Simplification by Cooperating Decision Procedures

Abstract

A method for combining decision procedures for several theories into a single decision procedure for their combination is described, and a simplifier based on this method is discussed. The simplifier finds a normal form for any expression formed from individual variables, the usual Boolean connectives, the equality predicate =, the conditional function if-then-else, the integers, the arithmetic functions and predicates +, -, and _<, the Lisp functions and predicates car, cdr, cons, and atom, the functions store and select for storing into and selecting from arrays, and uninterpreted function symbols. If the expression is a theorem it is simplified to the constant true, so the simplifier can be used as a decision procedure for the quantifier-free theory containing these functions and predicates. The simplifier is currently used in the Stanford Pascal Verifier.

Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures

“…For instance, 5 Tinelli and Zarba [26] have shown how to combine an arbitrary theory with one satisfying requirements which are stronger than stableinfiniteness. Thus, contrary to the combination schema by Nelson-Oppen [16], such a schema is asymmetric in the sense that the requirements on the component theories are not the same.…”

Formal Verification of a Combination Decision Procedure

“…There are two basic methods for building decision procedures for combinations of disjoint theories. Nelson and Oppen's method [NO79] combines decision procedures for the individual theories by allowing them to share specific kinds of equality information. Shostak's method [Sho84] extends congruence closure to equational theories that are canonizable and solvable.…”

Improving the Efficiency of Reasoning Through Structure-Based Reformulation

“…Work on theorem proving has focused on decomposition for parallel implementations [8,5,15,43] and has followed decomposition methods guided by lookahead and subgoals, neglecting the types of structural properties we used here. Another related line of work focuses on combining logical systems (e.g., [32,40,3,36,44]). Contrasted with this work, we focus on interactions between theories with overlapping signatures, the efficiency of reasoning, and automatic decomposition.…”