  The tree automaton completion is an algorithm used for proving safety
properties of systems that can be modeled by a term rewriting system. This
representation and verification technique works well for proving properties of
infinite systems like cryptographic protocols or more recently on Java Bytecode
programs. This algorithm computes a tree automaton which represents a (regular)
over approximation of the set of reachable terms by rewriting initial terms.
This approach is limited by the lack of information about rewriting relation
between terms. Actually, terms in relation by rewriting are in the same
equivalence class: there are recognized by the same state in the tree
automaton.
  Our objective is to produce an automaton embedding an abstraction of the
rewriting relation sufficient to prove temporal properties of the term
rewriting system.
  We propose to extend the algorithm to produce an automaton having more
equivalence classes to distinguish a term or a subterm from its successors
w.r.t. rewriting. While ground transitions are used to recognize equivalence
classes of terms, epsilon-transitions represent the rewriting relation between
terms. From the completed automaton, it is possible to automatically build a
Kripke structure abstracting the rewriting sequence. States of the Kripke
structure are states of the tree automaton and the transition relation is given
by the set of epsilon-transitions. States of the Kripke structure are labelled
by the set of terms recognized using ground transitions. On this Kripke
structure, we define the Regular Linear Temporal Logic (R-LTL) for expressing
properties. Such properties can then be checked using standard model checking
algorithms. The only difference between LTL and R-LTL is that predicates are
replaced by regular sets of acceptable terms.
