Automated reasoning

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Automated reasoning is an area of cognitive science (involves knowledge representation and reasoning) and metalogic 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 sub-field of artificial intelligence, it also has connections with theoretical computer science, and even philosophy.

Contents

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).[ citation needed ] Extensive work has also been done in reasoning by analogy using induction and abduction. [1]

Other important topics include reasoning under uncertainty and non-monotonic 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 [2] 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 many less formal ad hoc techniques.

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. [3]

Some consider the Cornell Summer meeting of 1957, which brought together many logicians and computer scientists, as the origin of automated reasoning, or automated deduction. [4] 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). [5] Automated reasoning, although a significant and popular area of research, went through an "AI winter" in the eighties and early nineties. The field subsequently revived, however. 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. [4]

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. [6]

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 fifty-two theorems from chapter two of Principia Mathematica, proving thirty-eight of them. [7] 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." [8]

Examples of Formal Proofs

YearTheoremProof SystemFormalizerTraditional Proof
1986 First Incompleteness Boyer-Moore Shankar [9] Gödel
1990 Quadratic Reciprocity Boyer-Moore Russinoff [10] Eisenstein
1996 Fundamental- of Calculus HOL Light HarrisonHenstock
2000 Fundamental- of Algebra Mizar MilewskiBrynski
2000 Fundamental- of Algebra Coq Geuvers et al.Kneser
2004 Four Color Coq Gonthier Robertson et al.
2004 Prime Number Isabelle Avigad et al. Selberg-Erdős
2005 Jordan Curve HOL Light HalesThomassen
2005 Brouwer Fixed Point HOL Light HarrisonKuhn
2006 Flyspeck 1 Isabelle Bauer- NipkowHales
2007 Cauchy Residue HOL Light HarrisonClassical
2008 Prime Number HOL Light HarrisonAnalytic proof
2012 Feit-Thompson Coq Gonthier et al. [11] Bender, Glauberman and Peterfalvi
2016 Boolean Pythagorean triples problem Formalized as SAT Heule et al. [12] None

Proof systems

Boyer-Moore 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:
  1. the use of Lisp as a working logic.
  2. the reliance on a principle of definition for total recursive functions.
  3. the extensive use of rewriting and "symbolic evaluation".
  4. an induction heuristic based the failure of symbolic evaluation. [13]
HOL Light
Written in OCaml, HOL Light is designed to have a simple and clean logical foundation and an uncluttered implementation. It is essentially another proof assistant for classical higher order logic. [14]
Coq
Developed in France, Coq is another automated proof assistant, which can automatically extract executable programs from specifications, as either Objective CAML or Haskell source code. Properties, programs and proofs are formalized in the same language called the Calculus of Inductive Constructions (CIC). [15]

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. [16]

See also

Conferences and workshops

Journals

Communities

Related Research Articles

Automated theorem proving is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Isabelle (proof assistant) software

The Isabelle automated theorem prover is an interactive theorem prover, a higher order logic (HOL) theorem prover. It is an LCF-style theorem prover. It is thus based on small logical core (kernel) to increase the trustworthiness of proofs without requiring explicit proof objects. Isabelle is generic: it provides a meta-logic, which is used to encode object logics like first-order logic (FOL), higher-order logic (HOL) or Zermelo–Fraenkel set theory (ZFC). The most widely used object logic is Isabelle/HOL, although significant set theory developments were completed in Isabelle/ZF. Isabelle's main proof method is a higher-order version of resolution, based on higher-order unification. Though interactive, Isabelle also features efficient automatic reasoning tools, such as a term rewriting engine and a tableaux prover, various decision procedures, and, through the sledgehammer interface, external Satisfiability modulo theories (SMT) solvers and resolution-based theorem provers.

Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

The QED manifesto was a proposal for a computer-based database of all mathematical knowledge, strictly formalized and with all proofs having been checked automatically.

Logic in computer science Academic discipline

Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas:

A formal proof or derivation is a finite sequence of sentences, each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof.

Proof assistant software tool to assist with the development of formal proofs by human-machine collaboration

In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer.

Metamath is a formal language and an associated computer program for archiving, verifying, and studying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others.

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

SNARK, , is a theorem prover for multi-sorted first-order logic intended for applications in artificial intelligence and software engineering, developed at SRI International.

Logic Theorist is a computer program written in 1956 by Allen Newell, Herbert A. Simon and Cliff Shaw. It was the first program deliberately engineered to mimic the problem-solving techniques of a human being and is called "the first artificial intelligence program". It would eventually prove 38 of the first 52 theorems in Whitehead and Russell's Principia Mathematica, and find new and more elegant proofs for some.

In mathematical logic, a judgment or assertion is a statement or enunciation in the metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.

Geoff Sutcliffe Australian computer scientist

Geoff Sutcliffe is a US-based computer scientist working in the field of automated reasoning. He is of both British and Australian nationality. He was born in the former British colony of Northern Rhodesia , grew up in South Africa, and earned his Ph.D. in Australia. He works at the University of Miami. He is the developer of the Thousands of Problems for Theorem Provers (TPTP) problem library, and of the TPTP language for formal specification of Automated theorem proving problems and solutions. Since 1996 he has been organizing the annual CADE ATP System Competition (CASC), associated with the Conference on Automated Deduction and International Joint Conference on Automated Reasoning. He has been a co-organizer of several Automated reasoning challenges, including The Modal Logic $100 Challenge, The MPTP $100 Challenges, and The SUMO $100 Challenges. Together with Stephan Schulz, Sutcliffe founded and has been organizing the ES* Workshop series, a venue for presentation and publishing of practically oriented Automated Reasoning research.

In information technology a reasoning system is a software system that generates conclusions from available knowledge using logical techniques such as deduction and induction. Reasoning systems play an important role in the implementation of artificial intelligence and knowledge-based systems.

In propositional logic, tautology is one of two commonly used rules of replacement. The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs. They are:

Interactive Theorem Proving (ITP) is an annual international academic conference on the topic of automated theorem proving, proof assistants and related topics, ranging from theoretical foundations to implementation aspects and applications in program verification, security, and formalization of mathematics.

Logic Study of inference and truth

Logic is the systematic study of the form of valid inference, and the most general laws of truth. A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. In ordinary discourse, inferences may be signified by words such as therefore, thus, hence, ergo, and so on.

References

  1. Defourneaux, Gilles, and Nicolas Peltier. "Analogy and abduction in automated deduction." IJCAI (1). 1997.
  2. John L. Pollock
  3. C. Hales, Thomas "Formal Proof", University of Pittsburgh. Retrieved on 2010-10-19
  4. 1 2 "Automated Deduction (AD)", [The Nature of PRL Project]. Retrieved on 2010-10-19
  5. Martin Davis, "The Prehistory and Early History of Automated Deduction," in Automation of Reasoning, eds. Siekmann and Wrightson, vol. 1, 1-28 at p. 15
  6. "Principia Mathematica", at Stanford University. Retrieved 2010-10-19
  7. "The Logic Theorist and its Children". Retrieved 2010-10-18
  8. Shankar, Natarajan Little Engines of Proof , Computer Science Laboratory, SRI International. Retrieved 2010-10-19
  9. Shankar, N. (1994), Metamathematics, Machines, and Gödel's Proof, Cambridge, UK: Cambridge University Press, ISBN   9780521585330
  10. Russinoff, David M. (1992), "A Mechanical Proof of Quadratic Reciprocity", J. Autom. Reason., 8 (1): 3–21, doi:10.1007/BF00263446
  11. Gonthier, G.; et al. (2013), "A Machine-Checked Proof of the Odd Order Theorem", in Blazy, S.; Paulin-Mohring, C.; Pichardie, D. (eds.), Interactive Theorem Proving (PDF), Lecture Notes in Computer Science, 7998, pp. 163–179, doi:10.1007/978-3-642-39634-2_14, ISBN   978-3-642-39633-5
  12. Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016). "Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer". Theory and Applications of Satisfiability Testing – SAT 2016. Lecture Notes in Computer Science. 9710. pp. 228–245. arXiv: 1605.00723 . doi:10.1007/978-3-319-40970-2_15. ISBN   978-3-319-40969-6.
  13. The Boyer- Moore Theorem Prover . Retrieved on 2010-10-23
  14. Harrison, John HOL Light: an overview . Retrieved 2010-10-23
  15. Introduction to Coq . Retrieved 2010-10-23
  16. Automated Reasoning , Stanford Encyclopedia. Retrieved 2010-10-10