Automated deduction with built-in theories: completeness results and constraint solving techniques

Postprint (published version

Decision Problems in Ordered Rewriting

A term rewrite system (TRS) terminates iff its rules are contained in a reduction ordering ?. In order to deal with any set of equations, including inherently non-terminating ones (like commutativity), TRS have been generalised to ordered TRS (E; ?), where equations of E are applied in whatever direction agrees with ?. The confluence of terminating TRS is well-known to be decidable, but for ordered TRS the decidability of confluence has been open. Here we show that the confluence of ordered TRS is decidable if ordering constraints for ? can be solved in an adequate way, which holds in particular for the class of LPO orderings. For sets E of constrained equations, confluence is shown to be undecidable. Finally, ground reducibility is proved undecidable for ordered TRS. 1 Introduction Term rewrite systems (TRS) have been applied to many problems in symbolic computation, automated theorem proving, program synthesis and verification, and logic programming among others. Two fundamental pr..

Decision Problems in Ordered Rewriting

Introduction Since the Knuth and Bendix landmark paper [12], a lot of work has been devoted to the completion of term rewriting systems. The basic idea of completion, as stated in [2], is to add consequences of equational axioms in order to simplify the equational proofs, according to a given well-founded ordering. Ultimately, the simplest proofs are rewrite proofs. Such proofs can always be obtained from canonical rewrite systems, yielding a decidable word problem. There are however several drawbacks in the original Knuth-Bendix completion procedure: first, it can run forever and second, it may fail. The first weakness cannot be avoided as the word problem is in general undecidable for equational theories. On the other hand, the failure of Knuth-Bendix completion occurs when the procedure encounters an equation which cannot be oriented using the given ordering. A typical example is the commutativity axiom x + y