Albert Rubio

Theorem proving with ordering and equality constrained clauses

Abstract Deduction methods for first-order constrained clauses with equality are described within an abstract framework: constraint strategies , consisting of an inference system, a constraint inheritance strategy and redundancy criteria for clauses and inferences. We give simple conditions for such a constraint strategy to be complete (refutationally and in the sense of Knuth-Bendix-like completion). This allows to prove in a uniform way the completeness of several instantiations of the framework with concrete strategies. For example, strategies in which equality constraints are inherited are basic : no inferences are needed on subterms introduced by unifiers of previous inferences. Ordering constraints reduce the search space by inheriting the ordering restrictions of previous inferences and increase the expressive power of the logic.

The higher-order recursive path ordering

This paper extends the termination proof techniques based on reduction orderings to a higher-order setting, by adapting the recursive path ordering definition to terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols. The obtained ordering is well-founded, compatible with p-reductions and with polymorphic typing, monotonic with respect to the function symbols, and stable under substitution. It can therefore be used to prove the strong normalization property of higher-order calculi in which constants can be defined by higher-order rewrite rules. For example, the polymorphic version of Godels recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Several other non-trivial examples are given which exemplify the expressive power of the ordering.

The Barcelogic SMT Solver

This is the first system description of the Barcelogic SMT solver, which implements all techniques that our group has been developing over the last four years as well as state-of-the-art features developed by other research groups. We pay special attention to the theory solvers and to functionalities that are not common in SMT solvers.

Basic superposition is complete

We define equality constrained equations and clauses and use them to prove the completeness of what we have called basic superposition: a restricted form of superposition in which only the subterms not created in previous inferences is superposed upon. We first apply our results to the equational case and define an unfailing Knuth-Bendix completion procedure that uses basic superposition as inference rule. Second, we extend the techniques to completion of full first-order clauses with equality. Moreover, we prove the refutational completeness of a new simple inference system.

Theorem Proving with Ordering Constrained Clauses

We use clauses with ordering constraints to reduce the search space in ordered inference systems for clauses with or without equality, such as ordered resolution or superposition. In our completion procedure for ordering constrained clauses redundant inferences can be ignored and redundant clauses can be deleted without loosing refutational completeness. Two new results needed for fast ordering constraint solving and incrementality of the set of function symbols are given. We discuss the use of our methods for reasoning about infinite sets of clauses defined by a finite number of ordering constrained ones.

Complete Monotonic Semantic Path Orderings

Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS).

A Precedence-Based Total AC-Compatible Ordering

Like Narendran and Rusinowitch [NR91], we define a simplification ordering which is AC-compatible and total on non-AC-equivalent ground terms, without any restrictions on the signature like the number of AC-symbols or free symbols.

A Fully Syntactic AC-RPO

We present the first fully syntactic (i.e., non-interpretation-based) AC-compatible recursive path ordering (RPO). It is simple, and hence easy to implement, and its behaviour is intuitive as in the standard RPO. The ordering is AC-total, and defined uniformly for both ground and non-ground terms, as well as for partial precedences. More importantly, it is the first one that can deal incrementally with partial precedences, an aspect that is essential, together with its intuitive behaviour, for interactive applications like Knuth-Bendix completion.

Polymorphic higher-order recursive path orderings

This article extends the termination proof techniques based on reduction orderings to a higher-order setting, by defining a family of recursive path orderings for terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols. These relations can be generated from two given well-founded orderings, on the function symbols and on the type constructors. The obtained orderings on terms are well founded, monotonic, stable under substitution and include β-reductions. They can be used to prove the strong normalization property of higher-order calculi in which constants can be defined by higher-order rewrite rules using first-order pattern matching. For example, the polymorphic version of Gödels recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Many nontrivial examples are given that exemplify the expressive power of these orderings. All have been checked by our implementation. This article is an extended and improved version of Jouannaud and Rubio [1999]. Polymorphic algebras have been made more expressive than in our previous framework. The intuitive notion of a polymorphic higher-order ordering has now been made precise. The higher-order recursive path ordering itself has been made much more powerful by replacing the congruence on types used there by an ordering on types satisfying some abstract properties. Besides, using a restriction of Dershowitzs recursive path ordering for comparing types, we can integrate both orderings into a single one operating uniformly on both terms and types.

AC-Superposition with Constraints: No AC-Unifiers Needed

We prove the completeness of (basic) deduction strategies with constrained clauses modulo associativity and commutativity (AC). Here each inference generates one single conclusion with an additional equality s=ACt in its constraint (instead of one conclusion for each minimal AC-unifier, i.e. exponentially many). Furthermore, computing AC-unifiers is not needed at all. A clause C〚 T〛 is redundant if the constraint T is not AC-unifiable. If C is the empty clause this has to be decided to know whether C〚 T 〛 denotes an inconsistency. In all other cases any sound method to detect unsatisfiable constraints can be used.