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The greatest common divisor: A case study for program extraction from classical proofs [chapter]

U. Berger, H. Schwichtenberg
1996 Lecture Notes in Computer Science  
Here we treat that example as a case study for program extraction from classical proofs. We apply H.  ...  The number c ∈ (a 1 , a 2 ) dividing a 1 and a 2 is the greatest common divisor since any common divisor of a 1 and a 2 must also be a divisor of c.  ...  We would like to thank Monika Seisenberger and Felix Joachimski for implementing the refined A-translation and doing the gcd example in the MINLOG system.  ... 
doi:10.1007/3-540-61780-9_60 fatcat:ohln4bijzrbzhndjx6gfg5vbaq

Greatest common divisors of polynomials given by straight-line programs

Erich Kaltofen
1988 Journal of the ACM  
Both the algorithm for that construction and the greatest common divisor algorithm are in random polynomial time for the usual coefftcient fields and output a straight-line program, which with controllably  ...  The second result shows how to find a straight-line program for the reduced numerator and denominator from one for the corresponding rational function.  ...  Introduction This study is concerned with complexity questions about performing operations, such as greatest common divisor (GCD) computation and factorization, on multivariate polynomials.  ... 
doi:10.1145/42267.45069 fatcat:hjux5gwgybe5nikgp3nozu7zzq

Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

Kaisa Matomäki, Maksym Radziwiłł, Terence Tao
2018 Proceedings of the London Mathematical Society  
Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated  ...  The Type d_4 sum is treated similarly using the classical L^2 mean value theorem and the classical van der Corput exponential sum estimates.  ...  We use (a, b) and [a, b] for the greatest common divisor and least common multiple of natural numbers a, b respectively, and write a|b if a divides b.  ... 
doi:10.1112/plms.12181 fatcat:z3m527zkojexrlywsfov5ccxza

Page 7528 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews  
The greatest common divisor: a case study for program extraction from classical proofs. Types for proofs and programs (Torino, 1995), 36-46, Lecture Notes in Comput.  ...  Here we treat that example as a case study for program extraction from classical proofs. We apply H. Friedman’s A-translation [in Higher set theory (Proc. Conf., Math.  ... 

Page 2070 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews  
Schwichtenberg, The greatest common divisor: a case study for program extraction from classical proofs (36-46); Ilya Beylin and Peter Dybjer, Extracting a proof of coherence for monoidal cat- egories from  ...  a proof of normalization for monoids (47-61); Jan Cederquist and Sara Negri, A constructive proof of the Heine- Borel covering theorem for formal reals (62-75); Thierry Coquand and Jan M.  ... 

Euclid's Number-Theoretical Work [article]

Shaohua Zhang
2010 arXiv   pre-print
Therefore, Euclid has preliminarily established Theory of Divisibility and the greatest common divisor. This is the foundation of Number Theory.  ...  We were quite surprised when we began to read the Elements for the first time.  ...  Acknowledgements I am happy to record my gratitude to my supervisor Professor Xiaoyun Wang who made a lot of helpful suggestions. I wish to thank the referees for a careful reading of this paper.  ... 
arXiv:0902.2465v2 fatcat:k2xehcvr4vdclpwvc332nhheqa

Foundations of Garside Theory [article]

Patrick Dehornoy , Daan Krammer, Jean Michel
2013 arXiv   pre-print
This text consists of the introduction, table of contents, and bibliography of a long manuscript (703 pages) that is currently submitted for publication.  ...  This manuscript develops an extension of Garside's approach to braid groups and provides a unified treatment for the various algebraic structures that appear in this context.  ...  the divisors of the fundamental element ∆ n : the point is that, if g is an element of B + n , then there exists a (unique) greatest common divisor g 1 for g and ∆ n and, moreover g = 1 implies g 1 = 1  ... 
arXiv:1309.0796v1 fatcat:iw46d4gqjnd4hpft55h7qw2tbq

The Logic of Computer Programming

Z. Manna, R. Waldinger
1978 IEEE Transactions on Software Engineering  
In our writing we have drawn on the wealth of material about the greatest common divisor and  ...  We are also grateful to the following members of the Theory of Computing Panel of the NSF Computer Science and Engineering Research Study (COSERS) --Richard Karp, A lbert Meyer, John Reynolds, Robert Ritchie  ...  As a new specimen for our study of recursion we will Introduce a recursive cousin of the greatest common divisor algorithm of Euclid, which appeared in his Elements over 2200 years ago.  ... 
doi:10.1109/tse.1978.231499 fatcat:ylzqmhqefjfyrnacqqozolyffi

A Formally Certified End-to-End Implementation of Shor's Factorization Algorithm [article]

Yuxiang Peng, Kesha Hietala, Runzhou Tao, Liyi Li, Robert Rand, Michael Hicks, Xiaodi Wu
2022 arXiv   pre-print
Techniques familiar to most programmers from the classical domain for avoiding, discovering, and diagnosing bugs do not easily transfer, at scale, to the quantum domain because of its unique characteristics  ...  The proof's validity is automatically confirmed -- certified -- by a "proof assistant".  ...  Acknowledgement We thank Andrew Childs, Steven Girvin, Liang Jiang, and Peter Shor for helpful feedback on the manuscript.  ... 
arXiv:2204.07112v1 fatcat:7guc72okn5gkboopsiop4zn2l4

Computers and Thought Lecture: The Ubiquity of Discovery

Douglas B. Lenat
1977 International Joint Conference on Artificial Intelligence  
One of them («2) says to look for motivation at the extreme cases of various known relationships. The relation in this case is "Divisors of".  ...  , new pieces of mass spectroscopy theory.These rules are then usable by the Dendral program, as if they had been extracted from a human expert.> The PECOS program [Barstow 1977] contains rules about computer  ... 
dblp:conf/ijcai/Lenat77 fatcat:dtqksvygdngqddw3ghtzyd6wkq

A Framework for the Verification of Certifying Computations

Eyad Alkassar, Sascha Böhme, Kurt Mehlhorn, Christine Rizkallah
2013 Journal of automated reasoning  
A checker for such a witness is usually much simpler than the original algorithmyet it is all the user has to trust.  ...  ., formal proofs that outputs for particular inputs are correct. We do so by combining the concept of certifying algorithms with methods for code verification and theorem proving.  ...  Moreover, we thank Lars Noschinski for developing a powerful graph library in Isabelle/HOL.  ... 
doi:10.1007/s10817-013-9289-2 fatcat:m5yhhyinprewdd6cc44zrks55y

A Framework for the Verification of Certifying Computations [article]

Eyad Alkassar and Sascha Böhme and Kurt Mehlhorn and Christine Rizkallah
2013 arXiv   pre-print
A checker for such a witness is usually much simpler than the original algorithm - yet it is all the user has to trust.  ...  We use the automatic verifier VCC for establishing the correctness of the checker and the interactive theorem prover Isabelle/HOL for high-level mathematical properties of algorithms.  ...  Moreover, we thank Lars Noschinski for developing a powerful graph library in Isabelle/HOL.  ... 
arXiv:1301.7462v1 fatcat:25jkpadgqrblbnr4jqrsxwjipq

Homology of gaussian groups

Patrick Dehornoy, Yves Lafont
2003 Annales de l'Institut Fourier  
x of M admits a unique decomposition x = xl ~ ~ ~ xp such that, for each i, xi is the greatest left divisor of xi ... xp lying in X.  ...  However, the rightmost cell does not start from 1, but from as shown in Figure 6 . Proof. The case n = 0 is obvious, so assume n > 1. We use induction on the length of the word w.  ... 
doi:10.5802/aif.1951 fatcat:rtfbjgxfknfktheaqcnv6boog4

Program development in constructive type theory

Didier Galmiche
1992 Theoretical Computer Science  
We present the program development concept in a logical framework including constructive type theory and then show how to use such theories to derive programs from proofs of formal specifications.  ...  We are interested in two important facts that are the mechanization of the proof construction and the possibility to express in the theory significant concepts for programming (like inductively defined  ...  Example 2 Another example is to determine the greatest common divisor of two nonzero naturals.  ... 
doi:10.1016/0304-3975(92)90037-g fatcat:cwdzm2e265awdpbribnyfeodey

QArith: Coq Formalisation of Lazy Rational Arithmetic [chapter]

Milad Niqui, Yves Bertot
2004 Lecture Notes in Computer Science  
This formalisation highlights the rôle of type theory both as a tool to verify hand-written programs and as a tool to generate verified programs.  ...  Here the function abs is the forgetful projection from Z onto nat (Coq natural numbers). Thus for p and q two integers, makeQ p q produces the signed binary sequence corresponding to p q .  ...  This work was completed during the first author's visit to INRIA Sophia Antipolis, made possible by a grant from the Dutch Organization for Scientific Research (NWO).  ... 
doi:10.1007/978-3-540-24849-1_20 fatcat:i4uwc4sggnbq7duac57togtyqq
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