Characterizing Conclusive Approximations by Logical Formulae

Abstract

Abstract. Considering an initial set of terms E, a rewriting relation R and a goal set of terms Bad, reachability analysis in term rewriting tries to answer to the following question: does there exists at least one term of Bad that can be reached from E using the rewriting relation R? Some of the approaches try to show that there exists at least one term of Bad reachable from E using the rewriting relation R by computing the set of reachable terms. Some others tackle the unreachability problem i.e. no term of Bad is reachable by rewriting from E. For the latter, over-approximations are computed. A main obstacle is to be able to compute an over-approximation precise enough that does not intersect Bad i.e. a conclusive approximation. This notion of precision is often defined by a very technical parameter of techniques implementing this over-approximation approach. In this paper, we propose a new characterization of conclusive approximations by logical formulae generated from a new kind of automata called symbolic tree automata. Solving a such formula leads automatically to a conclusive approximation without extra technical parameters.

Termination criteria for tree automata completion

“…Here our termination proof holds for any left-linear TRS provided that the set of equations satisfy some properties. The approach followed by [37,18] is very different. Starting from the TRS, the set of initial terms and a set of "bad" terms that should not be reachable, they tend to directly characterize a correct approximation automaton by constraint solving for the first and by finite model generation for the second.…”

“…Our first criterion permits to build exactly an automaton recognizing sets of reachable terms and does not relies on "bad" terms. In this case, algorithms of [37,18] cannot be applied. Our second criterion does not either require to define "bad" terms.…”

“…prove that delete(a,[(a|b)*])= [b*] means proving that from delete(a,[(a|b)*]) we cannot reach any constructor term in the complement of the language [b*]. The strength of [37,18] is that it does not require to invent approximation equations to prove it. Its weakness is that it may not terminate: there may not exists a regular overapproximation where "bad" terms are not reachable [38] and thus satisfiability checking may diverge.…”

“…Its weakness is that it may not terminate: there may not exists a regular overapproximation where "bad" terms are not reachable [38] and thus satisfiability checking may diverge. Moreover, when the TRS to analyze is big, the satisfiability checking problem becomes very difficult in practice [37,18]. In the last paper, the reverse([a+,b+])=[b+,a+]) problem is also studied and cannot be proven directly.…”

Equational Abstraction Refinement for Certified Tree Regular Model Checking

“…Completion with equational abstraction has been applied to very complex case studies such as the verification of (industrial) cryptography protocols [26,28] and Java bytecode applications [13]. CEGAR algorithms based on equational-abstraction completion exist [11,12], but are known to be inefficient.…”