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Revisiting Cut-Elimination: One Difficult Proof Is Really a Proof
[chapter]

*
Lecture Notes in Computer Science
*

We thus show that the original

doi:10.1007/978-3-540-70590-1_28
fatcat:bj36tpdcgnbshkmhdpodljd2we
*proof**is*indeed*a**proof*and that present automated proving technology*is*adequate for formalising such*difficult**proofs*. ... The first author used such*a*logical relation argument to establish strong normalising for*a**cut*-*elimination*procedure in classical logic. ... Sequent*Proofs*and*Cut*-*Elimination*The main idea behind the*cut*-*elimination*procedure presented in [14, 17]*is*to transport*one*subderivation of*a*commuting cut to the place(s) where the cut-formula ...##
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Cut Elimination for Monomial MALL Proof Nets

2008
*
Logic in Computer Science
*

This sharing leads to the definition of

doi:10.1109/lics.2008.31
dblp:conf/lics/LaurentM08
fatcat:kdd56hmag5etfcbh7dyfczk7mu
*a*strong*cut**elimination*procedure for MALL. ... We present*a*syntax for MALL (multiplicative additive linear logic without units)*proof*nets which refines Girard's*one*. ...*One*usually requires at least the first two properties to hold, otherwise it*is**really**difficult*to consider the proposed*proof*-net syntax as*a*real alternative to the sequent calculus. ...##
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On Formally Measuring and Eliminating Extraneous Notions in Proofs

2008
*
Philosophia Mathematica
*

In this paper I discuss extraneousness generally, and then consider

doi:10.1093/philmat/nkn015
fatcat:w4kto4hfmjbplnlkkt5aaf6sve
*a*specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's*cut*-*elimination*theorem. ... Many mathematicians and philosophers of mathematics believe some*proofs*contain elements extraneous to what*is*being proved. ... rule, it*is*also known as the "*cut*-*elimination*theorem". ...##
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Pomset logic: a logical and grammatical alternative to the Lambek calculus
[article]

2020
*
arXiv
*
pre-print

This classical calculus enjoys

arXiv:2001.02155v1
fatcat:p4f7qlru7ba55brlnygeu6ddsq
*a**proof*net calculus,*cut*-*elimination*, denotational semantics, but had no sequent calculus, despite my many attempts and the study of closely related deductive systems like ... We defined*a*grammatical formalism based*on*pomset logic, with partial*proof*nets as the deductive systems for parsing-as-deduction, with*a*lexicon mapping words to partial*proof*nets. ... Morphisms are defined by induction*on*the*proofs*and*one*has to check that the interpretations of*a**proof*before and after*one*step of*cut**elimination**is*unchanged. ...##
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Social processes, program verification and all that

2009
*
Mathematical Structures in Computer Science
*

We believe that the social nature of

doi:10.1017/s0960129509990041
fatcat:q4upavv6lfhfhdn4kqoouauvom
*proof*and program development*is*uncontroversial and ineluctable but formal verification*is*not antithetical to it. ... The recent, impressive achievements in the field of interactive theorem proving provide an interesting ground for*a*critical*revisiting*of those theses. ... In such*a*case, if the system enjoys the*cut**elimination*property, it*is*immediately consistent. ...##
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Relating Sequent Calculi for Bi-intuitionistic Propositional Logic

2011
*
Electronic Proceedings in Theoretical Computer Science
*

It

doi:10.4204/eptcs.47.7
fatcat:ozwsk32qijfk5cu74vym5kcc7m
*is*sometimes presented as*a*symmetric constructive subsystem of classical logic. ... Bi-intuitionistic logic*is*the conservative extension of intuitionistic logic with*a*connective dual to implication. ... Neither*is**cut**eliminable*in the sequent calculus of Rauszer [13] (the*proof*in the paper*is*incorrect). ...##
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The Consistency and Complexity of Multiplicative Additive System Virtual

2015
*
Scientific Annals of Computer Science
*

The

doi:10.7561/sacs.2015.2.245
fatcat:qwxnnjif6feorlnr5erxlgulwi
*cut**elimination**proof*involves*a*termination measure based*on*multisets of multisets of natural numbers to handle subtle interactions between operators of BV and MAV. ... Section 4 provides the*proof*theoretic devices that establish the consistency of MAV, via*a*generalised*cut**elimination*result. ... Acknowledgements The author*is*grateful to Bogdan Aman for assistance with*proof*reading, to*a*reviewer who helped identify issues regarding the termination of the splitting*proof*in the presence of*a*...##
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On the Computational Meaning of Axioms
[chapter]

2016
*
Logic, Epistemology, and the Unity of Science
*

This leads to

doi:10.1007/978-3-319-26506-3_5
fatcat:44oly6opwrhw5fupblpr7fag54
*a*general kind of*proof*theory, where the objects of study are not typed objects like deductions, but rather untyped*ones*, in which formulas have been replaced by geometrical configurations ... In order to deal with such computational aspects,*a*relaxation of syntax*is*shown to be necessary. ...*Cut**elimination*Now, there exists*a**cut*-*elimination*procedure*on**proof*structures, and this procedure*is*actually compatible with the interpretation of*proof*nets R as*a*couple consisting in an untyped ...##
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Least and Greatest Fixed Points in Linear Logic
[article]

2010
*
arXiv
*
pre-print

That second result provides

arXiv:0910.3383v4
fatcat:3znfpcjtd5e5rfoffvxkltibyq
*a*strong normal form for cut-free*proof*structures that can be used, for example, to help automate*proof*search. ... The resulting logic, which we call muMALL, satisfies two fundamental*proof*theoretic properties: we establish weak normalization for it, and we design*a*focused*proof*system that we prove complete. ... respect to*cut**elimination*. ...##
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On the Proof-Theoretic Foundations of Set Theory
[chapter]

2015
*
Advances in Proof-Theoretic Semantics
*

In this paper we discuss

doi:10.1007/978-3-319-22686-6_10
fatcat:fshso4uigjhlzniksz2ginvvfe
*a**proof*-theoretic foundation of set theory that focusses*on*set definitions in an open type free framework. ... What if we*revisit*the original idea without making strong assumptions*on*closure properties of the theoretical notion of*a*set? ... But consistency of course relates to issues of*cut**elimination*for sequent calculi, which relates to upward absoluteness. ...##
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A sequent calculus for a semi-associative law
[article]

2019
*
arXiv
*
pre-print

We establish

arXiv:1803.10080v2
fatcat:4epfs3yydzawng3ttxggvlwxou
*a*focusing property for this sequent calculus (*a*strengthening of*cut*-*elimination*), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly*one*...*A*new*proof*of the lattice property for the Tamari order, and 2.*A*new*proof*of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice Y_n. ...*Cut*-*elimination*theorems are*a*staple of*proof*theory, and often provide*a*rich source of information about*a*given logic. ...##
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Truth, modality and intersubjectivity

2007
*
Mathematical Structures in Computer Science
*

Quantum physics together with the experimental (and slightly controversial) quantum computing, induces

doi:10.1017/s0960129507006342
fatcat:eqeqgbr5kvgkzio3uhuvqs32ta
*a*twist in our vision of computation, thence -since computing and logic are intimately linked -in ... property (natural deduction,*proof*-nets) induces the compositionality of*proofs*, i.e., the existence of an underlying category. -3 : the*cut*-*elimination*process can be expressed as the solution of*a*linear ... The central result of*proof*-theory,*cut*-*elimination*, reads as follows in the three layers : -1 : the absurd sequent not being cut-free provable,*is*not provable at all, thence consistency. -2 : the Church-Rosser ...##
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Polarities & Focussing: a journey from Realisability to Automated Reasoning
[article]

2014
*
arXiv
*
pre-print

Arising from linear logic, they allow the construction of meaningful semantics for

arXiv:1412.6781v1
fatcat:sj4zl2ayu5bzfjaq76znvllsb4
*cut*-*elimination*in classical logic, some of which relate to the Call-by-Name and Call-by-Value disciplines of functional ... Its architecture, based*on*the concept of focussing, offers*a*platform where smart techniques from automated reasoning (or*a*user interface) can safely and trustworthily be implemented via the use of an ... We then described the implementation of*a**proof*-search engine called Psyche [Psy] whose architecture*is*based*on**a*kernel that interacts with plugins to be programmed via*a*specific API. ...##
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Introduction to linear logic and ludics, part II
[article]

2005
*
arXiv
*
pre-print

It

arXiv:cs/0501039v1
fatcat:dc4ltgacqrg43lm6py4jb3a4ie
*is*devoted to*proof*nets, in the limited, yet central, framework of multiplicative linear logic and to ludics, which has been recently developped in an aim of further unveiling the fundamental interactive ... We hope to offer*a*few computer science insights into this new theory. ... Acknowledgements I wish to thank Song Fangmin and the University of Nanjing for an invitation in October 2001, which offered me the occasion to give*a*few lectures*on*linear logic, and Aldo Ursini and ...##
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Enayat Theories
[article]

2019
*
arXiv
*
pre-print

In this paper we study solution attempts for

arXiv:1909.08877v1
fatcat:yzhp6yclxbaadnqcfh6blbygsa
*a*problem posed by Ali Enayat: can there be*a*finitely axiomatized consistent sequential theory that interprets itself plus the (sentential or non-uniform) ... We provide*a*basic framework for the study of this question and discuss some solution attempts. We connect the question with some interesting conjectures. ... Hence, we have*A*⊢ S N ′ → △ N ′ (m) S N .*Proof*. We have*A*⊢ S N K → KS N . So, by*cut*-*elimination*,*A*⊢ m S N K → KS N . Note that this makes sense only under our assumption*on*m. ...
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