Analogy in Automated Deduction: A Survey

We provide a survey of the main approaches for analogical reasoning in automated deduction. We start by proposing a general framework for reasoning by analogy based on a constrained sequent calculus in which higher-order variables denote first-order formulae. Then we briefly review the most successful approaches, ranging from early work in Horn logic to proof reuse in interactive higher-order theorem provers. With the help of many examples, we present the main ideas and basic features of each approach and briefly discuss their common points and differences.

[1]  Rance Cleaveland,et al.  Implementing mathematics with the Nuprl proof development system , 1986 .

[2]  Einar Broch Johnsen,et al.  Theorem Reuse by Proof Term Transformation , 2004, TPHOLs.

[3]  Alexander Leitsch,et al.  The Resolution Calculus , 1997, Texts in Theoretical Computer Science An EATCS Series.

[4]  Alan Robinson,et al.  The Inverse Method , 2001, Handbook of Automated Reasoning.

[5]  Jon Whittle,et al.  Analogy in Inductive Theorem Proving , 2004, Journal of Automated Reasoning.

[6]  Harald Ganzinger,et al.  Resolution Theorem Proving , 2001, Handbook of Automated Reasoning.

[7]  Christoph Kreitz,et al.  Building Proofs by Analogy via the Curry-Horward Isomorphism , 1992, LPAR.

[8]  Gilles Barthe,et al.  Type Isomorphisms and Proof Reuse in Dependent Type Theory , 2001, FoSSaCS.

[9]  Stefan Hetzl,et al.  A Sequent Calculus with Implicit Term Representation , 2010, CSL.

[10]  Régis Curien.  Second Order E-Matching as a Tool for Automated Theorem Proving , 1993, EPIA.

[11]  Alberto Martelli,et al.  An Efficient Unification Algorithm , 1982, TOPL.

[12]  Nicolas Peltier.  A General Method for Using Schematizations in Automated Deduction , 2001, IJCAR.

[13]  Tobias Nipkow,et al.  Isabelle/HOL , 2002, Lecture Notes in Computer Science.

[14]  Zhenyu Qian,et al.  Efficient Second-Order Matching , 1996, RTA.

[15]  Alexander Leitsch,et al.  Towards Algorithmic Cut-Introduction , 2012, LPAR.

[16]  Claude Kirchner,et al.  Theorem Proving Modulo , 2003, Journal of Automated Reasoning.

[17]  Ricardo Caferra,et al.  A Method for Building Models Automatically. Experiments with an Extension of OTTER , 1994, CADE.

[18]  Frank van Harmelen,et al.  The Oyster-Clam System , 1990, CADE.

[19]  Hubert Comon-Lundh,et al.  Equational Problems and Disunification , 1989, J. Symb. Comput..

[20]  Larry Wos.  The problem of reasoning by analogy , 2004, Journal of Automated Reasoning.

[21]  Georg Wilhelm Friedrich Hegel,et al.  Elements of the Philosophy of Right , 1820 .

[22]  W. W. Bledsoe,et al.  Non-Resolution Theorem Proving , 1977, Artif. Intell..

[23]  Nicolas Peltier,et al.  Semantic Generalizations for Proving and Disproving Conjectures by Analogy , 1998, Journal of Automated Reasoning.

[24]  Ricardo Caferra,et al.  Proof Analogy in Interactive Theorem Proving: A Method to Express and Use It via Second Order Pattern Matching , 1987, AAAI.

[25]  Tobias Nipkow,et al.  Term rewriting and all that , 1998 .

[26]  Nicolas Peltier,et al.  Partial Matching for Analogy Discovery in Proofs and Counter-Examples , 1997, CADE.

[27]  Amy P. Felty,et al.  Generalization and Reuse of Tactic Proofs , 1994, LPAR.

[28]  Stephen Owen,et al.  Analogy for automated reasoning , 1990, Perspectives in artificial intelligence.

[29]  Nicolas Peltier,et al.  Building Proofs or Counterexamples by Analogy in a Resoluton Framework , 1996, JELIA.

[30]  Rob Kling,et al.  A Paradigm for Reasoning by Analogy , 1971, IJCAI.

[31]  Thomas Kolbe,et al.  Proving theorems by reuse , 2000, Artif. Intell..

[32]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[33]  Warren D. Goldfarb,et al.  The Undecidability of the Second-Order Unification Problem , 1981, Theor. Comput. Sci..

[34]  Ricardo Caferra,et al.  A Method for Simultanous Search for Refutations and Models by Equational Constraint Solving , 1992, J. Symb. Comput..