Tree Automata and Term Rewrite Systems (Extended Abstract) Sophie Tison LIFL, Bˆ at M3, Universit´e Lille 1 F59655 Villeneuve d’Ascq cedex, France
[email protected]
Abstract. This tutorial is devoted to tree automata. We will present some of the most fruitful applications of tree automata in rewriting theory and we will give an outline of the current state of research on tree automata. We give here just a sketch of the presentation. The reader can also refer to the on-line book “Tree Automata and Their Applications” [CDG+ 97].
1
Introduction
Tree Automata theory and term rewriting theory are strongly connected [Dau94, Ott99]. On one side, tree automata can be viewed as a subclass of ground term rewrite systems [Bra69a, Bra69b]. On the other hand, tree automata have been used successfully as decision tools in rewriting theory. In this tutorial, we will present some of the most fruitful applications of tree automata in rewriting theory and we point out some promising research directions in this area. For definitions and properties of tree automata the reader can refer to [GS96, CDG+97]. Most of the results we mention here and more references can be found in [CDG+97].
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Classical Tree Automata and Rewrite Systems
If you want to use tree automata in rewriting theory, the ideal situation occurs when the reducibility relation is recognizable: a binary relation is said recognizable Iff the set of its encodings is a recognizable tree language; a couple is just encoded by overlapping its two terms: e.g. the [f(a,b), f(f(a,a,),a)] will be encoded into [f,f] ([a,f]([⊥,a],[⊥,a]),[b,a]). E.g., reducibility relations are recognizable for ground rewrite systems (more generally for linear term rewriting system such that left and right members of the rules do not share variables). Now, let us consider the following logical theory: the set of formulas is the set of all firstorder formulas using no function symbols and a single binary predicate symbol, the predicate symbol is interpreted as the reducibility relation associated with a given rewrite system. When the reducibility relation is recognizable, you get easily the decidability of the theory, thanks to the good closure and decision properties of tree automata; this implies decidability of any property expressible L. Bachmair (Ed.): RTA 2000, LNCS 1833, pp. 27–30, 2000. c Springer-Verlag Berlin Heidelberg 2000
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Sophie Tison
in this theory, like confluence. Furthermore, under some counditions, you can enrich the theory for expressing termination properties [DT90]. Clearly, recognizability of the reducibility relation is a very strong property restricted to limited subclasses of rewrite systems. Now, you can just require that the reducibility relation preserves recognizability, i.e. that the set of descendants (resp. the ancestors) of a recognizable tree language is recognizable. (Let us note that this is not the case even for linear systems). For example, reachability can then be easily reduced to membership to recognizable tree language. If preservation of regularity is undecidable [GT95], some conditions ensure it and it leads to decidability of reachability and joinability for some subclasses of term rewriting systems. ([Sal88, CDGV94, Jac96, FJSV98, NT99]). You can also require recognizability of the language of ground normal forms: it provides for example a very simple procedure for testing the ground reducibility of a term. Clearly, the set of ground normal forms is a regular tree language for left-linear rewrite systems. Moreover, recognizability of the set of normal forms has been proven decidable [VG92, Kou92]. Finally, recognizability of the set of normalizable terms (it’s clearly ensured when the set of ground normal forms is recognizable and the inverse reducibilty relation preseves recognizability) ensures decidability of the sequentiality of the system, when left-linear [Com95]. All the previous approaches provide good decision procedures but only for very restricted classes of rewrite systems. If you are interested in one particular rewrite system, you can also try to prove “experimentally” its good behavior w.r.t. to recognizability. E.g. J. Waldmann has proven by computer the recognizability of the set of normalizing S-terms [Wal98]. Some pointers to software for manipulating tree regular expressions and automata can be found in [CDG+97]. But a question rises: How far can we go in using tree automata when describing properties of term rewriting systems? Can we find new “interesting” classes of t.r.s. with good properties w.r.t. recognizability?
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How to Go beyond the Limits of Usual Tree Automata?
A way of going beyond the limits of the previous approaches is to consider approximation of rewrite systems. For example, Comon and Jacquemard study by these means reduction strategies and sequentiality [Com95, Jac96]. More recently, T. Genet and F. Klay compute regular over-approximations of the set of the descendants [Gen98] and use them for the verification of cryptographic protocols [GK00]. But to go beyond the limits of tree automata, you can also use extensions of tree automata. The idea is roughly to enrich the notion of tree automata for dealing with the non-linearity while keeping good closure and decision properties. Several classes have been defined in this view and have been sucessfully applied in term rewriting theory. For example, the reduction automata provide decision procedures for emptiness and finiteness of the language of ground normal forms for every term rewriting system and they give a new procedure for testing ground reducibility [Pla85, DCC95, CJ97, CJ94]. Tree automata with tests between
Tree Automata and Term Rewrite Systems
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brothers which have good decision properties allow also to get some new results for non-linear rewrite systems [CSTT99, STT99]. Let us finally cite also the powerful notion of tree t-uple synchronized grammars which has been used in unification theory end rewriting theory [LR98, LR99]. Of course, you can combine these two last approaches, e.g. by approximating the set of the descendants by extended recognizable tree languages. This opens new prospects and will require design of software for dealing with extended tree automata.
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