M. Dauchet, S. Tison, The Theory of Ground Rewrite Systems is Decidable, IEEE Symposium on Logic in Computer Science, Philadelphia, PA, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....each rigid equation either contains one variable, or has a ground left hand side and an equality between two variables as a right hand side. We show that SREU is decidable also in this restricted case. The proof is by reduction to the decidable first order theory of ground rewrite systems, or GRS [10]. In Section 7 we summarize the current status of SREU and list some open problems. 2 Preliminaries We will first establish some notation and terminology. We follow Chang and Keisler [4] regarding first order languages and structures. For the purposes of this paper it is enough to assume that ....

....two variables x and y. SREU restricted to systems with the united one variable property is called united one variable SREU. The main result of this section is that the united one variable SREU is decidable. The proof is by reduction to the decidable first order theory of ground rewrite systems [10]. 5.1 The Decidable Theory GRS Now we formally define the theory of ground rewrite systems or GRS. Consider a signature Sigma that contains all the function symbols and constants that we are going to need in the sequel. Let Gamma be the following signature constructed from Sigma . For ....

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M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. IEEE Conference on Logic in Computer Science (LICS), pages 242--248. IEEE Computer Society Press, 1990.

....techniques to decide certain properties of (ground) rewrite systems. Decidability of con uence of ground term rewriting systems was rst shown in [16] using tree automata techniques. Subsequently, the result was improved and the ( rstorder) theory of ground rewrite system was shown to be decidable [17]. The approach in this paper further re nes the notion of a ground tree transducer by additional conditions that make the rewrite relation convergent . It is then easily observed that reachability problem for ground term rewriting systems has a polynomial time solution, and so does the problem ....

....closure, we can show that we can reach a state consisting of all persisting rules using derivations of length O(n 2 n c 1 ) where n is the size of the input and c is the maximum arity of any symbol in . It is known that the rst order theory of ground rewrite systems is decidable [17]. The proof is based on associating a ground tree transducer with the rewrite relation induced by a ground rewrite system and associating inductively an automaton of a more general class (RR) with any formula. Existence of a nite rewrite closure for an arbitrary set of ground equations and rules ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc of the 5th IEEE Symposium on Logic in Computer Science, pages 242-248, Philadelphia, PA, June 1990. IEEE Computer Society Press.

....language. This was proven in [Buc64] and extended in [Cau92] Applications to the modelchecking of pushdown automata have been proposed in [FWW97, BEM97] The decidability of the first order theory of the rewrite relation induced by a ground term rewrite system relies on ground tree transducers [DT90] note that PA is defined by a conditional ground rewrite system) 2 Among the applications we develop for our regularity theorems, several have been suggested by Mayr s work on PA [May97c, May97b] and or our earlier work on RPPS [KS97a, KS97b] 1 Regular tree languages and tree automata We ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th IEEE Symp. Logic in Computer Science (LICS'90), Philadelphia, PA, USA, June 1990, pages 242--248, 1990.

.... pushdown automata have been proposed in [FWW97,BEM97] A parallel variant exists: the reachable con gurations of a BPP process form a semilinear set [Esp97] The transitive closure of the rewrite relation induced by a ground term rewrite system is recognizable by ground tree transducers [DT90] Note that PA is de ned by a conditional ground rewrite system, and in fact the induced reachability relation is a rational tree relation in the sense of Raoult [Rao97] see x 5.3) Among the applications we develop for our regularity theorems, several have been suggested by Mayr s work on PA ....

....(L) A natural question is to ask whether the relation (i.e. f(t; u) j t ug, a subset of E PA E PA ) is recognizable in some sense. Ground tree transducers. For this question, the most relevant notion of recognizability is based on ground tree transducers, GTT s for short, see [DT90,CDG 99] for details. It can be shown that the relation induced by a ground rewrite system is recognizable by a GTT. In the case of PA processes, the rules are ground rewrite rules with simple left hand sides, but with a contextual restriction on when a rule may be applied (re ecting the ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th IEEE Symp. Logic in Computer Science (LICS'90), Philadelphia, PA, USA, June 1990, pages 242-248, 1990.

....with abstraction ff 1 is the following: Norm ff 1 (f(g(q 1 ; f(a) q 0 ) ff(q 2 ) q 0 ; g(q 1 ; q 3 ) q 2 ; f(q 4 ) q 3 ; a q 4 g. 2 Approximation Technique For a regular set of terms E T (F) although there exists some restricted classes of TRSs R such that R (E) is regular (see [5, 21, 4, 12]) this is not the case in general [11, 12] In [9] for any tree automaton A (s.t. L(A) E) and for any leftlinear TRS R, it is proposed to build an approximation automaton TR (A) such that L(TR (A) R (E) The quality of the approximation highly depends on an approximation function ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th LICS Symp., Philadelphia (Pa., USA), pages 242--248, June 1990.

....of properties like encompassment, known to be decidable due to (Caron et al. 1993, Dauchet, Caron Coquid e 1995) would follow from the general decidability of theories of one step rewriting. Recall also that the first order theories of one step rewriting in finite ground systems are decidable (Dauchet Tison 1990). On the other hand, the transitive closure of the one step reducibility relation seems to be inexpressible in the theories of one step rewriting (the opposite would immediately lead to their undecidability) All these facts motivated the quest for the solution to the above problem and for the ....

Dauchet, M. & Tison, S. (1990), The theory of ground rewrite systems is decidable, in `Proc 5th IEEE Symp Logic in Computer Science', pp. 242-- 256.

....2 F such that w E 7 w 0 g. Buchi [3] has studied word rewriting systems that rewrite only at the end of words, that is to say ground rewriting systems for words. Specifically, he has proved that the language obtained by all possible derivations from a word is recognizable. Dauchet and al. [6] have studied ground term rewriting systems ; they have introduced the notion of ground tree transducer (GTT) which point out the link between ground rewriting and automata. In 3 particular, the rewriting relation E 7 is equivalent to a GTT so if F is a regular set of words then Im(F ) is ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. pages 242--248. LICS 90, 1990.

....with abstraction ff 1 is the following: Norm ff 1 (f(g(q 1 ; f(a) q 0 ) ff(q 2 ) q 0 ; g(q 1 ; q 3 ) q 2 ; f(q 4 ) q 3 ; a q 4 g. 2 Approximation Technique For a regular set of terms E T (F) although there exists some restricted classes of TRSs R such that R (E) is regular (see [5, 21, 4, 12]) this is not the case in general [11, 12] In [9] for any tree automaton A (s.t. L(A) E) and for any left linear TRS R, it is proposed to build an approximation automaton TR (A) such that L(TR (A) R (E) The quality of the approximation highly depends on an approximation function ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings 5th IEEE Symposium on Logic in Computer Science, Philadelphia (Pa., USA), pages 242248, June 1990.

....2 C Q 0 t , and if c 2 C Q 0 t then t c s 0 for some terminated s 0 such that Q 0 t s 0 . The proof (omitted) is by structural induction and is similar to the proof of the regularity theorems in [LS99] 8 4 Tree automata and n ary relations Products of trees. We follow [DT90] Given two terms s; t 2 TF , the pair (s; t) can be seen as one term over a product alphabet F def = F[f g) F[f g) f g where is a new symbol with arity 0. In F the arity of fg is the maximum of the arities of f and g. Formally we de ne s t as the term in TF given recursively by f(s ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th IEEE Symp. Logic in Computer Science (LICS'90), Philadelphia, PA, USA, June 1990, pages 242-248, 1990.

....such that Q t s 0 . 3. Q 0 t s i t s and s is terminated. Furthermore, if t c s then c 2 C Q 0 t , and if c 2 C Q 0 t then t c s 0 for some terminated s 0 such that Q 0 t s 0 . 4 Tree automata and n ary relations Products of trees. We follow [DT90] Given two terms s; t 2 TF , the pair (s; t) can be seen as one term over a product alphabet F def = F [f g) F[f g) f g where is a new symbol with arity 0. In F the arity of fg is the maximum of the arities of f and g. Formally we de ne s t as the term in TF given recursively by f(s ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th IEEE Symp. Logic in Computer Science , pages 242-248, 1990.

....with abstraction ff 1 is the following: Norm ff 1 (f(g(q 1 ; f(a) q 0 ) ff(q 2 ) q 0 ; g(q 1 ; q 3 ) q 2 ; f(q 4 ) q 3 ; a q 4 g. 2 Approximation Technique For a regular set of terms E T (F) although there exists some restricted classes of TRSs R such that R (E) is regular (see [5, 21, 4, 12]) this is not the case in general [11, 12] In [10] for any tree automaton A (s.t. L(A) E) and for any left linear TRS R, it is proposed to build an approximation automaton TR (A) such that L(TR (A) R (E) The quality of the approximation highly depends on an approximation function ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th LICS Symp., Philadelphia (Pa., USA), pages 242--248, June 1990.

....on words, which is a particular case of ground term rewriting. First, Buchi [3] has studied word rewriting systems that rewrite only at the end of words. Specifically, he has proven that the language obtained by all possible derivations from a word is recognizable. Moreover, Dauchet and al. [5] have studied ground term rewriting systems ; they have introduced the notion of ground tree transducer (GTT) which point out the link between ground rewriting and automata. With GTT, they have proven the decidability of the theory of ground term rewriting systems. It is easy to prove with GTT ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. pages 242--248. LICS 90, 1990.

.... validation, the problem of expressing by a nite tree automaton the transitive closure of a regular set E of ground terms with respect to an equational system, as well as the related problem of expressing the set of descendants of E with respect to a rewrite system, have already been investigated [1, 5, 13, 4, 9] 1 . All those papers assume that the right hand sides (both sides when dealing with equational systems) of rewrite rules are shallow, up to slight di erences. Shallow means that every variable appears at depth at most one. On the other hand, the possibility of approximating the set of ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc., Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242-248, Philadelphia, Pennsylvania, 1990. IEEE Computer Society Press.

....which have been considered steps towards the decidability of one step rewriting. ffl In case of a signature containing constants and unary function symbols only (so called string rewriting systems or semi Thue systems) the decidability of the theory of one step rewriting is a consequence of [DT90]. This decidability result can also be obtained by a direct translation into WS1S [Jac95] which is known to be decidable (see [Tho90] for a survey) ffl The first order theory of rewriting by a ground term rewriting system has been shown decidable in [DT90] To be precise, for a given ground ....

.... one step rewriting is a consequence of [DT90] This decidability result can also be obtained by a direct translation into WS1S [Jac95] which is known to be decidable (see [Tho90] for a survey) ffl The first order theory of rewriting by a ground term rewriting system has been shown decidable in [DT90]. To be precise, for a given ground term rewriting system R (that is all left and right hand sides of the rules are ground) the structure of all ground terms, where all terms are available as constants, with the predicates x y , x y and x jj y (parallel one step rewriting) is ....

Max Dauchet and Sophie Tison. The theory of ground rewrite systems is decidable. In Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242--256. IEEE Computer Society Press, 1990.

....and a procedure for building a regular tree grammar (resp. a tree automaton) producing (resp. recognising) IRR(R) can be found in [CR87] However, R (E) is not necessarily a regular tree language, even if E is. The language R (E) is regular if E is regular and if R is either a ground TRS [DT90] a right linear and monadic TRS [Sal88] a linear and semi monadic TRS [CDGV91] or an inversely growing TRS [Jac96] where inverselygrowing means that every right hand side is either a variable, or a term f(t 1 ; t n ) where f 2 F , ar(f) n, and 8i = 1; n, t i is a ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings 5th IEEE Symposium on Logic in Computer Science, Philadelphia (Pa., USA), pages 242--248, June 1990.

....reduct (are joinable ) Is unicity of normal forms (UN) a modular property of standard conditional systems 2.3 Confluence Problem 12. What is the complexity of the decision problem for the confluence of ground (variable free) term rewriting systems Decidability was shown in [22, 78] see also [23]. Problem 13 (J. J. L evy) By a lemma of G. Huet [38] left linear term rewriting systems are confluent if, for every critical pair t s (where t = u[roe] u[loe] g d = s, for some rules l r and g d) we have t k s (t reduces in one parallel step to s) The condition t k s ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings of the Fifth IEEE Symposium on Logic in Computer Science, pp. 242--248, Philadelphia, PA, June 1990.

....subtrees u 1 ; u k of t 1 by subtrees v 1 ; v k , respectively, where for every i there exists j such that (u i ; v i ) 2 (L j ; R j ) In the terminology of [DauTis1, DHLT] every ground term rewriting system can be simulated by a ground tree transducer. This result was used in [DauTis1, DHLT, DauTis2] to give an elegant proof of the decidability of confluence of a ground term rewriting system (also proved in [Oya] and, more generally, of the decidability of the first order theory of ground term rewriting. At the end of the paper we discuss this decidability result, together with the ....

....the remainder of this section we discuss confluence and termination of ground rewrite systems. An extended ground rewrite system P over Sigma is confluent if for all trees t; u; v 2 T Sigma with t P u and t P v, there is a tree w 2 T Sigma such that u P w and v P w. In [DauTis1, DHLT, DauTis2] it is shown on the basis of Theorem 5 that confluence is decidable for extended ground rewrite systems. The nicest proof is the one in [DauTis2] where it is even shown that the firstorder theory of extended ground rewrite systems is decidable. This first order theory includes properties such as ....

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M.Dauchet, S.Tison; The theory of ground rewrite systems is decidable, Proc. 5th Ann. IEEE Symp. on Logic in Computer Science (LICS), Philadelphia, 1990, pp.242--248

....of ground axioms (i.e. equations without variables) there is a conservative extension T (F 0 ) E 0 of T (F ) E where E 0 is quasi free. This shows that the first order theory of T (F ) E is decidable in this case. This should not be confused with results on the theory of ground systems [DT90]. In the latter case, the structure considered is indeed richer in one sense (there are predicate symbols other than equality) but poorer in some other respects (no function symbols, no equality predicate) Our decidability results are proved by rewriting the formulas in equivalent formulas ....

Max Dauchet and Sophie Tison. The theory of ground rewrite systems is decidable. Research Report IT 182, Laboratoire d'Informatique Fondamentale de Lille, Universit'e des Sciences et Techniques de Lille Flandres Artois, France, March 1990. Also in Proc. 5th IEEE LICS, Philadelphia.

....2 Similarly, we have the decidability of NV sequentiality (as defined in [17] it suffices to start with T instead of starting the construction with NF (R) Note that this gives a much simpler proof than in [17] and in a more general case. We could also prove this result using ideas similar to [5, 6]: 4.3.1 An indirect (very short) proof of lemma 21: Using a construction similar to that of lemma 14, the relation Gamma Gamma R V is recognized by a Ground Tree Transducer, as defined in [5] and hereafter called GTT. Such a construction is only valid for rewrite systems such that the right ....

....and left hand sides do not share variables. As shown in [5] the class of binary relations which are accepted by a GTT is closed under transitive closure: Gamma Gamma R V is recognized by a GTT, hence recognizable as a set of pairs: Gamma Gamma R V is definable in WSkS (see e.g. [6]) Now, NF (R) is also definable in WSkS (lemma 16) hence NR V is also definable in WSkS. 2 Note the tricks of this proof: there are (at least) three notions of recognizability for sets of pairs of trees (i.e. binary relations on T [ T Omega ) ffl Rec 1 is the class of cartesian products of ....

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Max Dauchet and Sophie Tison, The theory of ground rewrite systems is decidable, Proc. 5th IEEE Symp. Logic in Computer Science, Philadelphia, 1990.

....2 Similarly, we have the decidability of NV sequentiality (as defined in [17] it suffices to start with T instead of starting the construction with NF (R) Note that this gives a much simpler proof than in [17] and in a more general case. We could also prove this result using ideas similar to [5, 6]: 4.3.1 An indirect (very short) proof of lemma 4.17: From lemma 4.14, the relation Gamma Gamma R V is recognized by a Ground Tree Transducer (as defined in [5] and hereafter called GTT) As shown in [5] the class of binary relations which are accepted by a GTT is closed under transitive ....

.... (as defined in [5] and hereafter called GTT) As shown in [5] the class of binary relations which are accepted by a GTT is closed under transitive closure: Gamma Gamma R V is recognized by a GTT, hence recognizable as a set of pairs: Gamma Gamma R V is definable in WSkS (see e.g. [6]) Now, NF (R) is also definable in WSkS (lemma 4.15) hence NR V is also definable in WSkS. 2 Note the tricks of this proof: there are (at least) three notions of recognizability for sets of pairs of trees (i.e. binary relations on T [ T Omega ) ffl Rec 1 is the class of cartesian products of ....

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M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th IEEE Symp. Logic in Computer Science, Philadelphia, 1990.

....(The restriction to the same alphabet is essential, since confluence is in general not preserved under the addition of function symbols, not even for left linear systems. Problem 63 (M. Oyamaguchi) Is confluence of right ground term rewriting systems decidable Compare [ Oyamaguchi, 1987; Dauchet et al. 1990; Dauchet and Tison, 1990; Oyamaguchi and Ohta, 1993 ] Problem 64. Is confluence of ordered rewriting (using the intersection of one step replacement of equals and a reduction ordering that is total on ground terms) decidable when the (existential fragment of the) ordering is This question was ....

....to the same alphabet is essential, since confluence is in general not preserved under the addition of function symbols, not even for left linear systems. Problem 63 (M. Oyamaguchi) Is confluence of right ground term rewriting systems decidable Compare [ Oyamaguchi, 1987; Dauchet et al. 1990; Dauchet and Tison, 1990; Oyamaguchi and Ohta, 1993 ] Problem 64. Is confluence of ordered rewriting (using the intersection of one step replacement of equals and a reduction ordering that is total on ground terms) decidable when the (existential fragment of the) ordering is This question was raised in [ ....

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M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings of the Fifth Symposium on Logic in Computer Science, pages 242--248, Philadelphia, PA, June 1990.

....systems was shown in [15] In [17] a polynomial time algorithm for reachability of ground systems is given. In [6] the decidability of reachability for ground systems and ground rewriting modulo associativity, commutativity is studied. Decidability of the theory of ground rewriting is studied in [4] and the word problem for right ground systems in [16] 2 Preliminaries We assume familiarity with basic notions of rewriting (see [5, 9] Let V be a countable set of variables and Sigma be a countable set of function symbols with Sigma V = T is the set of all terms of a first order ....

....= property of left linear, right ground systems using ideas from [16] This may be used as a decidable sufficient condition ensuring UN = for left linear systems using approximation techniques. We remark that decidability of UN and UN = for ground rewrite systems follows from the work of [4]. However, their work does not address the complexity issue, and polynomial bounds are generally not possible using tree automata techniques used by [3, 4] because exponential time algorithms for determinization are usually needed. The algorithm makes use of the Nelson Oppen congruence closure ....

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M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. LICS, 1990.

....results for tree theories, as Theorem 3.1 does for theories of successor (i.e, fragments of arithmetic) In [TW68] and [Don70] the weak monadic theory of the binary infinite tree was shown to be decidable, using the decidability of the emptiness problem for tree automata. Dauchet and Tison [DT90] applied tree automata in the spirit of the decidability proof for Presburger arithmetic (as discussed in the previous section) Here an n ary relation of finite trees with label alphabet A is captured by a set of trees over the alphabet A n (possibly extended by a dummy label in the individual ....

....over the alphabet A n (possibly extended by a dummy label in the individual components if tuples of trees with different domains are to be handled) In analogy to the case of word relations, the j th components code the j th tree of the n tuple. Three relations between trees are considered in [DT90], each of them given by a finite tree rewriting system S ( ground rewriting system ) The first relation R 1 collects all tree pairs (s; t) such that t is obtained from s by application of a rule from S, the second relation R 2 contains all pairs (s; t) where such rewriting steps are applied in ....

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M. Dauchet, S. Tison, The theory of ground rewrite systems is decidable, Proc. 5th IEEE Symp. on Logic in Computer Science, 1990, pp. 242-248.

....axioms. Finite automata techniques would then be used in order to deterministically get rid of contexts before to use the rules described here. Considering ground rules rather than equations, all interesting questions are again decidable, e.g. termination [14] confluence [8, 28] and reachability [9]. Moreover, a set of ground equations can always be transformed into a finite convergent set of ground rules in polynomial time [13, 30] Unfortunately, shallow rewriting does not enjoy all properties of ground rewriting: termination is indeed decidable, but the first order theory of the embedding ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. Research Report IT 182, Laboratoire d'Informatique Fondamentale de Lille, Universit'e des Sciences et Techniques de Lille Flandres Artois, France, Mar. 1990. Also in Proc. 5th IEEE Symp. Logic in Computer Science, Philadelphia.

....[10] and a procedure for building a regular tree grammar (resp. a tree automaton) producing (resp. recognising) IRR(R) can be found in [5] However, R (E) is not necessarily a regular tree language, even if E is. The language R (E) is regular if E is regular and if R is either a ground TRS [7], a right linear and monadic TRS [25] a linear and semi monadic TRS [6] or an inversely growing TRS [16] where inversely growing means that every righthand side is either a variable, or a term f(t 1 ; t n ) where f 2 F , ar(f) n, and 8i = 1; n, t i is a variable, a ground ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th LICS Symp., Philadelphia (Pa., USA), pages 242--248, June 1990.

....Noetherian rewrite systems. Another improvement was obtained by F. Seynhaeve and M. Tommasi and R. Treinen [50] who proved undecidability of the 9 8 part of the theory for linear shallow systems. Some decidability results have been obtained for very restricted classes of rewrite systems [9, 13]. In particular, the first order theory of one step rewriting was shown to be decidable if for ground term rewriting systems and for systems which are left linear and right ground. In this section we first, in Subsection 3.1 give a simple proof of the result of Treinen, and then in Subsection 3.2 ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings of 5th IEEE LICS, pages 242--256. IEEE Press, 1990.

....4.13 NV sequentiality is decidable for left linear (possibly overlapping) rewrite systems. Proof: This is a consequence of theorem 3.3 and lemma 4.11. 2 Note that this gives a much simpler proof than in [15] and in a more general case. We could also prove this result using ideas similar to [5]. 4.4 Sorted systems All above results can be extended to order sorted rewrite systems. In such systems, variables are restricted to range over some regular sets of trees 2 . In particular, we find again some decidability results of [12] as well as their extension to arbitrary left linear ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th IEEE Symp. Logic in Computer Science, Philadelphia, 1990.

....automata have been proposed in [FWW97,BEM97] over PA terms is similar to the transitive closure of relations defined by ground rewrite systems. Because the sequential composition operator in PA implies a certain form of prefix rewriting, the ground tree transducers of Dauchet and Tison [DT90] cannot recognize . It turns out that can be seen as a rational tree relation as defined by Raoult [Rao97] Regarding the applications we develop for our regularity theorems, most have been suggested by Mayr s work on PA [May97c,May97b] and or our earlier work on RPPS [KS97a,KS97b] 1 ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th IEEE Symp. Logic in Computer Science (LICS'90), Philadelphia, PA, USA, June 1990, pages 242--248, 1990.

....language. This was proven in [B#c64] and extended in [Cau92] Applications to the modelchecking of pushdown automata have been proposed in [FWW97, BEM97] The decidability of the rst order theory of the rewrite relation induced by a ground term rewrite system relies on ground tree transducers [DT90] note that PA is de ned by a conditional ground rewrite system) Among the applications we develop for our regularity theorems, several have been suggested by Mayr s work on PA [May97c, May97b] and or our earlier work on RPPS [KS97a, KS97b] 1 Regular tree languages and tree automata We recall ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proc. 5th IEEE Symp. Logic in Computer Science (LICS'90), Philadelphia, PA, USA, June 1990, pages 242248, 1990.

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M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242248. IEEE Computer Society Press, 47 June 1990.

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M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242248. IEEE Computer Society Press, 47 June 1990.

No context found.

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242248. IEEE Computer Society Press, 47 June 1990.

No context found.

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242248. IEEE Computer Society Press, 47 June 1990.

No context found.

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242248. IEEE Computer Society Press, 47 June 1990.

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M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242248. IEEE Computer Society Press, 47 June 1990.

....theory. Early works in this area are also mentioned in the bibliographic notes of Chapter 1. The reader is also referred to the survey [GT95] The rst order theory of the binary (many steps) reduction relation w.r.t. a ground rewrite system has been shown decidable by. M. Dauchet and S. Tison [DT90] Extensions of the theory, including some function symbols, or other predicate symbols like the parallel rewriting or the termination predicate (Terminate(t) holds if there is no in nite reduction sequence starting from t) or fair termination etc. remain decidable [DT90] Mauvaise citation ....

....Dauchet and S. Tison [DT90] Extensions of the theory, including some function symbols, or other predicate symbols like the parallel rewriting or the termination predicate (Terminate(t) holds if there is no in nite reduction sequence starting from t) or fair termination etc. remain decidable [DT90] Mauvaise citation See also the exercises. Both the theory of one step and the theory of many steps rewriting are undecidable for arbitrary R [Tre96] Reduction strategies for term rewriting have been rst studied by Huet and Lvy in 1978 [HL91] They show here the decidability of strong ....

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M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242248. IEEE Computer Society Press, 47 June 1990.

....theorem (in the case of infinite trees) 14] In the word case this construction was chosen by C. Frougny and J. Sakarovitch [12] to study rational relations with bounded delay, and in the tree case by M. Dauchet and S. Tison to prove the decidability of the theory of ground rewrite systems [5], 18] For any relabeling T , for any couple of trees (t; u) 2 b T , t and u have the same skeleton. So, to encode (t; u) in a tree, denoted by [t; u] we just superpose the trees. For instance, t; u] b; fi] a; ff] a; ff] c; fl] is the code of (t; u) b(a(a) c) fi(ff(ff) ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. Proceedings of 5 th IEEE Symposium on Logic in Computer Sciences, Philadelphia, June 4-7, pp 242-248, 1990.

....some new restrictions on linearities on left or right hand sides of rules are now required. Another improvement is to try to compute the set R (L) of descendants (or the set of normal forms) of a regular language L by a rewrite system R. This topic has been studied in many papers ( Sal88,DT90,CDGV94] and all constructions rely on similar arguments. We are able to compute R (L) with our method under some restrictions. In fact, we construct a sequence of languages (L k ) k2N with L 0 = L such that: 8k 2 N; R(L k ) L k 1 R (L k ) These languages are regular when R is ....

....the language R (L) where R is a TRS of n rules and L is a regular language: R (L) ft j 9s 2 L; s R tg: Generally, this language is not regular even when the language R(L) is. However some authors proved that R (L) is regular for some classes of TRS: M. Dauchet et S. Tison [DT90] for ground TRS, i.e. systems whose left and right hand sides of rules are ground terms; K. Salomaa [Sal88] for right linear monadic TRS, a rule l r being monadic if Height(l) 1 and Height(r) 1; J. L. Coquid et al. CDGV94] for semi monadic TRS, a rule l r being semi monadic if height(l) ....

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242248. IEEE Computer Society Press, 1990.

No context found.

M. Dauchet, S. Tison, The Theory of Ground Rewrite Systems is Decidable, IEEE Symposium on Logic in Computer Science, Philadelphia, PA, 1990.

No context found.

Dauchet, M., and Tison, S. (1990), The theory of ground rewrite systems is decidable, in "Proc 5th IEEE Symp Logic in Computer Science," pp. 242--256.

No context found.

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings of the Fifth Symposium on Logic in Computer Science, pages 242#248, Philadelphia, PA, June 1990.

No context found.

M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings of the Fifth Symposium on Logic in Computer Science, pages 242--248, Philadelphia, PA, June 1990.

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