Abstract
This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andrews, P.B., Bishop, M., Issar, S., Nesmith, D., Pfenning, F., Xi, H.: TPS: A theorem proving system for classical type theory. Journal of Automated Reasoning 16, 321–353 (1996)
Appel, K., Haken, W.: Every planar map is four colorable. Bulletin of the American Mathematical Society 82, 711–712 (1976)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)
Bachmair, L., Dershowitz, N., Plaisted, D.A.: Completion without failure. In: Aït-Kaci, H., Nivat, M. (eds.) Resolution of Equations in Algebraic Structures Rewriting Techniques, vol. 2, pp. 1–30. Academic Press, San Diego (1989)
Ball, T., Bounimova, E., Cook, B., Levin, V., Lichtenberg, J., McGarvey, C., Ondrusek, B., Rajamani, S., Ustuner, A.: Thorough static analysis of device drivers. In: Proceedings of EuroSys 2006, the European Systems Conference (2006)
Benacerraf, P., Putnam, H.: Philosophy of mathematics: selected readings, 2nd edn. Cambridge University Press, Cambridge (1983)
Beth, E.W.: Semantic entailment and formal derivability. Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, new series 18, 309–342 (1955)
Biggs, N.L., Lloyd, E.K., Wilson, R.J.: Graph Theory, pp. 1736–1936. Clarendon Press, Oxford (1976)
Birtwistle, G., Subrahmanyam, P.A. (eds.): VLSI Specification, Verification and Synthesis. International Series in Engineering and Computer Science, vol. 35. Kluwer, Dordrecht (1988)
Bledsoe, W.W.: Some automatic proofs in analysis. In: Bledsoe, W.W., Loveland, D.W. (eds.) Automated Theorem Proving: After 25 Years, Contemporary Mathematics, vol. 29, pp. 89–118. American Mathematical Society (1984)
Boole, G.: The calculus of logic. The Cambridge and Dublin Mathematical Journal 3, 183–198 (1848)
Bourbaki, N.: Theory of sets. In: Elements of mathematics Translated from French Théorie des ensembles in the series Eléments de mathématique, originally published by Hermann, Addison-Wesley, London (1968)
Boyer, R.S., Moore, J.S.: A Computational Logic. ACM Monograph Series. Academic Press, San Diego (1979)
Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers C-35, 677–691 (1986)
Bryant, R.E.: A method for hardware verification based on logic simulation. Journal of the ACM 38, 299–328 (1991)
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Mathematisches Institut der Universität Innsbruck, 1965. English translation to appear in Journal of Symbolic Computation (2006)
Buchberger, B.: Ein algorithmisches Kriterium fur die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes Mathematicae, vol. 4, 374–383, 1970, English translation An Algorithmical Criterion for the Solvability of Algebraic Systems of Equations in [18] pp. 535–545 (1970)
Buchberger, B., Winkler, F. (eds.): Gröbner Bases and Applications. London Mathematical Society Lecture Note Series, vol. 251. Cambridge University Press, Cambridge (1998)
Bumcrot, R.: On lattice complements. In: Proceedings of the Glasgow Mathematical Association 7, 22–23 (1965)
Bundy, A.: A science of reasoning. In: Lassez, J.-L., Plotkin, G. (eds.) Computational Logic: Essays in Honor of Alan Robinson, pp. 178–198. MIT Press, Cambridge (1991)
Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and monographs in symbolic computation. Springer, Heidelberg (1998)
Chou, S.-C.: An introduction to Wu’s method for mechanical theorem proving in geometry. Journal of Automated Reasoning 4, 237–267 (1988)
Church, A.: An unsolvable problem of elementary number-theory. American Journal of Mathematics 58, 345–363 (1936)
Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching-time temporal logic. In: Kozen, D. (ed.) Logics of Programs. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1981)
Clarke, E.M., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)
Cohn, A.: A proof of correctness of the VIPER microprocessor: The first level. In: Birtwistle and Subrahmanyam [9], pp. 27–71
Corless, R.M., Jeffrey, D.J.: Well... it isn’t quite that simple. SIGSAM Bulletin 26(3), 2–6 (1992)
Corless, R.M., Jeffrey, D.J.: The unwinding number. SIGSAM Bulletin 30(2), 28–35 (1996)
Davis, M.: A computer program for Presburger’s algorithm. In: Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, pp. 215–233. Institute for Defense Analyses, Princeton, NJ, Reprinted in [94], pp. 41–48 (1957)
Davis, M. (ed.): The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Raven Press, NY (1965)
Davis, M., Putnam, H.: A computing procedure for quantification theory. Journal of the ACM 7, 201–215 (1960)
de Bruijn, N.G.: The mathematical language AUTOMATH, its usage and some of its extensions. In: Laudet, et al. [67], pp. 29–61
de Bruijn, N.G.: A survey of the project AUTOMATH. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry: Essays in Combinatory Logic, Lambda Calculus, and Formalism, pp. 589–606. Academic Press, San Diego (1980)
de Nivelle, H.: Ordering Refinements of Resolution. PhD thesis, Technische Universiteit Delft (1995)
Dijkstra, E.W.: Formal techniques and sizeable programs (EWD563). In: Dijkstra, E.W. (ed.) Selected Writings on Computing: A Personal Perspective Paper prepared for Symposium on the Mathematical Foundations of Computing Science, Gdansk, pp. 205–214. Springer, Heidelberg (1976)
Feigenbaum, E.A., Feldman, J. (eds.): Computers & Thought. AAAI Press / MIT Press (1995)
Franzén, T.: Gödel’s Theorem. An Incomplete Guide to its Use and Abuse. A. K. Peters (2005)
Gelerntner, H.: Realization of a geometry-theorem proving machine. In: Proceedings of the International Conference on Information Processing, UNESCO House, pp. 273–282, 1959 Also appears in [94], pp. 99–117 and in [36], pp. 134–152 (1959)
Gilmore, P.C.: A proof method for quantification theory: Its justification and realization. IBM Journal of research and development 4, 28–35 (1960)
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatshefte für Mathematik und Physik, vol. 38, 173–198, English translation, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, I In: [52], pp. 592–618 or [30], pp. 4–38 (1931)
Goldberg, E., Novikov, Y.: BerkMin: a fast and robust Sat-solver. In: Kloos, C.D., Franca, J.D. (eds.) DATE 2002. Design, Automation and Test in Europe Conference and Exhibition, Paris, France, pp. 142–149. IEEE Computer Society Press, Los Alamitos (2002)
Gordon, M.J.C., Melham, T.F.: Introduction to HOL: a theorem proving environment for higher order logic. Cambridge University Press, Cambridge (1993)
Gordon, M.J.C., Milner, R., Wadsworth, C.P.: Edinburgh LCF. LNCS, vol. 78. Springer, Heidelberg (1979)
Graham, B.T.: The SECD Microprocessor: A verification case study. Kluwer international series in engineering and computer science, vol. 178. Kluwer Academic Publishers, Boston (1992)
Grothendieck, A.: Éléments de Géométrie Algébraique IV: Étude locale de schémas et des morphismes de schémas, vol. 20 of Publications Mathématiques. IHES (1964)
Guard, J.R., Oglesby, F.C., Bennett, J.H., Settle, L.G.: Semi-automated mathematics. Journal of the ACM 16, 49–62 (1969)
Hales, T.C.: The Kepler conjecture (1998), available at http://front.math.ucdavis.edu/math.MG/9811078
Harrison, J.: Proof style. In: Giménez, E., Paulin-Mohring, C. (eds.) TYPES 1996. LNCS, vol. 1512, pp. 154–172. Springer, Heidelberg (1998)
Harrison, J.: Floating-point verification using theorem proving. In: Bernardo, M., Cimatti, A. (eds.) SFM 2006. LNCS, vol. 3965, pp. 211–242. Springer, Heidelberg (2006)
Harrison, J., Théry, L.: A sceptic’s approach to combining HOL and Maple. Journal of Automated Reasoning 21, 279–294 (1998)
Heawood, P.J.: Map-colour theorem. Quarterly Journal of Pure and Applied Mathematics Reprinted in [8] 24, 332–338 (1890)
Heijenoort, J.v. (ed.): From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931. Harvard University Press, Cambridge (1967)
Hilbert, D.: Die logischen Grundlagen der Mathematik. Mathematische Annalen 88, 151–165 (1922)
Hintikka, J.: Form and content in quantification theory. Acta Philosophica Fennica — Two papers on Symbolic Logic 8, 8–55 (1955)
Hobbes, T.: Leviathan. Andrew Crooke (1651)
Huet, G.: A complete proof of correctness of the Knuth-Bendix completion procedure. Journal of Computer and System Sciences 23, 11–21 (1981)
Hunt, W.A.: A Verified Micrprocessor. PhD thesis, University of Texas, 1985. In: Hunt Jr., W.A. (ed.) FM8501: A Verified Microprocessor. LNCS, vol. 795, Springer, Heidelberg (1994)
Hurd, J.: Integrating Gandalf and HOL. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 311–321. Springer, Heidelberg (1999)
Joyce, J.J.: Formal verification and implementation of a microprocessor. In: Birtwistle and Subrahmanyam [9], pp. 129–158
Kandri-Rody, A., Kapur, D., Narendran, P.: An ideal-theoretic approach to word problems and unification problems over finitely presented commutative algebras. In: Jouannaud, J.-P. (ed.) Rewriting Techniques and Applications. LNCS, vol. 202, pp. 345–364. Springer, Heidelberg (1985)
Kapur, D.: Automated geometric reasoning: Dixon resultants, Gröbner bases, and characteristic sets. In: Wang, D. (ed.) Automated Deduction in Geometry. LNCS, vol. 1360, Springer, Heidelberg (1998)
Kempe, A.B.: On the geographical problem of the four colours. American Journal of Mathematics Reprinted in [8] 2, 193–200 (1879)
Knuth, D., Bendix, P.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, Pergamon Press, New York (1970)
Kowalski, R.A., Kuehner, D.: Linear resolution with selection function. Artificial Intelligence 2, 227–260 (1971)
Kreisel, G.: Hilbert’s programme. Dialectica Revised version in [6] 12, 346–372 (1958)
Lam, C.W.H.: How reliable is a computer-based proof? The Mathematical Intelligencer 12, 8–12 (1990)
Laudet, M., Lacombe, D., Nolin, L., Schützenberger, M. (eds.): Symposium on Automatic Demonstration. Lecture Notes in Mathematics, vol. 125. Springer, Heidelberg (1970)
Lecat, M.: Erreurs de Mathématiciens des origines à nos jours. Ancne Libraire Castaigne et Libraire Ém Desbarax, Brussels (1935)
Littlewood, J.E.: Littlewood’s Miscellany Edited by Bela Bollobas. Cambridge University Press, Cambridge (1986)
Loveland, D.W.: Mechanical theorem-proving by model elimination. Journal of the ACM 15, 236–251 (1968)
Loveland, D.W.: A linear format for resolution. In: Laudet, et al. [67], pp. 147–162
Luckham, D.: Refinements in resolution theory. In: Laudet, et al. [67], pp. 163–190
Mac Lane, S.: Mathematics: Form and Function. Springer, Heidelberg (1986)
MacKenzie, D.: Mechanizing Proof: Computing, Risk and Trust. MIT Press, Cambridge (2001)
Marciszewski, W., Murawski, R.: Mechanization of Reasoning in a Historical Perspective, vol. 43 of Poznań Studies in the Philosophy of the Sciences and the Humanities. Rodopi, Amsterdam (1995)
McCune, W.: Solution of the Robbins problem. Journal of Automated Reasoning 19, 263–276 (1997)
McCune, W., Padmanabhan, R.: Automated Deduction in Equational Logic and Cubic Curves. LNCS, vol. 1095. Springer, Heidelberg (1996)
Moore, J.S., Lynch, T., Kaufmann, M.: A mechanically checked proof of the correctness of the kernel of the AMD5 K 86 floating-point division program. IEEE Transactions on Computers 47, 913–926 (1998)
Moskewicz, W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: DAC 2001. Proceedings of the 38th Design Automation Conference, pp. 530–535. ACM Press, New York (2001)
Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Transactions on Programming Languages and Systems 1, 245–257 (1979)
Newell, A., Simon, H.A.: The logic theory machine. IRE Transactions on Information Theory 2, 61–79 (1956)
O’Leary, J., Zhao, X., Gerth, R., Seger, C.-J.H.: Formally verifying IEEE compliance of floating-point hardware. Intel Technology Journal, 1999-Q1, 1–14 (1999), available on the Web as http://developer.intel.com/technology/itj/q11999/articles/art_5.htm
Pratt, V.R.: Anatomy of the Pentium bug. In: Mosses, P.D., Schwartzbach, M.I., Nielsen, M. (eds.) CAAP 1995, FASE 1995, and TAPSOFT 1995. LNCS, vol. 915, pp. 97–107. Springer, Heidelberg (1995)
Prawitz, D., Prawitz, H., Voghera, N.: A mechanical proof procedure and its realization in an electronic computer. Journal of the ACM 7, 102–128 (1960)
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Sprawozdanie z I Kongresu metematyków slowiańskich, Warszawa 1929, pp. 92–101, 395. Warsaw, 1930. Annotated English version by [101] (1929)
Queille, J.P., Sifakis, J.: Specification and verification of concurrent programs in CESAR. In: Dezani-Ciancaglini, M., Montanari, U. (eds.) International Symposium on Programming. LNCS, vol. 137, pp. 195–220. Springer, Heidelberg (1982)
Robinson, J.A.: Automatic deduction with hyper-resolution. International Journal of Computer Mathematics 1, 227–234 (1965)
Robinson, J.A.: A machine-oriented logic based on the resolution principle. Journal of the ACM 12, 23–41 (1965)
Robu, J.: Geometry Theorem Proving in the Frame of Theorema Project. PhD thesis, RISC-Linz (2002)
Rusinoff, D.: A mechanically checked proof of IEEE compliance of a register-transfer-level specification of the AMD-K7 floating-point multiplication, division, and square root instructions. LMS Journal of Computation and Mathematics 1, 148–200 (1998), available on the Web at http://www.onr.com/user/russ/david/k7-div-sqrt.html
Seger, C.-J.H., Bryant, R.E.: Formal verification by symbolic evaluation of partially-ordered trajectories. Formal Methods in System Design 6, 147–189 (1995)
Sheeran, M., Stålmarck, G.: A tutorial on Stålmarck’s proof procedure for propositional logic. In: Gopalakrishnan, G.C., Windley, P. (eds.) FMCAD 1998. LNCS, vol. 1522, pp. 82–99. Springer, Heidelberg (1998)
Shostak, R.: Deciding combinations of theories. Journal of the ACM 31, 1–12 (1984)
Siekmann, J., Wrightson, G. (eds.): Automation of Reasoning — Classical Papers on Computational Logic, vol. I, pp. 1957–1966. Springer, Heidelberg (1983)
Simmons, H.: The solution of a decision problem for several classes of rings. Pacific Journal of Mathematics 34, 547–557 (1970)
Simpson, S.: Partial realizations of Hilbert’s program. Journal of Symbolic Logic 53, 349–363 (1988)
Slagle, J.R.: Automatic theorem proving with renamable and semantic resolution. Journal of the ACM 14, 687–697 (1967)
Smullyan, R.M.: Gödel’s Incompleteness Theorems. Oxford Logic Guides, vol. 19. Oxford University Press, Oxford (1992)
Srivas, M.K., Miller, S.P.: Applying formal verification to the AAMP5 microprocessor: A case study in the industrial use of formal methods. Formal Methods in System Design 8, 31–36 (1993)
Stålmarck, G., Säflund, M.: Modeling and verifying systems and software in propositional logic. In: Daniels, B.K. (ed.) SAFECOMP 1990. Safety of Computer Control Systems, Gatwick, UK, pp. 31–36. Pergamon Press, New York (1990)
Stansifer, R.: Presburger’s article on integer arithmetic: Remarks and translation. Technical Report CORNELLCS:TR84-639, Cornell University Computer Science Department (1984)
Suttner, C.B., Sutcliffe, G.: The TPTP problem library. Technical Report AR-95-03, Institut für Infomatik, TU München, Germany, Also available as TR 95/6 from Dept. Computer Science, James Cook University, Australia, and on the Web (1995)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry. University of California Press, 1951. Previous version published as a technical report by the RAND Corporation, 1948; prepared for publication by J. C. C. McKinsey. Reprinted in [21], pp. 24–84 (1951)
Trybulec, A.: The Mizar-QC/6000 logic information language. ALLC Bulletin (Association for Literary and Linguistic Computing) 6, 136–140 (1978)
Trybulec, A., Blair, H.A.: Computer aided reasoning. In: Parikh, R. (ed.) Logics of Programs. LNCS, vol. 193, pp. 406–412. Springer, Heidelberg (1985)
Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society (2) 42, 230–265 (1936)
Wang, H.: Toward mechanical mathematics. IBM Journal of research and development 4, 2–22 (1960)
Wen-tsün, W.: On the decision problem and the mechanization of theorem proving in elementary geometry. Scientia Sinica 21, 157–179 (1978)
Whitehead, A.N., Russell, B.: Principia Mathematica (3 vols). Cambridge University Press, Cambridge (1910)
Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)
Wos, L., Pieper, G.W.: A Fascinating Country in the World of Computing: Your Guide to Automated Reasoning. World Scientific, Singapore (1999)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Harrison, J. (2007). A Short Survey of Automated Reasoning. In: Anai, H., Horimoto, K., Kutsia, T. (eds) Algebraic Biology. AB 2007. Lecture Notes in Computer Science, vol 4545. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73433-8_24
Download citation
DOI: https://doi.org/10.1007/978-3-540-73433-8_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73432-1
Online ISBN: 978-3-540-73433-8
eBook Packages: Computer ScienceComputer Science (R0)