolfgang Bibel for his helpful comments on a preliminary version of this paper. nanoCoP: A Non-clausal Connection Prover  ... 
doi:10.1007/978-3-319-40229-1_21 fatcat:rhrl3tvvnrhfhloiwuz245wuxu

Tractable Transformations from Modal Provability Logics into First-Order Logic [chapter]

Stéephane Demri, Rajeev Goré
1999 Lecture Notes in Computer Science  
Theorem provers for classical rst-order logic can then be used to mechanise deduction in these \psuedo-displayable second order" modal logics.  ...  Using Belnap's proof-theoretical framework Display Logic we characterise the \pseudo-displayable" subclass of LF and show how to de ne polynomial-time transformations from each such LF into the corresponding  ...  That is, these indices are represented in binary writing. The map g is a generalisation of the map from Grz into S4 de ned in DG99a].  ... 
doi:10.1007/3-540-48660-7_2 fatcat:h72prlqi6rgdrf4zotnkt3krta

nanoCoP: Natural Non-clausal Theorem Proving

Jens Otten
2017 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence  
Most efficient fully automated theorem provers implement proof search calculi that require the input formula to be in a clausal form, i.e. disjunctive or conjunctive normal form.  ...  This paper presents a non-clausal automated theorem prover for classical first-order logic. It is based on a non-clausal connection calculus and implemented with a few lines of Prolog code.  ...  Acknowledgements The author would like to thank Wolfgang Bibel for his helpful comments on a preliminary version of this paper.  ... 
doi:10.24963/ijcai.2017/695 dblp:conf/ijcai/Otten17 fatcat:e7twk7ckbnhjvee46egwmol7km

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

Angelo Brillout, Daniel Kroening, Philipp Rümmer, Thomas Wahl
2011 Journal of automated reasoning  
We prove our calculus to be sound and complete.  ...  We introduce an interpolation approach based on a sequent calculus for QFPA that determines interpolants by annotating the steps of an unsatisfiability proof with partial interpolants.  ...  Acknowledgments.We thank Jérôme Leroux, Vijay D'Silva, Georg Weissenbacher, and the anonymous referees for their comments.  ... 
doi:10.1007/s10817-011-9237-y fatcat:e53cvfqtmvgrlj6i6is6zaquhm

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic [chapter]

Angelo Brillout, Daniel Kroening, Philipp Rümmer, Thomas Wahl
2010 Lecture Notes in Computer Science  
We prove our calculus to be sound and complete.  ...  We introduce an interpolation approach based on a sequent calculus for QFPA that determines interpolants by annotating the steps of an unsatisfiability proof with partial interpolants.  ...  Acknowledgments.We thank Jérôme Leroux, Vijay D'Silva, Georg Weissenbacher, and the anonymous referees for their comments.  ... 
doi:10.1007/978-3-642-14203-1_33 fatcat:evdehmnl45btlcqpwt5hf75f4i

Universal Proof Theory: Semi-analytic Rules and Craig Interpolation [article]

Amirhossein Akbar Tabatabai, Raheleh Jalali
2018 arXiv   pre-print
In [6], Iemhoff introduced the notion of a focused axiom and a focused rule as the building blocks for a certain form of sequent calculus which she calls a focused proof system.  ...  Using this relationship, we first present a uniform method to prove interpolation for different logics from sub-structural logics FL_e, FL_ec, FL_ew and IPC to their appropriate classical and modal extensions  ...  We are grateful to Rosalie Iemhoff for bringing this interesting line of research to our attention, for her generosity in sharing her ideas on the subject that we call universal proof theory and for the  ... 
arXiv:1808.06256v1 fatcat:qg4fhyswv5ddpptbrm7zcycwei

Simultaneous rigid E-unification and other decision problems related to the Herbrand theorem

Andrei Voronkov
1999 Theoretical Computer Science  
If all variables of a formula cp are x, we denote by 3~ the formula 3x(p. Some of results depend on the signature and logic in which formulas are written.  ...  A sequent is an expression r 4 d, where r and d are multisets of formulas. In  ...  I am also grateful to the editor of this volume Sergei Adian for his patience and to an anonymous referee for helpful comments.  ... 
doi:10.1016/s0304-3975(98)00317-x fatcat:f2gi56tez5hlnck3ech6uj26nu

A theorem proving framework for the formal verification of Web Services Composition

Petros Papapanagiotou, Jacques D. Fleuriot
2011 Electronic Proceedings in Theoretical Computer Science  
Our approach is based on the proofs-as-processes paradigm that enables inference rules of Classical Linear Logic (CLL) to be translated into pi-calculus processes.  ...  In this setting, composition is achieved by representing available web services as CLL sentences, proving the requested composite service as a conjecture, and then extracting the constructed pi-calculus  ...  In CLL we can represent the output of selectSki as (PRICE USD ⊕ EXCEPT ION). • Linear negation (· ⊥ ) in CLL obeys similar laws to those of classical negation.  ... 
doi:10.4204/eptcs.61.1 fatcat:ahbkprrdg5dobce6naxze65r6a

The Sequent Calculus of Skew Monoidal Categories [article]

Tarmo Uustalu, Niccolò Veltri, Noam Zeilberger
2020 arXiv   pre-print
In this calculus, antecedents consist of a stoup (an optional formula) followed by a context, and the connectives behave like in the standard monoidal sequent calculus except that the left rules may only  ...  We present a sequent calculus for skew monoidal categories, building on the recent formulation by one of the authors of a sequent calculus for the Tamari order (skew semigroup categories).  ...  pY b Zq ÝÑ pX b pI b Y qq b Z is derivable in the monoidal sequent calculus.  ... 
arXiv:2003.05213v1 fatcat:likx7pfi3bdlda5jvgeiszaiz4

On the π-calculus and linear logic

G. Bellin, P.J. Scott
1994 Theoretical Computer Science  
On the n-calculus and linear logic, Theoretical Computer Science 135 (1994) 11-65.  ...  We give a detailed treatment of information flow in proof-nets and show how to mirror various evaluation strategies for proof normalization.  ...  -from classical MLL sequent proofs (thus from IMLL proofs) to proof nets (cf. also [20] ) which determines a bijection between the inferences of a derivation 9 and the links of Y'=(g)-.  ... 
doi:10.1016/0304-3975(94)00104-9 fatcat:fo4hu5wjpvfstmcotl57lpj2sa

A decision procedure revisited: notes on direct logic, linear logic and its implementation

Gianluigi Bellin, Jussi Ketonen
1992 Theoretical Computer Science  
We will focus on the structure of direct predicate calculus as defined in Ketonen and Weyhrauch (1984) in the light of the recent work of Girard (Girard , 1988 ) on linear logic.  ...  In addition, the implementation of a decision procedure for direct predicate logic is sketched.  ...  S is derivable in classical sequent calculus LK if there is a sequent S* in 2, derivable in direct sequent calculus such that TT(S*) = S. Basic proof theory Theorem (Cut elimination).  ... 
doi:10.1016/0304-3975(92)90069-r fatcat:zz4x4y4mobfchgtm6eskc5dygu

Interpreting HOL in the calculus of constructions

Jonathan P. Seldin
2004 Journal of Applied Logic  
The purpose of this paper is to consider a representation of the HOL theorem-prover in the calculus of constructions with the property that consistency results from the calculus of constructions imply  ...  such results in HOL.  ...  Acknowledgements I would like to thank Martin Bunder, Roger Hindley, and the anonymous referees for their helpful comments and suggestions.  ... 
doi:10.1016/j.jal.2004.02.003 fatcat:annw2dbd55cv7kfktab3jkcwqa
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