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

Christian Urban, Bozhi Zhu
Lecture Notes in Computer Science  
We thus show that the original 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  ... 
doi:10.1007/978-3-540-70590-1_28 fatcat:bj36tpdcgnbshkmhdpodljd2we

Cut Elimination for Monomial MALL Proof Nets

Olivier Laurent, Roberto Maieli
2008 Logic in Computer Science  
This sharing leads to the definition of 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.  ... 
doi:10.1109/lics.2008.31 dblp:conf/lics/LaurentM08 fatcat:kdd56hmag5etfcbh7dyfczk7mu

On Formally Measuring and Eliminating Extraneous Notions in Proofs

A. Arana
2008 Philosophia Mathematica  
In this paper I discuss extraneousness generally, and then consider 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".  ... 
doi:10.1093/philmat/nkn015 fatcat:w4kto4hfmjbplnlkkt5aaf6sve

Pomset logic: a logical and grammatical alternative to the Lambek calculus [article]

Christian Retoré
2020 arXiv   pre-print
This classical calculus enjoys 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.  ... 
arXiv:2001.02155v1 fatcat:p4f7qlru7ba55brlnygeu6ddsq

Social processes, program verification and all that

ANDREA ASPERTI, HERMAN GEUVERS, RAJA NATARAJAN
2009 Mathematical Structures in Computer Science  
We believe that the social nature of 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.  ... 
doi:10.1017/s0960129509990041 fatcat:q4upavv6lfhfhdn4kqoouauvom

Relating Sequent Calculi for Bi-intuitionistic Propositional Logic

Luís Pinto, Tarmo Uustalu
2011 Electronic Proceedings in Theoretical Computer Science  
It 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).  ... 
doi:10.4204/eptcs.47.7 fatcat:ozwsk32qijfk5cu74vym5kcc7m

The Consistency and Complexity of Multiplicative Additive System Virtual

Ross Horne
2015 Scientific Annals of Computer Science  
The 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  ... 
doi:10.7561/sacs.2015.2.245 fatcat:qwxnnjif6feorlnr5erxlgulwi

On the Computational Meaning of Axioms [chapter]

Alberto Naibo, Mattia Petrolo, Thomas Seiller
2016 Logic, Epistemology, and the Unity of Science  
This leads to 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  ... 
doi:10.1007/978-3-319-26506-3_5 fatcat:44oly6opwrhw5fupblpr7fag54

Least and Greatest Fixed Points in Linear Logic [article]

David Baelde
2010 arXiv   pre-print
That second result provides 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.  ... 
arXiv:0910.3383v4 fatcat:3znfpcjtd5e5rfoffvxkltibyq

On the Proof-Theoretic Foundations of Set Theory [chapter]

Lars Hallnäs
2015 Advances in Proof-Theoretic Semantics  
In this paper we discuss 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.  ... 
doi:10.1007/978-3-319-22686-6_10 fatcat:fshso4uigjhlzniksz2ginvvfe

A sequent calculus for a semi-associative law [article]

Noam Zeilberger
2019 arXiv   pre-print
We establish 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.  ... 
arXiv:1803.10080v2 fatcat:4epfs3yydzawng3ttxggvlwxou

Truth, modality and intersubjectivity

JEAN-YVES GIRARD
2007 Mathematical Structures in Computer Science  
Quantum physics together with the experimental (and slightly controversial) quantum computing, induces 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  ... 
doi:10.1017/s0960129507006342 fatcat:eqeqgbr5kvgkzio3uhuvqs32ta

Polarities & Focussing: a journey from Realisability to Automated Reasoning [article]

Stéphane Graham-Lengrand
2014 arXiv   pre-print
Arising from linear logic, they allow the construction of meaningful semantics for 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.  ... 
arXiv:1412.6781v1 fatcat:sj4zl2ayu5bzfjaq76znvllsb4

Introduction to linear logic and ludics, part II [article]

Pierre-Louis Curien
2005 arXiv   pre-print
It 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  ... 
arXiv:cs/0501039v1 fatcat:dc4ltgacqrg43lm6py4jb3a4ie

Enayat Theories [article]

Albert Visser
2019 arXiv   pre-print
In this paper we study solution attempts for 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.  ... 
arXiv:1909.08877v1 fatcat:yzhp6yclxbaadnqcfh6blbygsa
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