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Program Extraction from Normalization Proofs

Ulrich Berger, Stefan Berghofer, Pierre Letouzey, Helmut Schwichtenberg
2006 Studia Logica: An International Journal for Symbolic Logic  
From the formal proofs programs are machine-extracted that implement variants of the well-known normalization-by-evaluation algorithm.  ...  The case study is used to test and compare the program extraction machineries of the three proof assistants in a non-trivial setting.  ...  As to be expected, the change from strong to weak normalization does not affect the extracted programs.  ... 
doi:10.1007/s11225-006-6604-5 fatcat:6e35esi3gvdrzevonerxrmiw7u

Program Extraction from Proofs of Weak Head Normalization

Malgorzata Biernacka, Olivier Danvy, Kristian Støvring
2005 BRICS Report Series  
Subsequently we use Kreisel's modified realizability to extract evaluation algorithms from the proofs, following Berger; the proofs are based on Tait-style reducibility predicates, and hence the extracted  ...  We formalize two proofs of weak head normalization for the simply typed lambda-calculus in first-order minimal logic: one for normal-order reduction, and one for applicative-order reduction in the object  ...  They also outlined a process of "program extraction" with which their normalization algorithms can be obtained from simple instances of a normalization proof due to Martin-Löf, and noticed the connection  ... 
doi:10.7146/brics.v12i12.21878 fatcat:atitabzthzgwxmzqmyura5feu4

Program Extraction From Proofs of Weak Head Normalization

Małgorzata Biernacka, Olivier Danvy, Kristian Støvring
2006 Electronical Notes in Theoretical Computer Science  
Subsequently we use Kreisel's modified realizability to extract evaluation algorithms from the proofs, following Berger; the proofs are based on Tait-style reducibility predicates, and hence the extracted  ...  We formalize two proofs of weak head normalization for the simply typed lambda-calculus in firstorder minimal logic: one for normal-order reduction, and one for applicative-order reduction in the object  ...  They also outlined a process of "program extraction" with which their normalization algorithms can be obtained from simple instances of a normalization proof due to Martin-Löf, and noticed the connection  ... 
doi:10.1016/j.entcs.2005.11.056 fatcat:eum43xmwb5erhmh7pm5ilobcwe

Program extraction from normalization proofs [chapter]

Ulrich Berger
Lecture Notes in Computer Science  
From the formal proofs programs are machine-extracted that implement variants of the well-known normalization-by-evaluation algorithm.  ...  The case study is used to test and compare the program extraction machineries of the three proof assistants in a non-trivial setting.  ...  As to be expected, the change from strong to weak normalization does not affect the extracted programs.  ... 
doi:10.1007/bfb0037100 dblp:conf/tlca/Berger93 fatcat:4pfm6c3u7rbm5huslujnqiyrua

Synthesis of ML programs in the system Coq

Christine Paulin-Mohring, Benjamin Werner
1993 Journal of symbolic computation  
From a constructive proof formalized in Coq, one extracts a functional program which can be compiled and executed in ML. This paper describes how to obtain ML programs from proofs in Coq.  ...  We study the specification of the problem, the development of the proof and the extraction of the executable ML program.  ...  ACKNOWLEDGMENTS We thank Xavier Leroy and Chet Murthy for their help in analysing the behavior of the optimized normalization function.  ... 
doi:10.1016/s0747-7171(06)80007-6 fatcat:rtmd5gsei5g3jnya4x2tp76mda

Extracting a Normalization Algorithm in Isabelle/HOL [chapter]

Stefan Berghofer
2006 Lecture Notes in Computer Science  
We present a formalization of a constructive proof of weak normalization for the simply-typed λ-calculus in the theorem prover Isabelle/HOL, and show how a program can be extracted from it.  ...  Thus, the program obtained from this proof is typable in simply-typed higher-order logic as implemented in Isabelle/HOL, and a proof of its correctness can automatically be derived within the system.  ...  We show how a provably correct program can automatically be extracted from this proof using Isabelle's framework for program extraction [9, 10] .  ... 
doi:10.1007/11617990_4 fatcat:pe2szgcl2vbqzemfy2ukhtjlne

The Specialization and Transformation of Constructive Existence Proofs

Peter Madden
1989 International Joint Conference on Artificial Intelligence  
The transformation of constructive program synthesis proofs is discussed and compared with the more traditional approaches to program transformation.  ...  An example system for adapting programs to special situations by transforming constructive synthesis proofs has been reconstructed and is compared with the original implementation [Goad, 1980b, Goad, 1980a  ...  then, once completed, extract an algorithm from the transformed proof.  ... 
dblp:conf/ijcai/Madden89 fatcat:qwqnsbe4ujhfxn5l2ticrtsl6e

Towards the theory of programming in constructive logic [chapter]

A. A. Voronkov
1990 Lecture Notes in Computer Science  
We develop an approach to the theory of extracting programs from proofs based on constructive semantics of the first order formulas called consTrucTive TruLb. The underlying ideas are discussed.  ...  There are essentially two groups of methods of extracting programs from proofs.  ...  Thus the algorithm for extracting programs from proofs should be based on this notion.  ... 
doi:10.1007/3-540-52592-0_78 fatcat:wy7miwvy5jgmxdpzkw5mmhlcly

APPLICATIONS OF INDUCTIVE DEFINITIONS AND CHOICE PRINCIPLES TO PROGRAM SYNTHESIS [chapter]

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.  ...  in term rewriting theory for termination proofs [CTB94,Tou02].  ...  a program from the latter proof.  ... 
doi:10.1093/acprof:oso/9780198566519.003.0008 fatcat:n4z453pujrenfgte7abt4tkjhi

Protocols between Programs and Proofs [chapter]

Iman Poernomo, John N. Crossley
2001 Lecture Notes in Computer Science  
The methodology we use for automatically extracting "correct" programs from proofs is a development of the well-known Curry-Howard process.  ...  In this paper we describe a new protocol that we call the Curry-Howard protocol between a theory and the programs extracted from it.  ...  Theorem 1 above shows that we can extract a new program that satisfies the Skolemized version of the formula from whose proof we extracted the program.  ... 
doi:10.1007/3-540-45142-0_2 fatcat:y62xq3w6infltbvlws55qlc4ea

Synthesis of Moduli of Uniform Continuity by the Monotone Dialectica Interpretation in the Proof-system MinLog

Mircea-Dan Hernest
2007 Electronical Notes in Theoretical Computer Science  
The automated synthesis of such moduli proceeds from a proof of the hereditarily extensional equality (≈) of t to itself, hence a proof in a weakly extensional variant of Berger-Buchholz-Schwichtenberg's  ...  We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel's T of type (N → N) → (N → N).  ...  Schwichtenberg for having suggested to us that the already available formulations of the Monotone (or Bounded) Dialectica may not be satisfying enough from the computer-applied viewpoint.  ... 
doi:10.1016/j.entcs.2007.01.023 fatcat:mbhog62lvzdplpemv3mwh4lwgu

Eliminating Proofs from Programs

Femke van Raamsdonk, Paula Severi
2002 Electronical Notes in Theoretical Computer Science  
This paper presents a step in the development of an operational approach to program extraction in type theory. In order to get a program from a lambda term, the logical parts need to be removed.  ...  In the general setting the properties confluence, subject reduction, and strong normalization are studied.  ...  Conclusions Evaluating a program -i.e. applying β-reduction -to a program that contains logical parts is rather inefficient.  ... 
doi:10.1016/s1571-0661(04)80505-x fatcat:n7i3v2v2prh5dmtvgsled2e2l4

Light Dialectica Program Extraction from a Classical Fibonacci Proof

Mircea-Dan Hernest
2007 Electronical Notes in Theoretical Computer Science  
We demonstrate program extraction by the Light Dialectica Interpretation (LDI) on a minimal logic proof of the classical existence of Fibonacci numbers.  ...  Although syntactically different, it also has the same computational complexity as the original program yielded by the refined A-translation from the undistorted input classical Fibonacci proof.  ...  Introduction There has been quite some work in the last years in the field of program extraction from classical proofs.  ... 
doi:10.1016/j.entcs.2006.10.050 fatcat:ozqg6y3sbne4xc3xpycrd32iea

A Context-based Approach to Proving Termination of Evaluation

Małgorzata Biernacka, Dariusz Biernacki
2009 Electronical Notes in Theoretical Computer Science  
For each of the proofs we present its computational content that takes the form of an evaluator in continuation-passing style and is an instance of normalization by evaluation.  ...  Acknowledgement We are grateful to Olivier Danvy for his willingness to proof-read the article at a short notice, and for his helpful, comprehensive comments.  ...  Some of the proofs have been formalized in proof assistants and normalizers have been extracted from them in the form of functional programs [4, 6] .  ... 
doi:10.1016/j.entcs.2009.07.090 fatcat:firz4v2zebfmxpvqkiyatry5fe

Uniform Heyting arithmetic

Ulrich Berger
2005 Annals of Pure and Applied Logic  
programs from a rather large class of classical first-order proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program.  ...  We present an extension of Heyting arithmetic in finite types called Uniform Heyting Arithmetic (HA u ) that allows for the extraction of optimized programs from constructive and classical proofs.  ...  From [4] we have taken the uniform quantifier {∀x} introduced there to extract the well-known normalization-by-evaluation program from the Tait/Troelstra proof of strong normalization of the simply typed  ... 
doi:10.1016/j.apal.2004.10.006 fatcat:36ejf4m4cjd47cq42436lk6efy
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