Propositional Tree Automata

Abstract

Abstract. In the paper, we introduce a new tree automata framework, called propositional tree automata, capturing the class of tree languages that are closed under an equational theory and Boolean operations. This framework originates in work on developing a sufficient completeness checker for specifications with rewriting modulo an equational theory. Propositional tree automata recognize regular equational tree languages. However, unlike regular equational tree automata, the class of propositional tree automata is closed under Boolean operations. This extra expressiveness does not affect the decidability of the membership problem. This paper also analyzes in detail the emptiness problem for propositional tree automata with associative theories. Though undecidable in general, we present a semi-algorithm for checking emptiness based on machine learning that we have found useful in practice.

On the Completeness of Context-Sensitive Order-Sorted Specifications

“…In the left-linear case, we are able to reduce the µ-canonical completeness and µ-sufficient completeness properties to an emptiness problem for Propositional Tree Automata, a class of tree automata introduced in [12] which is closed under Boolean operations and an equational theory. We are further able to use the results of Theorem 5 to have sufficient conditions for showing the µ-semantic completeness of R when R is left-linear, µ-weakly normalizing, µ-canonically complete, ground confluent, and ground sort-decreasing.…”

“…We now define Propositional Tree Automata, first introduced in [12]. We extend the definition of [12] from unsorted signatures to many-sorted signatures.…”

“…We extend the definition of [12] from unsorted signatures to many-sorted signatures. We also use production rules α := f (β 1 , .…”

“…, β n ) → α in the definition to reflect a change in how the rules are interpreted. The definition using rewrite rules in [12] and the definition below are equivalent when the equations in the signature are linear. They are not equivalent, in general, when considering non-linear equations such as idempotence f (x, x) = x.…”

“…Given that the CS programs we consider perform rewriting modulo axioms and are order-sorted, our methods are based on Propositional Tree Automata [12], a kind of equational tree automata, that can take into account both sort information and reasoning modulo axioms. These decision procedures have been implemented in an extension of Maude's Sufficient Completeness Checker (SCC) [10], and we use several Maude programs to illustrate both the basic ideas and the use of SCC in verifying CS completeness properties.…”