Sufficient Completeness Checking with Propositional Tree Automata

Su cient completeness means that enough equations have been speci ed, so that the functions of an equational speci cation are fully de ned on all relevant data. This is important for both debugging and formal reasoning. In this work we extend su cient completeness methods to handle expressive speci cations involving: (i) partiality; (ii) conditional equations; and (iii) deductionmodulo axioms. Speci cally, we give useful characterizations of the su cient completeness property for membership equational logic (MEL) speci cations having features (i){ (iii). We also propose a kind of equational tree automata [18, 22], called propositional tree automata (PTA) and identify a class of MEL speci cations (called PTA-checkable) whose su cient completeness problem is equivalent to the emptiness problem of their associated PTA. When the reasoning modulo involves only symbols that are either associative and commutative (AC) or free, we further show that the emptiness of AC-PTA is decidable, and therefore that the su cient completeness of AC-PTAcheckable speci cations is decidable. The methods presented here can serve as a basis for building a next-generation su cient completeness tool for MEL speci cations having features (i){(iii). These features are widely used in practice, and are supported by languages such as Maude and other advanced speci cation and equational programming languages.