A Tree Automata Theory for Uni cation Modulo Equational Rewriting

An extension of tree automata framework, called equational tree automata, is presented. This theory is useful to deal with uni cation modulo equational rewriting. In the manuscript, we demonstrate how equational tree automata can be applied to uni cation problems. 1 Equational Tree Languages Uni cation modulo equational theory is a central topic in automated reasoning. Tree automata are the powerful technique for handling uni cation modulo rewriting [2]. On the other hand, to model some network security problems like Di e-Hellman key exchange algorithm, rewrite rules and equations (e.g. associativity and commutativity axioms) have to be separately dealt with in the underlying theory, but it causes the situation where the standard tree automata technique is useless. In our recent papers [5, 6], we have proposed an extension of tree automata, which is called equational tree automata. This framework subsumes Petri nets (Example 1). In a practical example, equational tree automata can be used to verify a security problem of Di e-Hellman protocol (Example 2). We start this section with basics of tree automata and the equational extension. A tree automaton (TA for short) A is de ned by the 4-tuple (F ;Q;Q n ; ): each of those components is a signature F (a nite set of function symbols with xed arities), a nite set Q of states (special constants with F\Q = ?), a subset Q n of Q consisting of so-called nal states and a nite set of transition rules in the following form: { f(p 1 ; : : : ; p n )! t for some f 2 F with arity(f) = n and p 1 ; : : : ; p n 2 Q and t 2 T (ffg [ Q). Note that function symbols, except state symbols, in the right-hand side must be the same as one in the left-hand side. An equational tree automaton (ETA for short) A=E is the combination of a TA A and an equational system (a set of equations; ES for short) E over the same signature F . An ETA A=E is called { regular if the right-hand side t is a single state q, { monotone if the right-hand side t is a single state q or a term f(q 1 ; : : : ; q n ) for every transition rule f(p 1 ; : : : ; p n ) ! t in . Note that equational tree automata de ned in [5, 6] are in the above monotone case. A term t is accepted by A=E if t 2 T (F) and t ! A=E q for some q 2 Q n . Elements in the set L(A=E) are ground terms accepted by A=E . A tree language (TL for short) L over F is some subset of T (F). A TL L is E-recognizable if there exists A=E such that L = L(A=E). Similarly, L is called E-monotone (E-regular) if A=E is monotone (regular). If L is E-recognizable with E = ?, we say L is recognizable. Likewise, we say L is monotone (regular) if L is ?-monotone (?-regular). We say A=E is a C-TA (A-TA, AC-TA) if E = C (E = A, E = AC, respectively). Lemma 1. Every C-recognizable tree language is regular. Proof. Let F be the signature. Suppose L is a tree language recognizable with a C-TA A=C, where A = (F ;Q;Q n ; ). We de ne B = (F ;Q;Q n ; 0 ), where 0 = f(p 1 ; : : : ; p n )! q f(q 1 ; : : : ; q n )! r 2 such that f(p 1 ; : : : ; p n ) C f(q 1 ; : : : ; q n ) and r ! A=C q : Then it can be proved that the regular TA B recognizes L. ut Lemma 2. The following language hierarchy holds if E = A: regular TL ( E-regular TL ( E-monotone TL ( E-recognizable TL Proof. For the rst inclusion, we take the tree language L = ft j jtj a = jtj b g over the signature F = ff; a; bg, where arity(f) = 2 and a; b are constant symbols. If F A = ffg then L is A-regular (Lemma 8, [5]), but is not regular. The second inclusion is proved in [6]. For the third inclusion we suppose F = F 0 [ ffg with F A = ffg. Here F 0 denotes a set of constant symbols. Then, a (word) language W over F 0 is context-sensitive if and only if an A-monotone tree language is maximal for W . A tree language L is called maximal for a language W if for all terms t in T (F), leaf(t) 2 W if and only if t 2 L. Similarly, it holds that a language W is recursively enumerable if and only if an A-recognizable tree language is maximal for W . It is known that recursively enumerable languages strictly include context-sensitive languages. ut Question 1. Does the above hierarchy hold also for E = AC? Note that the rst inclusion is trivial. Kudlek and Mitrana [3] showed in word case that multiset regular languages and multiset arbitrary languages are closed under homomorphisms and substitutions but multiset monotone languages are not. However, it does not imply the strict hierarchy of tree case, because of di erent de nitions. 2 AC-Tree Automata for Uni cation Problems In this section we discuss the applications of equational tree automata for unication problems. Our examples rely on the following decidability result. Theorem 1 (Reachable property problem). Given a ground AC-TRSR=AC and tree languages L 1 ; L 2 over F with F AC . If L 1 and L 2 are AC-recognizable tree languages, it is decidable whether there exist some s in L 1 and t in L 2 such that s! R=AC t, i.e. (! R=AC )[L 1 ] \ L 2 6= ? is a computable question.