Conditional Equational Specifications of Data Types with Partial Operations for Inductive Theorem Proving

We propose a specification language for the formalization of data types with partial or non-terminating operations as part of a rewrite-based framework for inductive theorem proving. The language requires constructors for designating data items and admits positive/negative conditional equations as axioms in specifications. The (total algebra) semantics for such specifications is based on so-called data models. We develop admissibility conditions that guarantee the unique existence of a distinguished data model. Since admissibility of a specification requires confluence of the induced rewrite relation, we provide an effectively testable confluence criterion which does not presuppose termination.

[1]  David A. Plaisted,et al.  Semantic Confluence Tests and Completion Methods , 1985, Inf. Control..

[2]  Jan A. Bergstra,et al.  Conditional Rewrite Rules: Confluence and Termination , 1986, J. Comput. Syst. Sci..

[3]  Claus-Peter Wirth,et al.  Inductive theorem proving in theories specified by positive/negative-conditional equations , 1999 .

[4]  Dieter Hutter,et al.  INKA: The Next Generation , 1996, CADE.

[5]  Robert S. Boyer,et al.  Computational Logic , 1990, ESPRIT Basic Research Series.

[6]  Hartmut Ehrig,et al.  Fundamentals of Algebraic Specification 1 , 1985, EATCS Monographs on Theoretical Computer Science.

[7]  Jürgen Avenhaus,et al.  Theorem Proving in Hierarchical Causal Specifications , 1997, Advances in Algorithms, Languages, and Complexity.

[8]  Bernhard Gramlich,et al.  A Constructor-Based Approach to Positive/Negative-Conditional Equational Specifications , 1994, J. Symb. Comput..

[9]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[10]  Bernhard Gramlich,et al.  On Notions of Inductive Validity for First-Oder Equational Clauses , 1994, CADE.

[11]  Nachum Dershowitz,et al.  Confluence of Conditional Rewrite Systems , 1988, CTRS.

[12]  Claus-Peter Wirth.  Syntactic Confluence Criteria for Positive/Negative-Conditional Term Rewriting Systems , 2009, ArXiv.

[13]  Martin Wirsing,et al.  Algebraic Specification , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.