OBJ as a Theorem Prover with Applications to Hardware Verification

Abstract

This paper states, justifies, and illustrates some new techniques for proving theorems with a functional programming language, like OBJ, that has a rigorous algebraic semantics. These techniques avoid the complexities of both higher order logic and Knuth-Bendix completion, and instead support a style of user interaction in which: (1) OBJ’s powerful reduction engine does the routine work automatically, in such a way that partially successful proofs return information that often suggests what to try next; and (2) OBJ’s flexible facilities for hierarchical and generic modules describe complex proof strategies in a style that is familiar from experience with programming languages. New results in this paper include: a simple extension of first order equational logic that allows universal quantification over functions instead of just over ground elements; a technique for eliminating both first and second order universal quantifiers; a completeness theorem; a technique for transforming conditional equations to unconditional equations; some very useful structural induction principles; and some techniques for reasoning about parameterized modules. These tools for reasoning about (first order) functions are powerful enough to justify algorithms that automatically generate OBJ code for verifying some rather broad classes of (second order) assertions. The paper features some hardware verification examples, but also includes some inductive proofs for the natural numbers and an OBJ-style parameterized module verification. Parameterized module verification supports a flexible style of proof reuse, and this example also illustrates the rigorous treatment of exceptions in OBJ. The hardware verification examples illustrate what seems a very promising approach to both combinational and sequential circuits, using standard gates and/or bidirectional components (such as MOS transistors), possibly with “don’t care” conditions. Order sorted algebra plays an important role in many of the examples.