J.P. Jouannaud and W. Sad . Strong Sequentiality of Left-Linear Overlapping Rewrite Systems. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:235-246, Springer-Verlag, Berlin, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... necessary (1) to provide a method to decide whether a redex is needed and (2) to identify the class of TRSs ensuring that every reducible term has a redex for which the previous method succeeds [DM97] Decidable approximations to neededness have been extensively explored [Com00, DM97, HL91, Jac96, JS94, KM91, NST95, NT99, Oya93, TKS00, Toy92] Recently, we have investigated the use of these approximations to capture root neededness for almost orthogonal TRSs [Luc98b] We have demonstrated that, among them, NV sequentiality [Oya93] hence strong sequentiality [HL91] a particular case of ....

....(t) rather than I nf (t) Strong indices can also be effectively computed by using Omega Gammang 26231 in Section 2. 1: Given a fresh symbol ffl and p 2 (t) we have that p 2 I s (t) iff (t[ffl] p )j p = ffl [KM91, Toy92] In fact, we take this result as a (re )definition of strong index [JS94, Toy92] f(x,a) c g(a,x) c Note that t = x) is an Omega normal form. Position 2:1 corresponds to a strong index, since the reduction step ,x) g(ffl,x) g(ffl,x) computes the normal form (t[ffl] 2:1 ) of t[ffl] 2:1 (remember that is confluent) which does contain ....

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J.P. Jouannaud and W. Sadfi. Strong Sequentiality of LeftLinear Overlapping Rewrite Systems. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:235-246, SpringerVerlag, Berlin, 1995.

.... (1) decidable approximations of neededness and (2) decidable properties of TRSs which ensure that every reducible term has a needed redex identied by (1) Starting with the seminal work of Huet and L#vy [8] on strong sequentiality, these issues have been extensively investigated in the literature [1, 9, 10, 13, 15, 16, 19]. In all these works Huet and L#vy s notions of index, reduction, and sequentiality gure prominently. We present an approach to decidable call by need computations to normal form in which issues (1) and (2) above are addressed directly. Besides facilitating understanding this enables us to ....

J.-P. Jouannaud and W. Sad, Strong Sequentiality of Left-Linear Overlapping Rewrite Systems, Proc. 4th CTRS, LNCS 968, pp. 235246, 1995.

....neededness 1 [HL79,HL91] have been extensively explored (see, for instance Work partially supported by Spanish CICYT under grant TIC 98 0445 C03 01. 1 A redex in a term is needed if the redex (itself or one of its descendants) is reduced in each rewriting sequence leading to a normal form. [Com95,DM97,HL91,Jac96,JS94,KM91,NST95,NT99,Oya93,Tha87,Toy92]) Recently, we have investigated the use of these approximations to capture rootneededness in almost orthogonal TRSs [Luc98] We have demonstrated that NVsequentiality [Oya93] is the most general approximation to root neededness and the only one which is adequate for infinitary normalization. ....

....show that (surprisingly) there are two different (but related) notions of strong sequentiality of left linear (possibly overlapping) TRSs in the literature. The first one is Toyama s extension of strong sequentiality from orthogonal to left linear, possibly overlapping TRSs which was developed in [JS94,Toy92]. The second one follows Comon s definition of strong sequentiality of a left linear TRS and arises as a natural extension of the classical definition of sequentiality [Com00,Com95,HL91,Jac96] As we show in this paper, the first approach (strictly) includes the second one. Hence, we prove that ....

[Article contains additional citation context not shown here]

J.P. Jouannaud and W. Sadfi. Strong Sequentiality of Left-Linear Overlapping Rewrite Systems. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:235-246, Springer-Verlag, Berlin, 1995.

.... necessary (1) to provide a method to decide whether a redex is needed and (2) to identify the class of TRSs ensuring that every reducible term has a redex for which the previous method succeeds [DM97] Decidable approximations to neededness have been extensively explored [Com95, DM97, HL91, Jac96, JS94, KM91, NST95, NT99, Oya93, Toy92] Recently, we have investigated the use of these approximations to capture root neededness for almost orthogonal TRSs [Luc98b] We have demonstrated that NV sequentiality [Oya93] hence strong sequentiality [HL91] a particular case of NVsequentiality) is the ....

....of a term t is denoted by I s (t) rather than I nf (t) Strong indices can also be effectively computed by using Omega Gammang 17952 in Section 2. 1: Given a fresh symbol ffl and p 2 Pos Omega (t) we have that p 2 I s (t) iff (t[ffl] p )j p = ffl [KM91, Toy92] In fact, by following [JS94, Toy92] we take this result as a (re )definition of strong index. A TRS R is strongly sequential if I s (t) 6= for every Omega Gammay 3828 form t. Strong sequentiality has been proven decidable for left linear TRSs in [JS94] The following properties are used later. Proposition 7.2 [JS94, ....

[Article contains additional citation context not shown here]

J.P. Jouannaud and W. Sadfi. Strong Sequentiality of Left-Linear Overlapping Rewrite Systems. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:235-246, Springer-Verlag, Berlin, 1995.

.... (1) decidable approximations of neededness and (2) decidable properties of TRSs which ensure that every reducible term has a needed redex identi ed by (1) Starting with the seminal work of Huet and L#vy [8] on strong sequentiality, these issues have been extensively investigated in the literature [1, 9, 10, 13, 15, 16, 19]. In all these works Huet and L#vy s notions of index, reduction, and sequentiality gure prominently. We present an approach to decidable call by need computations to normal form in which issues (1) and (2) above are addressed directly. Besides facilitating understanding this enables us to ....

J.-P. Jouannaud and W. Sadø, Strong Sequentiality of Left-Linear Overlapping Rewrite Systems, Proc. 4th CTRS, LNCS 968, pp. 235246, 1995.

....form. In general, sequential indices are not computable, but Huet and L evy give a class of TRSs, the strongly sequential TRSs (without rules l r such that l 2 V ) for which this can be done [HL91] It is also decidable whether a TRS (without rules l r such that l 2 V ) is strongly sequential [HL91, JS94, KM91, Toy92]. The calculus of a strongly sequential index is performed by using the function which is defined by means of a reduction relation Omega [KM91] Given a TRS R, C[t] Omega C[ Omega Gamma if t 6= Omega and t s for some redex s. The relation Omega is confluent and terminating (see ....

J.-P. Jouannaud and W. Sadfi. Strong Sequentiality of Left-Linear Overlapping Rewrite Systems. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of 4th Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:235-246, Springer-Verlag, Berlin, 1995.

.... decidable approximations of neededness and (2) decidable properties of TRSs which ensure that every reducible term has a needed redex identified by (1) Starting with the seminal work of Huet and L evy [8] on strong sequentiality, these issues have been extensively investigated in the literature [1, 9, 10, 13, 15, 16, 19]. In all these works Huet and L evy s notions of index, reduction, and sequentiality figure prominently. We present an approach to decidable call by need computations to normal form in which issues (1) and (2) above are addressed directly. Besides facilitating understanding this enables us to ....

J.-P. Jouannaud and W. Sadfi, Strong Sequentiality of Left-Linear Overlapping Rewrite Systems, Proc. 4th CTRS, LNCS 968 (1995) 235--246.

....of a same variable. The original proof is quite intricate. J. W. Klop and A. Middeldorp [14] give a simpler proof to the price of an increased complexity. The case of linear, possibly overlapping rewrite systems was considered first by Toyama [24] and later shown decidable by Jouannaud and Sadfi [10]. M. Oyamaguchi defines NV sequentiality a property intermediate between sequentiality and strong sequentiality, which is also decidable for orthogonal rewrite systems [17] In this paper we use another quite simple approach, though less elementary: we show that the sequentiality of P is definable ....

....one is a 5 lines proofs, relying on previous results by Dauchet et al. and the other is a 1 page direct construction. The decidability of NVNF sequentiality implies in particular the decidability of strong sequentiality of possibly overlapping left linear rewrite systems (a result proved in [10]) Then we show in section 4.4 that sequentiality (which is in general undecidable) is decidable for shallow rewrite system, again as an application of theorem 3.10. 4.1 Strong sequentiality of left linear term rewriting systems Let NR be the predicate symbol on T Omega [ T which holds true iff ....

[Article contains additional citation context not shown here]

J.-P. Jouannaud and W. Sadfi. Strong sequentiality of left-linear overlapping rewrite systems. In Workshop on Conditional Term Rewriting systems, Jerusalem, July 1994.

....of a same variable. The original proof is quite intricate. J. W. Klop and A. Middeldorp [13] give a simpler proof to the price of an increased complexity. The case of linear, possibly overlapping rewrite systems was considered first by Toyama [22] and later shown decidable by Jouannaud and Sadfi [9]. M. Oyamaguchi defines NV sequentiality a property intermediate between sequentiality and strong sequentiality, which is also decidable for orthogonal rewrite systems [15] In this paper we use another quite simple approach, though less elementary: we show that the sequentiality of P is definable ....

....: q t n ) q r and the states q r are propagated: we have the rules f(q ; q ; q r ; q ; q ) q r for all function symbols f . Finally, if not already present, we add the rules f(q ; q ) q . This yields as an easy consequence the following (known from [9]) decidability result: Corollary 4.4 The strong sequentiality of left linear (possibly overlapping) rewrite systems is decidable. Proof: This is a consequence of lemma 4.3 and theorem 3.3 since recognizable sets of terms are definable in WSkS [20] 2 Note that this doesn t work for non left ....

J.-P. Jouannaud and W. Sadfi. Strong sequentiality of left-linear overlapping rewrite systems. In Workshop on Conditional Term Rewriting systems, Jerusalem, July 1994.

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J.P. Jouannaud and W. Sad . Strong Sequentiality of Left-Linear Overlapping Rewrite Systems. In N. Dershowitz and N. Lindenstrauss, editors, Proc. of Workshop on Conditional Term Rewriting Systems, CTRS'94, LNCS 968:235-246, Springer-Verlag, Berlin, 1995.

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