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Automation of Mathematical Induction as part of the History of Logic [article]

J Strother Moore, Claus-Peter Wirth
2014 arXiv   pre-print
We review the history of the automation of mathematical induction  ...  To cover the history of the automation of mathematical induction, we need a wider scope in mathematics and more historical detail.  ...  A Snapshot of a Decisive Moment in History The automation of mathematical theorem proving for deductive first-order logic started in the 1950s, and it took about half a century to develop software systems  ... 
arXiv:1309.6226v5 fatcat:i7nzngq47zfd7its4kfynlfpti

Computer-assisted human-oriented inductive theorem proving by descente infinie--a manifesto

C.-P. Wirth
2012 Logic Journal of the IGPL  
In this short position paper we briefly review the development history of automated inductive theorem proving and computer-assisted mathematical induction.  ...  We think that the current low expectations on progress in this field result from a faulty narrow-scope historical projection.  ...  Moreover, it is a pleasure to thank Tobias for the wonderful years we shared working on QUODLIBET and descente infinie.  ... 
doi:10.1093/jigpal/jzr048 fatcat:qps74fzzmngmzaluxlcaqogt4y

A Survey of Automated Deduction [chapter]

Alan Bundy
1999 Lecture Notes in Computer Science  
A Potted History of Early Automated Deduction Deduction is the branch of reasoning formalised by and studied in mathematical logic.  ...  As can be seen from these examples, most of the early applications of automated deduction were to proving mathematical theorems.  ...  Throughout its development automated deduction has drawn on and fed back t o w ork in mathematics and especially logic. Most of the logics have come from mathematical or philosophical logic.  ... 
doi:10.1007/3-540-48317-9_6 fatcat:r7kti6fn2bcjbodyatqdq64wu4

A Very Mathematical Dilemma

A. Bundy
2006 Computer journal  
Mathematics is facing a dilemma at its heart: the nature of mathematical proof.  ...  The choice seems to be between either abandoning the investigation of theorems whose only proofs are enormous or changing traditional mathematical practice to include computer-aided proofs.  ...  Acknowledgements I am grateful to the organisers of, speakers at and participants in the Royal Society discussion meeting on the Nature of Mathematical Proof, for the stimulating interactions that led  ... 
doi:10.1093/comjnl/bxl021 fatcat:l6iza3rdnnhlfowmo4ycmkvuau

The automation of proof: a historical and sociological exploration

D. Mackenzie
1995 IEEE Annals of the History of Computing  
This article reviews the history of the use of computers to automate mathematical proofs.  ...  The first strand has been underpinned by com-~itment to the goal of artificial intelligence; practitioners of the sec-/ -~ ... ond strand have been drawn mainly from mathematical logic; and e third strand  ...  True, logic was part of the medieval university curriculum, but formal, mathematical logic as a distinct academic specialty stretches back only a few decades before the start of the period we focus on  ... 
doi:10.1109/85.397057 fatcat:rcaohi4jafaoxivpjmdqmhsqtq

Cooperative reasoning for automatic software verification

Andrew Ireland
2007 Proceedings of the second workshop on Automated formal methods - AFM '07  
Mathematical induction is essential for the synthesis and verification of the inductively defined predicates that arise within separation logic specifications.  ...  To achieve this would require the ability to automate proof by mathematical induction. We will return to this point later.  ... 
doi:10.1145/1345169.1345175 fatcat:2c5i67mz4fguzogyeoflfoe2li

Book announcements

1996 Discrete Applied Mathematics  
Early twentieth century logic, Deduction and computation. Recent automation of logic and PROLOG. The future. Appendix B: A Genea- logical Database. Bibliography. History of mathematics.  ...  History of logic. Mathematical logic. Intuitionistic, modal, and temporal logics. Logic and computation. Index of Symbols. Index of Terms. David Gries and Fred B.  ... 
doi:10.1016/0166-218x(96)90069-4 fatcat:eswucpcizbfdxbpbqejedzpsyu

Mathematical Logic in Computer Science [article]

Assaf Kfoury
2018 arXiv   pre-print
The article retraces major events and milestones in the mutual influences between mathematical logic and computer science since the 1950s.  ...  I owe special thanks to all of them.  ...  Acknowledgments: I updated the text whenever I received comments from colleagues who took time to read earlier drafts.  ... 
arXiv:1802.03292v1 fatcat:bqajf55ih5aw3mxxo6tt5tjzkq

What is a proof?

Alan Bundy, Mateja Jamnik, Andrew Fugard
2005 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
We will look particularly at the faulty proof of Euler's Theorem and Lakatos' rational reconstruction of the history of this proof.  ...  We introduce the concept of schematic proofs and argue that they offer a possible cognitive model for the human construction of proofs in mathematics.  ...  Acknowledgements The research reported in this paper was supported by EPSRC grants GR/S01771 and GR/S31099 (Bundy), an EPSRC Advanced Research Fellowship GR/R76783 (Jamnik), and a Swedish Institute Guest  ... 
doi:10.1098/rsta.2005.1651 pmid:16188611 fatcat:m2huagnkhfhq5ijcag63gppqr4

Proof Theory [article]

Jeremy Avigad
2017 arXiv   pre-print
Proof theory began in the 1920's as a part of Hilbert's program, which aimed to secure the foundations of mathematics by modeling infinitary mathematics with formal axiomatic systems and proving those  ...  The program thus viewed mathematics as a system of reasoning with precise linguistic norms, governed by rules that can be described and studied in concrete terms.  ...  It is common today to view mathematical reasoning as consisting of a properly mathematical part that is used in conjunction with more general forms of logical reasoning, though there are still debates  ... 
arXiv:1711.01994v2 fatcat:3zjao7da6vdljgkb66w2vdjl74

History of Interactive Theorem Proving [chapter]

John Harrison, Josef Urban, Freek Wiedijk
2014 Handbook of the History of Logic  
The helpful review of the official reader Larry Paulson as well as useful comments from Mike Gordon, Tom Hales and J Moore, have significantly improved the eventual form of the chapter.  ...  ACKNOWLEDGEMENTS The authors are grateful to Jörg Siekmann for inviting them to prepare this chapter and for his patience in the face of our lengthy delays.  ...  -is to a large extent automated and user-programmable, allowing the automation of large parts of the mundane mathematical reasoning. 12 Where such automation fails, the RECONSIDER statement can be used  ... 
doi:10.1016/b978-0-444-51624-4.50004-6 fatcat:sngnxnl3ivectj5taj3dgsaa7m

The Importance of Applying Computational Creativity to Scientific and Mathematical Domains

Alison Pease, Simon Colton, Chris Warburton, Athanasios Nathanail, Irina Preda, Daniel Arnold, Daniel Winterstein, Mike Cook
2019 Zenodo  
We identify a key challenge in Automated Reasoning – that it has not achieved widespread adoption by mathematicians; and one in Automated Scientific Discovery – the need for communicability of automatically  ...  Science and mathematics are currently underrepresented in the computational creativity (CC) community.  ...  The eighth author is supported by the Royal Academy of Engineering under the Research Fellowship scheme.  ... 
doi:10.5281/zenodo.3951572 fatcat:pqnhpxib4rdc7ci6ub5kxrbzzy

Automation of Higher-Order Logic [chapter]

Christoph Benzmüller, Dale Miller
2014 Handbook of the History of Logic  
Besides the readers of this chapter, we thank Zakaria Chihani, Julian Röder, Leon Weber, and Max Wisnieswki for proofreading the document.  ...  The first author has been supported by the German Research Foundation under Heisenberg grant BE2501/9-1 and the second author has been supported by the ERC Advanced Grant ProofCert.  ...  Introduction Early efforts to formalize mathematics in order to make parts of it more rigorous and to show its consistency started with the codification of parts of logic.  ... 
doi:10.1016/b978-0-444-51624-4.50005-8 fatcat:jfcztdvymjfujg3bzb2rq2qyzy

View from the Fringe of the Fringe [chapter]

Steven D. Johnson
2001 Lecture Notes in Computer Science  
Although it may seem relatively academic to some, it is vital that this the so-called "theorem proving approach" continue to be as vigorously explored as approaches favoring highly automated reasoning.  ...  Formal analysis remains outside the mainstream of system design practice. Interactive methods and tools are regarded by some to be on the margin of useful research in this area.  ...  mathematical induction based on a representation of streams as functions over the natural numbers and extensionality.  ... 
doi:10.1007/3-540-44755-5_2 fatcat:7ijmucz2ffbhfiyn7mow7fxpeu

View from the Fringe of the Fringe [chapter]

Steven D. Johnson
2001 Lecture Notes in Computer Science  
Although it may seem relatively academic to some, it is vital that this the so-called "theorem proving approach" continue to be as vigorously explored as approaches favoring highly automated reasoning.  ...  Formal analysis remains outside the mainstream of system design practice. Interactive methods and tools are regarded by some to be on the margin of useful research in this area.  ...  mathematical induction based on a representation of streams as functions over the natural numbers and extensionality.  ... 
doi:10.1007/3-540-44798-9_1 fatcat:qqijippmtjay7oucg7lo5yf4jy
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