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Some results concerning proofs of statements about programs

R.J. Orgass
1970 Journal of computer and system sciences (Print)  
three concepts of the equivalence of computations and of programs, this system can be used to show that two computations or programs are or are not equivalent. (2) Given a program and a finite number  ...  That is, it is shown that certain relations of equivalence among programs and the relation of a program to the functions whose computation it specifies probably obey the law of excluded middle in this  ...  INTRODUCTION A number of writers have considered the problem of using various systems of logic to make and prove statements about computations and programs.  ... 
doi:10.1016/s0022-0000(70)80013-7 fatcat:lwjolmjgnfdabfeujnsn7m52ai

Page 4105 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews  
“We obtain two basic results: one concerns the characterization of the well-founded model in terms of (generalized) proof trees and as the least fixed point of some operator acting on proof tree roots.  ...  The second basic result concerns an application: a sound and complete proof method of partial correctness and weak  ... 

A Note on the Decidability of the Necessity of Axioms [article]

Merlin Carl
2014 arXiv   pre-print
A typical kind of question in mathematical logic is that for the necessity of a certain axiom: Given a proof of some statement ϕ in some axiomatic system T, one looks for minimal subsystems of T that allow  ...  In particular, one asks whether, given some system T+ψ, T alone suffices to prove ϕ. We show that this problem is undecidable unless T+ψ is decidable.  ...  There are various places in mathematical logic where one is concerned with the necessity of certain axioms for the proof of a theorem; in set theory, typical questions are about the necessity of the axiom  ... 
arXiv:1408.5314v1 fatcat:tlkldohfvjccloe2kpyhq5ruzi

Why Do We Believe Theorems? [article]

Andrzej Pelc
2014 arXiv   pre-print
Our argument is based on considerations concerning length of formal derivations. We also discuss psychological and social factors that contribute to building belief in theorems.  ...  Opposing this opinion, the main claim of the present paper is that such a gain of confidence obtained from any link between proofs and formal derivations is, even in principle, impossible in the present  ...  Thanks are due to Jody Azzouni for enlightening discussions concerning the subject of this paper and to an anonymous referee whose important remarks permitted us to correct the arguments and improve presentation  ... 
arXiv:1411.4857v1 fatcat:gtwibgx4nbd5jhjuiimqvyem2y

Fifty years of Hoare's Logic [article]

Krzysztof R. Apt, Ernst-Ruediger Olderog
2019 arXiv   pre-print
We present a history of Hoare's logic.  ...  The second one concerns reasoning about time complexity of the programs.  ...  Subsequently Hoare introduced an axiom to reason about the assignment statement and the proof rules to reason about the program composition and the while statement.  ... 
arXiv:1904.03917v2 fatcat:kfpjuntx35g4vhnkvsnetmkdnm

Page 42 of Journal of Research and Practice in Information Technology Vol. 14, Issue 2 [page]

1982 Journal of Research and Practice in Information Technology  
The most important assertions concern the initial and final values of variables. Assertions about their intermediate values are included to simplify the proof.  ...  Some methods of program construction and some computer languages lead to easier proofs than others. These points are demonstrated convincingly in E.W.  ... 

Mathematical Proofs at a Crossroad? [chapter]

Cristian S. Calude, Solomon Marcus
2004 Lecture Notes in Computer Science  
In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but  ...  now changing radically the very essence of proof.  ...  For some a proof including computer programs is like a telephone directory, while a human proof may compete with a beautiful novel.  ... 
doi:10.1007/978-3-540-27812-2_2 fatcat:5dzfdx5ppfghlkjglfgnbkq5gu

Page 190 of American Society of Civil Engineers. Collected Journals Vol. 129, Issue 4 [page]

2003 American Society of Civil Engineers. Collected Journals  
While part of him wants to pursue the matter, he also realizes that exploring his doubts about the exist- ing room being explosion-proof will not only further stress out Dante, but could result in additional  ...  Results are tabulated at the end of the voting period and the statistical responses are reported, along with commentary and in some cases, the actual solution for the case.  ... 

Didactical Issues at the Interface of Mathematics and Computer Science [chapter]

Viviane Durand-Guerrier, Antoine Meyer, Simon Modeste
2019 The Mathematics Teacher in the Digital Era  
We will elaborate on the links between the concepts of algorithm, proof, and program, and will argue for their significance in a general reflection on didactical issues in mathematics and computer science  ...  In view of this, addressing the question of proof in mathematics and computer science is a central didactical issue, which we examine in this contribution.  ...  about the statement.  ... 
doi:10.1007/978-3-030-28483-1_6 fatcat:cgtoxvtatfdipczdeyocpkav5y

Computation, Proof, Machine: Mathematics Enters a New Age, Gilles Dowek, Cambridge University Press, 2015. Hardback, ISBN 978-0-521-11801-9, 152 pages

2017 Theory and Practice of Logic Programming  
In a short Chapter 11 a convincing case for the proof-checking programs is made: one of such programs discovered an, admittedly small, mistake in one of Newton's proofs concerning the motion of planets  ...  Paulson, Let me conclude with some minor complaints about the book under review.  ... 
doi:10.1017/s1471068417000011 fatcat:yiexizomkbf4fjlgbt5lcdmzcy

Fifty years of Hoare's logic

Krzysztof R. Apt, Ernst-Rüdiger Olderog
2019 Formal Aspects of Computing  
Subsequently Hoare introduced an axiom to reason about the assignment statement and the proof rules to reason about the program composition and the while statement.  ...  Precursors Turing The concern about correctness of computer programs is as old as computers themselves.  ...  The assertions used for partial correctness are similar to the ones used in the correctness proof of [dB75] of a program corresponding to Turing's example that uses both a while and a repeat statement  ... 
doi:10.1007/s00165-019-00501-3 fatcat:vzxfucuz45fozhicmxzkxdpdjm

Reasoning about call-by-value: a missing result in the history of Hoare's logic [article]

Krzysztof R. Apt, Frank S. de Boer
2019 arXiv   pre-print
, and in which the correctness proofs are linear in the length of the program.  ...  We provide a sound and relatively complete Hoare-like proof system for reasoning about partial correctness of recursive procedures in presence of local variables and the call-by-value parameter mechanism  ...  Acknowledgement We thank Ernst-Rüdiger Olderog and Stijn de Gouw for helpful comments about the contributions of a number of relevant references.  ... 
arXiv:1909.06215v1 fatcat:rrsfhxbsn5co5mlt7u7nb3zlsu

Javalight is type-safe---definitely

Tobias Nipkow, David von Oheimb
1998 Proceedings of the 25th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '98  
of the machine-checked proof of type soundness.  ...  We emphasize that by 'machine-checked' we do not just mean th& it has passed some type checker, but that some non-trivial properties of the language have been established with t.he help of a (semiautomatic  ...  We thank Sophia Drossopoulou and Donald Syme for the very helpful discussions about their work.  ... 
doi:10.1145/268946.268960 dblp:conf/popl/NipkowO98 fatcat:ezfyy5sbnzdf7msoa5g4e7mwpy

Mathematics and the mind [chapter]

Edward Nelson
2002 Advances in Consciousness Research  
This is a modest question that concerns the functioning of the mind in one area, rather than the fundamental nature of consciousness, but I expect it to be more fruitful than speculations about Platonic  ...  Such a program requires no debugging since it is a by-product of a proof that the objects have the properties stated in the theorem whose constructive proof has been verified.  ... 
doi:10.1075/aicr.33.08nel fatcat:jeh44nqfkfbztpsaanhyfrnl5e

Gödel and the Limits of Logic

John W. Dawson
1999 Scientific American  
: some other true statements about the natural numbers will remain unprovable.  ...  his refrigerator aroused continuing concern about his mental stability.  ...  Further reading Read more about mathematical proof in past Plus articles, including our series on the origins of proof from Euclid to Gödel to the proof of Fermat's Last Theorem.  ... 
doi:10.1038/scientificamerican0699-76 fatcat:nbdbeqxbrnbnto6x7lmyh3xj2y
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