Automated reasoning
automated reasoning, automated reasoning and theorem provingAutomated reasoning is an area of computer science and mathematical logic dedicated to understanding different aspects of reasoning The study of automated reasoning helps produce computer programs that allow computers to reason completely, or nearly completely, automatically Although automated reasoning is considered a subfield of artificial intelligence, it also has connections with theoretical computer science, and even philosophy
The most developed subareas of automated reasoning are automated theorem proving and the less automated but more pragmatic subfield of interactive theorem proving and automated proof checking viewed as guaranteed correct reasoning under fixed assumptions Extensive work has also been done in reasoning by analogy induction and abduction
Other important topics include reasoning under uncertainty and nonmonotonic reasoning An important part of the uncertainty field is that of argumentation, where further constraints of minimality and consistency are applied on top of the more standard automated deduction John Pollock's OSCAR system is an example of an automated argumentation system that is more specific than being just an automated theorem prover
Tools and techniques of automated reasoning include the classical logics and calculi, fuzzy logic, Bayesian inference, reasoning with maximal entropy and a large number of less formal ad hoc techniques
Contents
 1 Early years
 2 Significant contributions
 3 Proof systems
 4 Applications
 5 See also
 51 Conferences and workshops
 52 Journals
 53 Communities
 6 References
 7 External links
Early years
The development of formal logic played a big role in the field of automated reasoning, which itself led to the development of artificial intelligence A formal proof is a proof in which every logical inference has been checked back to the fundamental axioms of mathematics All the intermediate logical steps are supplied, without exception No appeal is made to intuition, even if the translation from intuition to logic is routine Thus, a formal proof is less intuitive, and less susceptible to logical errors
Some consider the Cornell Summer meeting of 1957, which brought together a large number of logicians and computer scientists, as the origin of automated reasoning, or automated deduction Others say that it began before that with the 1955 Logic Theorist program of Newell, Shaw and Simon, or with Martin Davis’ 1954 implementation of Presburger’s decision procedure which proved that the sum of two even numbers is even Automated reasoning, although a significant and popular area of research, went through an "AI winter" in the eighties and early nineties Luckily, it got revived after that For example, in 2005, Microsoft started using verification technology in many of their internal projects and is planning to include a logical specification and checking language in their 2012 version of Visual C
Significant contributions
Principia Mathematica was a milestone work in formal logic written by Alfred North Whitehead and Bertrand Russell Principia Mathematica  also meaning Principles of Mathematics  was written with a purpose to derive all or some of the mathematical expressions, in terms of symbolic logic Principia Mathematica was initially published in three volumes in 1910, 1912 and 1913
Logic Theorist LT was the first ever program developed in 1956 by Allen Newell, Cliff Shaw and Herbert A Simon to "mimic human reasoning" in proving theorems and was demonstrated on fiftytwo theorems from chapter two of Principia Mathematica, proving thirtyeight of them In addition to proving the theorems, the program found a proof for one of the theorems that was more elegant than the one provided by Whitehead and Russell After an unsuccessful attempt at publishing their results, Newell, Shaw, and Herbert reported in their publication in 1958, The Next Advance in Operation Research:
"There are now in the world machines that think, that learn and that create Moreover, their ability to do these things is going to increase rapidly until in a visible future the range of problems they can handle will be co extensive with the range to which the human mind has been applied"Examples of Formal Proofs
Year  Theorem  Proof System  Formalizer  Traditional Proof 

1986  First Incompleteness  BoyerMoore  Shankar  Gödel 
1990  Quadratic Reciprocity  BoyerMoore  Russinoff  Eisenstein 
1996  Fundamental of Calculus  HOL Light  Harrison  Henstock 
2000  Fundamental of Algebra  Mizar  Milewski  Brynski 
2000  Fundamental of Algebra  Coq  Geuvers et al  Kneser 
2004  Four Color  Coq  Gonthier  Robertson et al 
2004  Prime Number  Isabelle  Avigad et al  SelbergErdős 
2005  Jordan Curve  HOL Light  Hales  Thomassen 
2005  Brouwer Fixed Point  HOL Light  Harrison  Kuhn 
2006  Flyspeck 1  Isabelle  Bauer Nipkow  Hales 
2007  Cauchy Residue  HOL Light  Harrison  Classical 
2008  Prime Number  HOL Light  Harrison  analytic proof 
2012  FeitThompson  Coq  Gonthier et al  Bender, Glauberman and Peterfalvi 
2016  Boolean Pythagorean triples problem  Formalized as SAT  Heule et al  none 
Proof systems
BoyerMoore Theorem Prover NQTHM The design of NQTHM was influenced by John McCarthy and Woody Bledsoe Started in 1971 at Edinburgh, Scotland, this was a fully automatic theorem prover built using Pure Lisp The main aspects of NQTHM were: the use of Lisp as a working logic
 the reliance on a principle of definition for total recursive functions
 the extensive use of rewriting and "symbolic evaluation"
 an induction heuristic based the failure of symbolic evaluation
Applications
Automated reasoning has been most commonly used to build automated theorem provers Oftentimes, however, theorem provers require some human guidance to be effective and so more generally qualify as proof assistants In some cases such provers have come up with new approaches to proving a theorem Logic Theorist is a good example of this The program came up with a proof for one of the theorems in Principia Mathematica that was more efficient requiring fewer steps than the proof provided by Whitehead and Russell Automated reasoning programs are being applied to solve a growing number of problems in formal logic, mathematics and computer science, logic programming, software and hardware verification, circuit design, and many others The TPTP Sutcliffe and Suttner 1998 is a library of such problems that is updated on a regular basis There is also a competition among automated theorem provers held regularly at the CADE conference Pelletier, Sutcliffe and Suttner 2002; the problems for the competition are selected from the TPTP library
See also
 Automated theorem proving
 Reasoning system
 Semantic reasoner
 Program analysis computer science
 Applications of artificial intelligence
 Outline of artificial intelligence
 Casuistry • Casebased reasoning
 Abductive reasoning
 Duck test
 I know it when I see it
 Commonsense reasoning
 Purposeful omission • Iceberg Theory • Show, don't tell • Concision
Conferences and workshops
 International Joint Conference on Automated Reasoning IJCAR
 Conference on Automated Deduction CADE
 International Conference on Automated Reasoning with Analytic Tableaux and Related Methods
Journals
 Journal of Automated Reasoning
Communities
 Association for Automated Reasoning AAR
References
 ^ John L Pollock
 ^ C Hales, Thomas "Formal Proof", University of Pittsburgh Retrieved on 20101019
 ^ a b "Automated Deduction AD", Retrieved on 20101019
 ^ Martin Davis, "The Prehistory and Early History of Automated Deduction," in Automation of Reasoning, eds Siekmann and Wrightson, vol 1, 128 at p 15
 ^ "Principia Mathematica", at Stanford University Retrieved 20101019
 ^ "The Logic Theorist and its Children" Retrieved 20101018
 ^ Shankar, Natarajan Little Engines of Proof, Computer Science Laboratory, SRI International Retrieved 20101019
 ^ Shankar, N 1994, Metamathematics, Machines, and Gödel's Proof, Cambridge, UK: Cambridge University Press
 ^ Russinoff, David M 1992, "A Mechanical Proof of Quadratic Reciprocity", J Autom Reason, 8 1: 3–21, doi:101007/BF00263446
 ^ Gonthier, G; et al 2013, "A MachineChecked Proof of the Odd Order Theorem", in Blazy, S; PaulinMohring, C; Pichardie, D, Interactive Theorem Proving, Lecture Notes in Computer Science, 7998, pp 163–179, doi:101007/9783642396342_14, ISBN 9783642396335
 ^ https://arxivorg/abs/160500723
 ^ The Boyer Moore Theorem Prover Retrieved on 20101023
 ^ Harrison, John HOL Light: an overview Retrieved 20101023
 ^ Introduction to Coq Retrieved 20101023
 ^ Automated Reasoning, Stanford Encyclopedia Retrieved 20101010
External links
 International Workshop on the Implementation of Logics
 Workshop Series on Empirically Successful Topics in Automated Reasoning



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