A New Procedural Interpretation of Horn Clauses with Equality

We introduce the equality elimination method which is a new procedure for dealing with Horn clause logic programs with equality. The method combines SLD-resolution with a bottom-up equation solving. By solving equations, we try to transform a logic program with equality to a logic program without equality. The transformation uses basic superposition as the main operation. We prove soundness and completeness of the equality elimination method. We also show that approaches based on complete sets of E-uni ers are not satisfying. In particular, we provide a negative solution to the open problem of completeness of SLDE+-resolution.

[1]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[2]  Akihiro Yamamoto.  A Theoretical Combination of SLD-Resolution and Narrowing , 1987, ICLP.

[3]  Stéphane Kaplan,et al.  Simplifying Conditional Term Rewriting Systems: Unification, Termination and Confluence , 1987, J. Symb. Comput..

[4]  Roberto Barbuti,et al.  LEAF: A Language which Integrates Logic, Equations and Functions , 1986, Logic Programming: Functions, Relations, and Equations.

[5]  Paliath Narendran,et al.  Theorem proving using equational matings and rigid E-unification , 1992, JACM.

[6]  Heinrich Hußmann,et al.  Unification in Conditional Equational Theories , 1985, European Conference on Computer Algebra.

[7]  John W. Lloyd,et al.  A Logical Reconstruction of Prolog II , 1984, J. Log. Program..

[8]  James B. Morris,et al.  E-Resolution: Extension of Resolution to Include the Equality Relation , 1969, IJCAI.

[9]  Gerald E. Peterson,et al.  A Technique for Establishing Completeness Results in Theorem Proving with Equality , 1980, SIAM J. Comput..

[10]  Wayne Snyder,et al.  Basic Paramodulation and Superposition , 1992, CADE.

[11]  Albert Rubio,et al.  Theorem Proving with Ordering Constrained Clauses , 1992, CADE.

[12]  Laurent Fribourg,et al.  Oriented Equational Clauses as a Programming Language , 1984, J. Log. Program..

[13]  Michaël Rusinowitch,et al.  A New Method for Establishing Refutational Completeness in Theorem Proving , 1986, CADE.

[14]  Laurent Fribourg,et al.  SLOG: A Logic Programming Language Interpreter Based on Clausal Superposition and Rewriting , 1985, SLP.

[15]  Alexander Herold.  Narrowing Techniques Applied to Idempotent Unification , 1987, GWAI.

[16]  Andrei Mantsivoda,et al.  Flang: A Functional-Logic Language , 1991, PDK.

[17]  Harald Ganzinger.  A Completion Procedure for Conditional Equations , 1987, CTRS.

[18]  Laurent Vieille,et al.  Recursive Query Processing: The Power of Logic , 1989, Theor. Comput. Sci..

[19]  Michael Hanus,et al.  The Integration of Functions into Logic Programming: From Theory to Practice , 1994, J. Log. Program..

[20]  Jean-Marie Hullot,et al.  Canonical Forms and Unification , 1980, CADE.

[21]  Valentin M. Antimirov,et al.  Consistency of Equational Enrichments , 1992, LPAR.

[22]  Richard C. T. Lee,et al.  Symbolic logic and mechanical theorem proving , 1973, Computer science classics.

[23]  Uday S. Reddy,et al.  Narrowing as the Operational Semantics of Functional Languages , 1985, SLP.

[24]  Michaël Rusinowitch.  Theorem-Proving with Resolution and Superposition , 1991, J. Symb. Comput..

[25]  José Meseguer,et al.  Models and Equality for Logical Programming , 1987, TAPSOFT, Vol.1.

[26]  Bernhard Beckert,et al.  An Improved Method for Adding Equality to Free Variable Semantic Tableaux , 1992, CADE.

[27]  E. W. Elcock.  Prolog: Subsumption of Equality Axioms by the Homogeneous Form , 1989, J. Log. Program..

[28]  J. Lloyd.  Foundations of Logic Programming , 1984, Symbolic Computation.

[29]  Mario Rodríguez-Artalejo,et al.  On the Completeness of Narrowing as the Operational Semantics of Functional Logic Programming , 1992, CSL.

[30]  Robert Anderson.  Completeness results for e-resolution , 1970, AFIPS '70 (Spring).

[31]  José Meseguer,et al.  EQLOG: Equality, Types, and Generic Modules For Logic Programming , 1986, Logic Programming: Functions, Relations, and Equations.

[32]  John W. Lloyd,et al.  Foundations of Logic Programming, 1st Edition , 1984 .

[33]  Corrado Moiso,et al.  Narrowing vs. SLD-Resolution , 1988, Theor. Comput. Sci..

[34]  L. Wos,et al.  Paramodulation and Theorem-Proving in First-Order Theories with Equality , 1983 .

[35]  Claude Kirchner,et al.  NARROWER: A New Algorithm for Unification and Its Application to Logic Programming , 1985, RTA.

[36]  Malcolm C. Harrison,et al.  Equality-based binary resolution , 1986, JACM.

[37]  Werner Nutt,et al.  Basic Narrowing Revisited , 1989, J. Symb. Comput..

[38]  M. H. van Emden,et al.  Logic Programming with Equations , 1987, J. Log. Program..

[39]  Aart Middeldorp,et al.  Counterexamples to completeness results for basic narrowing , 1991 .

[40]  Rolf Socher-Ambrosius,et al.  A Goal Oriented Strategy Based on Completion , 1992, ALP.

[41]  Corrado Moiso,et al.  A Completeness Result for E-unification Algorithms Based on Conditional Narrowing , 1986, Foundations of Logic and Functional Programming.

[42]  Gérard P. Huet,et al.  Proofs by induction in equational theories with constructors , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[43]  Deepak Kapur,et al.  First-Order Theorem Proving Using Conditional Rewrite Rules , 1988, CADE.

[44]  Michaël Rusinowitch,et al.  On Word Problems in Horn Theories , 1988, J. Symb. Comput..

[45]  Pierre Réty.  Improving Basic Narrowing Techniques , 1987, RTA.

[46]  Michael J. Maher,et al.  A Theory of Complete Logic Programs with Equality , 1984, J. Log. Program..

[47]  Jean H. Gallier,et al.  Extending SLD Resolution to Equational Horn Clauses using E-Unification , 1989, J. Log. Program..

[48]  Larry Wos,et al.  Automated reasoning - 33 basic research problems , 1988 .

[49]  Laurent Fribourg,et al.  A Strong Restriction of the Inductive Completion Procedure , 1986, J. Symb. Comput..

[50]  Harald Ganzinger,et al.  Rewrite-Based Equational Theorem Proving with Selection and Simplification , 1994, J. Log. Comput..

[51]  Stig Kanger,et al.  A Simplified Proof Method for Elementary Logic , 1959 .

[52]  R SlagleJames.  Automated Theorem-Proving for Theories with Simplifiers Commutativity, and Associativity , 1974 .

[53]  Andrei Voronkov,et al.  Theorem Proving in Non-Standard Logics Based on the Inverse Method , 1992, CADE.

[54]  C. Cordell Green,et al.  The Application of Theorem Proving to Question-Answering Systems , 1969, Outstanding Dissertations in the Computer Sciences.

[55]  Philip T. Cox,et al.  Incorporating equality into logic programming via surface deduction , 1986, Ann. Pure Appl. Log..

[56]  Steffen Hölldobler.  From Paramodulation to Narrowing , 1988, ICLP/SLP.

[57]  Albert Rubio,et al.  Basic Superposition is Complete , 1992, ESOP.