Termination criteria for tree automata completion

Abstract

This paper presents two criteria for the termination of tree automata completion. Tree automata completion is a technique for computing a tree automaton recognizing or over-approximating the set of terms reachable w.r.t. a term rewriting system. The first criterion is based on the structure of the term rewriting system itself. We prove that for most of the known classes of linear rewriting systems preserving regularity, the tree automata completion is terminating. Moreover, it outputs a tree automaton recognizing exactly the set of reachable terms. When the term rewriting system is outside of such classes, the set of reachable terms can be approximated using a set of equations defining an abstraction. The second criterion, which holds for any left-linear term rewriting system, defines sufficient restrictions on the set of equations for the tree automata completion to terminate. We then show how to take advantage of this second criterion to use completion as a new static analysis technique for functional programs. Some examples are demonstrated using the Timbuk completion tool.

Verifying Higher-Order Functions with Tree Automata

“…Termination of the tree automata completion algorithm is not ensured in general [19]. For instance, if R * (L(A)) is not regular, it cannot be represented as a tree automaton.…”

“…Equations make TAC powerful enough to verify first-order functional programs [19]. However, state-of-the-art TAC has two short-comings.…”

“…It has been shown in [19] that it is possible to tune the precision of the approximation. For a given TRS R, initial state automaton A and set of equations E, the termination of the completion algorithm is undecidable in general, even with the use of equations.…”

“…In this section, we show that termination of the completion algorithm with a set of equations E is ensured under the following conditions: if (i) A k is reduced -free and deterministic (written REFD in the rest of the paper) for all k; (ii) every term of A k can be rewritten into a term of a given language L ⊆ T (F) using R (for instance if R is terminating); (iii) L has a finite number of equivalence classes w.r.t E. Completion is known to preserve -reduceness and -determinism if E ⊇ E r ∪ E R [19] where …”

“…The tree automaton completion technique is one analysis technique able to verify first-order Java programs [4]. Until now, the completion algorithm was guaranteed to terminate only in the case of first-order functional programs [19].…”