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On the strength of proof-irrelevant type theories

2008
*
Logical Methods in Computer Science
*

We present a type

doi:10.2168/lmcs-4(3:13)2008
fatcat:5hzx7dhn25efnnej4muo54oumy
*theory*with some*proof*-irrelevance built into the conversion rule. ... We show that*in*these*theories*, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. ...*In*other words, a*proof*-system built upon a*theory*as the*one*presented here would allow the user to eciently exploit the*computational*behavior of a constructive*proof**in*order to prove new facts. ...##
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On the Proof Theory of Regular Fixed Points
[chapter]

2009
*
Lecture Notes in Computer Science
*

We provide a coinductive characterization of inclusion that yields a natural bridge to

doi:10.1007/978-3-642-02716-1_8
fatcat:5iou2rnduvh3jnfwwpowzf2jpe
*proof*-*theory*. ... This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic*proofs**in*particular. ... While the finite behavior case is trivially handled*in*the*proof*-*theory*of such logics, finite-state behaviors are not so well understood. ...##
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On the Strength of Proof-Irrelevant Type Theories
[chapter]

2006
*
Lecture Notes in Computer Science
*

We present a type

doi:10.1007/11814771_49
fatcat:wi45amls7fbabpqxloqkwzlurq
*theory*with some*proof*-irrelevance built into the conversion rule. ... Finally we show that*in*these*theories*, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. ...*In*other words, a*proof*-system built upon a*theory*as the*one*presented here would allow the user to eciently exploit the*computational*behavior of a constructive*proof**in*order to prove new facts. ...##
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Why Proof-Theory Matters in Specification-Based Testing

2020
*
Italian Conference on Theoretical Computer Science
*

We survey some recent developments

dblp:conf/ictcs/Momigliano20
fatcat:bkji3bx47bggfeahzzsotsvy4q
*in*giving a logical reconstruction of specification-based testing via the lenses of structural*proof*-*theory*. ... SBT as*Proof*-Search We use as a running examples a call-by-value λ-calculus (where values are lambdas and numerals) whose static and big-step dynamic semantics follows -readers should ... Acknowledgment A shout-out to my co-authors*in*this line of work: Rob Blanco, James Cheney, Francesco Komauli, Dale Miller and Matteo Pessina. ...##
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A 'Theory' Mechanism for a Proof-Verifier Based on First-Order Set Theory
[chapter]

2002
*
Lecture Notes in Computer Science
*

Through several examples, centered

doi:10.1007/3-540-45632-5_9
fatcat:c2a2sn43krf3vg3qyeizzpuy5i
*on*the finite summation operation, we illustrate the potential utility*in*large-scale*proof*-development of the '*theory*' mechanism: utility which stems*in*part from the ... We propose classical set*theory*as the core of an automated*proof*-verifier and outline a version of it, designed to assist*in**proof*development, which is indefinitely expansible with function symbols generated ... Acknowledgements We thank Ernst-Erich Doberkat (Universität Dortmund, D), who brought to our attention the text by Frege cited*in*the epigraph of this paper. ...##
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The Complete Proof Theory of Hybrid Systems

2012
*
2012 27th Annual IEEE Symposium on Logic in Computer Science
*

We now show that,

doi:10.1109/lics.2012.64
dblp:conf/lics/Platzer12a
fatcat:rylkaeazdbc3dizsyfaziliu54
*proof*-theoretically, this is not the case. We present a complete*proof*-theoretical alignment that interreduces the discrete dynamics and the continuous dynamics of hybrid systems. ... This fundamental cornerstone sheds light*on*the nature of hybridness and enables flexible and provably perfect combinations of discrete reasoning with continuous reasoning that lift to all aspects of hybrid ...*On*top of its theoretical value and the full provability alignment that our new result shows, our discrete completeness result is significant*in*that-*in**computer**science*and verification-programs are closer ...##
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Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic
[chapter]

2009
*
Lecture Notes in Computer Science
*

*On*the technical level this paper provides two new aspects of the

*proof*

*theory*of MLL2 with units. First, we give a novel

*proof*system

*in*the framework of the calculus of structures. ... The main feature of the new system is the consequent use of deep inference, which allows us to observe a decomposition which is a version of Herbrand's theorem that is not visible

*in*the sequent calculus ... Introduction The question of when two

*proofs*are the same is important for

*proof*

*theory*and its applications. ...

##
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Theorem Proving Based on Proof Scores for Rewrite Theory Specifications of OTSs
[chapter]

2014
*
Lecture Notes in Computer Science
*

. … • Equational

doi:10.1007/978-3-642-54624-2_31
fatcat:tssyw3urfbggrhutd2xfu44dgy
*theory*specification: Inductive*proof*with search (1) 8 … t t t c C 1 C 2 C n Let SPEC-T be a specification (module)*in*which there exists exactly*one*trans rule whose label is t and c ... 2011, SAS 2014] • Suggesting theorem proving that rewrite*theory*specifications enjoy invariant properties by writing*proof*scores*in*CafeOBJ • An authentication protocol is used to describe it ... Summary • NSLPK has been used to demonstrate that (some specific style/class of) rewrite*theory*specifications can be used for theorem proving based*on**proof*scores. ...##
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On the Convergence of Reduction-based and Model-based Methods in Proof Theory

2008
*
Electronical Notes in Theoretical Computer Science
*

*In*the recent past, the reduction-based and the model-based methods to prove cut elimination have converged, so that they now appear just as two sides of the same coin. ... Then, a simple induction

*on*

*proof*structure shows that,

*in*constructive natural deduction, the last rule of a cut free

*proof*

*in*the empty

*theory*is an introduction. ... to extract programs from such

*proofs*, rely

*on*cut elimination theorems, asserting that when a proposition is provable

*in*some

*theory*, it has a

*proof*of a special form: a cut free

*proof*. ...

##
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On linear rewriting systems for Boolean logic and some applications to proof theory

2017
*
Logical Methods in Computer Science
*

Linear rules have played an increasing role

doi:10.2168/lmcs-12(4:9)2016
fatcat:q52wxqmhvzcidg2jm76mq5qc24
*in*structural*proof**theory**in*recent years. ... We draw tools and concepts from term rewriting, Boolean function*theory*and graph*theory**in*order to access some required intermediate results. ... We would like to thank Paola Bruscoli, Kaustuv Chaudhuri, Alessio Guglielmi, Willem Heijltjes and others*in*the deep inference community for many fruitful discussions*on*these topics. ...##
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Proofs by induction in equational theories with constructors

1980
*
21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
*

The documents may come from teaching and research institutions

doi:10.1109/sfcs.1980.37
dblp:conf/focs/HuetH80
fatcat:pgky2s3itjdctcldbokbyg3tye
*in*France or abroad, or from public or private research centers. ... HAL Id: inria-00076533 https://hal.inria.fr/inria-00076533 Submitted*on*24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, ...##
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On the Continuity of Gelfond-Lifschitz Operator and Other Applications of Proof-Theory in ASP
[chapter]

2008
*
Lecture Notes in Computer Science
*

The introduction of the formula rΦP leads to a double-backtracking algorithm for

doi:10.1007/978-3-540-89982-2_25
fatcat:wjpyvtauo5hlflwqwy5cdm45bq
*computation*of stable models by reduction to satisfiability of suitably chosen propositional*theories*. ... This algorithm does not use the reduction via loop-formulas as proposed*in*[LZ02] or its extension proposed*in*[FLL06]. ... The P -*proof*schemes described*in*this paper represent*one*possible candidate of such a*proof*system for ASP. ...##
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A special issue on structural proof theory, automated reasoning and computation in celebration of Dale Miller's 60th birthday

2019
*
Mathematical Structures in Computer Science
*

*proof*

*theory*and its application to

*computation*and reasoning. ... The genesis of this special issue was

*in*a meeting that took place at Université Paris Diderot

*on*December 15 and 16, 2016. ...

*proof*

*theory*and its application to

*computation*and reasoning. ...

##
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Proof Theory for Kleene Algebra

*
20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05)
*

We develop an (infinitary)

doi:10.1109/lics.2005.37
dblp:conf/lics/Hardin05
fatcat:3nbvimxbgzbybbefjpcb3lj5ga
*proof*system, based*on*well-founded trees of finite automata, which is sound and complete for this*theory*. ... A small modification of this system yields a*proof*system which is sound and complete for the universal Horn*theory*of * -continuous Kleene algebras with tests (KAT * ). ... Introduction Kleene algebra (KA) arises*in*many areas of*computer**science*, such as automata*theory*, the design and analysis of algorithms, dynamic logic, and program semantics. ...##
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Intensionality, extensionality, and proof irrelevance in modal type theory

*
Proceedings 16th Annual IEEE Symposium on Logic in Computer Science
*

We develop a uniform type

doi:10.1109/lics.2001.932499
dblp:conf/lics/Pfenning01
fatcat:7ki2zmyxwbfa7fvnuhe4qetgge
*theory*that integrates intensionality, extensionality, and*proof*irrelevance as judgmental concepts. ... Our type*theory*contrasts with previous approaches that a priori distinguish propositions (whose*proofs*are all identified-only their existence is important) from specifications (whose implementations ... If our operational interpretation of type*theory*is based*on*staged*computation*[DP00] , then the modality is necessary to reason about staged programs. ...
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