Combining Shostak Theories

Abstract

Ground decision procedures for combinations of theories are used in many systems for automated deduction. There are two basic paradigms for combining decision procedures. The Nelson-Oppen method combines decision procedures for disjoint theories by exchanging equality information on the shared variables. In Shostak’s method, the combination of the theory of pure equality with canonizable and solvable theories is decided through an extension of congruence closure that yields a canonizer for the combined theory. Shostak’s original presentation, and others that followed it, contained serious errors which were corrected for the basic procedure by the present authors. Shostak also claimed that it was possible to combine canonizers and solvers for disjoint theories. This claim is easily verifiable for canonizers, but is unsubstantiated for the case of solvers. We show how our earlier procedure can be extended to combine multiple disjoint canonizable, solvable theories within the Shostak framework.

This work was funded by NSF Grant CCR-0082560, DARPA/AFRL Contract F33615-00-C-3043, and NASA Contract NAS1-00079. During a phone conversation with the first author on 2nd April 2001, Rob Shostak suggested that the problem of combining Shostak solvers could be solved through variable abstraction. His suggestion is the key inspiration for the combination of Shostak theories presented here. We thank Clark Barrett, Sam Owre, and Ashish Tiwari for their meticulous reading of earlier drafts. We also thank Harald Ganzinger for pointing out certain limitations of our original definition of solvability with respect to σ-models. The first author is grateful to the program committees and program chairs of the FME, LICS, and RTA conferences at FLoC 2002 for their kind invitation.