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Typed realizability for first-order classical analysis

Valentin Blot, Peter Selinger
2015 Logical Methods in Computer Science  
choice.  ...  This allows a direct interpretation of classical proofs, avoiding the usual negative translation to intuitionistic logic.  ...  Concerning extraction of programs from proofs, Friedman's trick is implemented through the use of an external µ-variable rather than through the replacement of the ⊥ formula by an existential statement  ... 
doi:10.2168/lmcs-11(4:22)2015 fatcat:mpy2aq4dvbdyjm2jzmjpjt2vdm

A computational interpretation of open induction

U. Berger
2004 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.  
Via modified realizability we obtain a new direct method for extracting programs from classical proofs of ¡ £ ¢ ¤ -formulas using open induction.  ...  It is a classical reformulation of Nash-Williams minimal-bad-sequence argument which is used in classical proofs of Kruskal's theorem and related theorems [18, 6].  ...  proofs of -formulas using dependent choice and classical logic to the problem of realizing update induction.  ... 
doi:10.1109/lics.2004.1319627 dblp:conf/lics/Berger04 fatcat:aihxty33a5evzetc57jlzwbiz4


Ulrich Berger, Monika Seisenberger
2005 From Sets and Types to Topology and Analysis  
We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles.  ...  We also discuss some proof-theoretic optimizations that were crucial for the formalization and implementation of this work in the interactive proof system Minlog. 1 Higman's lemma is used, for example,  ...  Classical dependent choice Now we show how to extract computational content directly from Nash-Williams' proof.  ... 
doi:10.1093/acprof:oso/9780198566519.003.0008 fatcat:n4z453pujrenfgte7abt4tkjhi

A Constructive Proof of Dependent Choice, Compatible with Classical Logic

Hugo Herbelin
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science  
choice, the axiom of dependent choice, and a form of bar induction in ways that make each of them computationally compatible with classical logic.  ...  Keywords-Dependent choice; classical logic; constructive logic; strong existential 3 Failure of subject reduction when combining strong existential quantification and computational classical logic was  ...  Acknowledgements Concepts coming from the programming languages side were instrumental for this work and I'm grateful to the community that designed them.  ... 
doi:10.1109/lics.2012.47 dblp:conf/lics/Herbelin12 fatcat:nesmi636cregfia2nfimv4zeze

Page 9528 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
Since the focus of this paper is to help get programs from proofs in classical analysis, and since the axioms of classical analysis are second-order arithmetic and the axiom of countable choice, the last  ...  Guillaume, “The 2,°-calculus”, in Proceedings of the 2nd Workshop on Explicit Sub- stitutions: Theory and Applications to Programs and Proofs, 2-13, 1999]) using a translation from typed 4,,; to proof  ... 

Hybrid realizability for intuitionistic and classical choice

Valentin Blot
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
Having both versions allows us to obtain efficient extracted programs while keeping the provability strength of classical logic.  ...  In this model two versions of the axiom of choice can co-exist in a single proof: intuitionistic choice and classical countable choice.  ...  Any proof in classical arithmetic with the axiom of countable choice can be interpreted as a program and a concrete value can be extracted from a proof of a Π 0 2 formula in this theory.  ... 
doi:10.1145/2933575.2934511 dblp:conf/lics/Blot16 fatcat:dc4eo7h4vjgc3jt275ztdbzgvy

Implementing program extraction from CL1-proofs [article]

Meixia Qu, Ke Chen, Daming Zhu, Junfeng Luan
2013 arXiv   pre-print
In this paper, we present a program that takes a CL1-proof of an arbitrary formula F, and extract a winning strategy for F from that proof then play F using that strategy.  ...  The logical vocabulary contains all of the operators of classical logic and choice operators, the atoms represent elementary games i.e. predicates of classical logic.  ...  In this paper, we present a program that takes a CL1proof of an arbitrary formula F , extract a winning strategy for F from that proof, and then play F using that strategy.  ... 
arXiv:1311.6542v1 fatcat:7b3dlped3zaozgwj54lfpbc6wq

Proof-search in type-theoretic languages: an introduction

Didier Galmiche, David J. Pym
2000 Theoretical Computer Science  
We do not claim to cover all of the theoretical and implementation issues in the study of proof-search in type-theoretic languages; rather, we present some key ideas and problems, starting from well-motivated  ...  The strong connections between di erent proof-search methods in logics, type theories and logical frameworks, together with their impact on programming and implementation issues, are central in this context  ...  Acknowledgements We are grateful to Maurice Nivat for encouraging us to produce this article and for acting as a referee. Any remaining errors are our responsibility.  ... 
doi:10.1016/s0304-3975(99)00169-3 fatcat:gy4nmmsupfeype23pw3jzbrvwu

Complements in Distributive Allegories [chapter]

Michael Winter
2009 Lecture Notes in Computer Science  
Furthermore, we will discuss possible modifications of the assumptions used in the that theorem. It is know in topos theory that axiom of choice implies that the topos is Boolean.  ...  Furthermore, we will discuss possible modifications of the assumptions used in the that theorem. Abstract. It is know in topos theory that axiom of choice implies that the topos is Boolean.  ...  Introduction The calculus of relations, and its categorical versions in particular, are often used to model programming languages, classical and non-classical logics and different methods of data mining  ... 
doi:10.1007/978-3-642-04639-1_23 fatcat:axwpxhsmdbhunlacdxmarnei6a

On the computational content of the axiom of choice

Stefano Berardi, Marc Bezem, Thierry Coquand
1998 Journal of Symbolic Logic (JSL)  
We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements.  ...  We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice.  ...  We thank Anne Troelstra for helpful comments and answers to various questions that arose during the preparation of this paper, Loi'c Colson for interesting discussions on the axiom of choice and impredicativity  ... 
doi:10.2307/2586854 fatcat:zpaddcad75hmjo5vxzxycusnaq

Extracting efficient exact real number computation from proofs in constructive type theory [article]

Michal Konečný, Sewon Park, Holger Thies
2022 arXiv   pre-print
We propose a new axiomatization of the real numbers in a dependent type theory with the goal of extracting certified exact real computation programs from constructive proofs.  ...  From the examples we have automatically extracted Haskell programs that use the exact real computation framework AERN for efficiently performing exact operations on real numbers.  ...  The implementation allows to extract Haskell programs from our proofs using Coq's code extraction mechanism.  ... 
arXiv:2202.00891v1 fatcat:gq7zenc7cjf6zd2auk2ssdp4ny

Fuzzy Kolmogorov Complexity Based on a Classical Description

Songsong Dai
2020 Entropy  
In the classical setting, the Kolmogorov complexity of a single finite string is the length of the shortest program that produces this string.  ...  The classical Kolmogorov complexity is extended to the fuzzy domain retaining classical descriptions.  ...  Proof. To prove this inequality. Suppose p is a shortest program to produce L 1 , and q is a shortest program to produce L 2 from L 1 . Similar to the above theorem, we have to identify p and q.  ... 
doi:10.3390/e22010066 pmid:33285841 fatcat:5bsdkboymzc5tgdechpxb3lzme

Mechanically Verified Calculational Abstract Interpretation [article]

David Darais, David Van Horn
2015 arXiv   pre-print
Calculational abstract interpretation, long advocated by Cousot, is a technique for deriving correct-by-construction abstract interpreters from the formal semantics of programming languages.  ...  This paper addresses the problem of deriving correct-by-verified-construction abstract interpreters with the use of a proof assistant.  ...  The mechanized verification of this formal development using a proof assistant can be foreseen with automatic extraction of a correct program from its correctness proof.  ... 
arXiv:1507.03559v1 fatcat:a2ho4ehqyjaqrprr5nujpaj7ri

What sequential games, the tychonoff theorem and the double-negation shift have in common

Martín Escardó, Paulo Oliva
2010 Proceedings of the third ACM SIGPLAN workshop on Mathematically structured functional programming - MSFP '10  
from topology, and (3) realizes the Double-Negation Shift from logic and proof theory.  ...  We discuss a higher-type functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a computational version of the Tychonoff Theorem  ...  To extract programs from classical proofs in PA ω using the axiom of choice, one needs to realize the K-translation of the axiom of choice, which is often referred to as the classical axiom of choice.  ... 
doi:10.1145/1863597.1863605 dblp:conf/icfp/EscardoO10 fatcat:y4rnbuktjzds3avoyxawcvd5s4

Constructive Game Logic [chapter]

Brandon Bohrer, André Platzer
2020 Lecture Notes in Computer Science  
the extraction of computational artifacts from game proofs.  ...  Game Logic is an excellent setting to study proofs-aboutprograms via the interpretation of those proofs as programs, because constructive proofs for games correspond to effective winning strategies to  ...  Rule M is used to execute programs right-to-left, giving shorter, more efficient proofs.  ... 
doi:10.1007/978-3-030-44914-8_4 fatcat:by5xt6i2g5gi7ds2kebai5otgu
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