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Typed realizability for first-order classical analysis

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 ...

##
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A computational interpretation of open induction

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

doi:10.1109/lics.2004.1319627
dblp:conf/lics/Berger04
fatcat:aihxty33a5evzetc57jlzwbiz4
*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. ...##
###
APPLICATIONS OF INDUCTIVE DEFINITIONS AND CHOICE PRINCIPLES TO PROGRAM SYNTHESIS
[chapter]

2005
*
From Sets and Types to Topology and Analysis
*

We describe two methods of extracting constructive content

doi:10.1093/acprof:oso/9780198566519.003.0008
fatcat:n4z453pujrenfgte7abt4tkjhi
*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*. ...##
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A Constructive Proof of Dependent Choice, Compatible with Classical Logic

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. ...

##
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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

2016
*
Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16
*

Having both versions allows

doi:10.1145/2933575.2934511
dblp:conf/lics/Blot16
fatcat:dc4eo7h4vjgc3jt275ztdbzgvy
*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. ...##
###
Implementing program extraction from CL1-proofs
[article]

2013
*
arXiv
*
pre-print

In this paper, we present a

arXiv:1311.6542v1
fatcat:7b3dlped3zaozgwj54lfpbc6wq
*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. ...##
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Proof-search in type-theoretic languages: an introduction

2000
*
Theoretical Computer Science
*

We do not claim to cover all of the theoretical and implementation issues in the study of

doi:10.1016/s0304-3975(99)00169-3
fatcat:gy4nmmsupfeype23pw3jzbrvwu
*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. ...##
###
Complements in Distributive Allegories
[chapter]

2009
*
Lecture Notes in Computer Science
*

Furthermore, we will discuss possible modifications of the assumptions

doi:10.1007/978-3-642-04639-1_23
fatcat:axwpxhsmdbhunlacdxmarnei6a
*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 ...##
###
On the computational content of the axiom of choice

1998
*
Journal of Symbolic Logic (JSL)
*

We also show how to compute witnesses

doi:10.2307/2586854
fatcat:zpaddcad75hmjo5vxzxycusnaq
*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 ...##
###
Extracting efficient exact real number computation from proofs in constructive type theory
[article]

2022
*
arXiv
*
pre-print

We propose a new axiomatization of the real numbers in a

arXiv:2202.00891v1
fatcat:gq7zenc7cjf6zd2auk2ssdp4ny
*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. ...##
###
Fuzzy Kolmogorov Complexity Based on a Classical Description

2020
*
Entropy
*

In the

doi:10.3390/e22010066
pmid:33285841
fatcat:5bsdkboymzc5tgdechpxb3lzme
*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. ...##
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Mechanically Verified Calculational Abstract Interpretation
[article]

2015
*
arXiv
*
pre-print

Calculational abstract interpretation, long advocated by Cousot, is a technique for deriving correct-by-construction abstract interpreters

arXiv:1507.03559v1
fatcat:a2ho4ehqyjaqrprr5nujpaj7ri
*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*. ...##
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What sequential games, the tychonoff theorem and the double-negation shift have in common

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*. ...

##
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Constructive Game Logic
[chapter]

2020
*
Lecture Notes in Computer Science
*

the extraction of computational artifacts

doi:10.1007/978-3-030-44914-8_4
fatcat:by5xt6i2g5gi7ds2kebai5otgu
*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*. ...
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