Reachability Analysis with State-Compatible Automata

Abstract

Abstract. Regular tree languages are a popular device for reachability analysis over term rewrite systems, with many applications like analysis of cryptographic protocols, or confluence and termination analysis. At the heart of this approach lies tree automata completion, first introduced by Genet for left-linear rewrite systems. Korp and Middeldorp introduced so-called quasi-deterministic automata to extend the technique to nonleft-linear systems. In this paper, we introduce the simpler notion of state-compatible automata, which are slightly more general than quasideterministic, compatible automata. This notion also allows us to decide whether a regular tree language is closed under rewriting, a problem which was not known to be decidable before. Several of our results have been formalized in the theorem prover Isabelle/HOL. This allows to certify automatically generated non-confluence and termination proofs that are using tree automata techniques.

CSI: New Evidence – A Progress Report

“…For non-confluence CeTA can check that, given derivations s → * t 1 and s → * t 2 , t 1 and t 2 cannot be joined. Here the justifications used by CSI are: using tcap [44] (i.e., test that tcap(t 1 σ) and tcap(t 2 σ) are not unifiable), and reachability analysis using tree automata [11]. Experimental results for certified confluence analysis are presented in Section 7.…”

Proving Looping and Non-Looping Non-Termination by Finite Automata

“…Our approach can be summarized as searching for non-termination proofs based on regular (tree) automata. Regular (tree) automata have been fruitfully applied to a wide rage of properties of term rewriting systems: for proving termination [26,21,28], infinitary normalization [12], liveness [29], and for analyzing reachability and deciding the existence of common reducts [24,13]. Local termination on regular languages, has been investigated in [9].…”

“…In [24] it has been shown that this monotonicity property is strong enough to characterize and decide the closure of the regular languages under rewriting. In particular, the language of a deterministic tree automaton is closed under rewriting if and only if there exists such a monotonic quasi-order on the states.…”

The Certification Problem Format

“…The former by joinability of critical pairs and the latter reusing all the available machinery for termination techniques. For nonconfluence CPF admits the syntactic criteria mentioned in [31], the tree-automata based techniques of [10], and the methods based on interpretations and orders of [1].…”