Proving and Computing: Applying Automated Reasoning to the Verification of Symbolic Computation Systems

The application of automated reasoning to the formal verification of symbolic computation systems is motivated by the need of ensuring the correctness of the results computed by the system, beyond the classical approach of testing. Formal verification of properties of the implemented algorithms require not only to formalize the properties of the algorithm, but also of the underlying (usually rich) mathematical theory. We show how we can use ACL2, a first-order interactive theorem prover, to reason about properties of algorithms that are typically implemented as part of symbolic computation systems. We emphasize two aspects. First, how we can override the apparent lack of expressiveness we have using a first-order approach (at least compared to higher-order logics). Second, how we can execute the algorithms (efficiently, if possible) in the same setting where we formally reason about their correctness. Three examples of formal verification of symbolic computation algorithms are presented to illustrate the main issues one has to face in this task: a Gr¨obner basis algorithm, a first-order unification algorithm based on directed acyclic graphs, and the Eilenberg-Zilber algorithm, one of the central components of a symbolic computation system in algebraic topology