M. O'Donnell, Computing in Systems described by Equations, Springer LNCS 58, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....allows to evaluate admissible graphs in a deterministic way by using rewriting strategies. For constructor based weakly orthogonal term rewriting systems, efficient strategies have been proposed in the literature. For example, O Donnell has shown that paralleloutermost strategy is normalizing [13], Sekar and Ramakrishnan [18] as well as Antoy [1] improved O Donnell s strategy by proposing strategies which rewrite in parallel a necessary set of redexes. The two last strategies are optimal with respect to strategies which have no knowledge about the right hand sides of the rules. Their ....

M. J. O'Donnell. Computing in Systems Described by Equations. LNCS 58, 1977.

....is normalized. A good survey of term rewriting appears in [DJ90] An implementation based on term rewriting can be complete if the input programs are guaranteed to be confluent or Church Rosser. For programming purposes, it is natural to do this by requiring input programs to be orthogonal 1 [OD77, HL79]. In such systems, the lhs of rules are linear (i.e. no variable in an lhs occurs more than once) and no two lhs unify except possibly at the root. If they do unify, then any term that is an instance of such overlapping rules must be rewritten into the same term by each of those rules. For ....

....of rules are linear (i.e. no variable in an lhs occurs more than once) and no two lhs unify except possibly at the root. If they do unify, then any term that is an instance of such overlapping rules must be rewritten into the same term by each of those rules. For orthogonal programs, O Donnell [OD77] showed that the parallel outermost strategy, which repeatedly replaces all the outermost redexes, is complete. Many of the reductions performed by this strategy could be wasteful in general. A lazy normalization algorithm that completely eliminated these wasteful reductions by reducing only ....

M.J. O'Donnell, Computing in Systems described by Equations, Springer LNCS 58, 1977.

....Then, 0 ) jfx;ug = fx 7 m1:b(n4:a,n4) u 7 n4:ag. 3 Constructor based Graph Rewriting Systems This section introduces the di erent classes of graph rewriting systems we consider (see also [9] For practical reasons, many declarative languages use constructor based signatures [19]. A constructor based signature is a triple = hS; C; Di where S is a set of sorts, C is an S indexed family of sets of constructor symbols whose r ole consists in building data structures and D is an S indexed family of sets of de ned operations such that C D = and hS; C [Di is a ....

M. J. O'Donnell. Computing in Systems Described by Equations. LNCS 58, 1977.

....a sound and complete narrowing strategy and implementing it efficiently. We limit the discussion to first order computations and the focus to strategies. Figure 1 summarizes the state of the art in this field. All the TRSs in the figure are constructor based (follow the constructor discipline [21]) and left linear. For the time being, we ignore whether they are conditional [8] We will see later that this is not a substantial characterization for our discussion. The inductively sequential TRSs are the first order component of functional languages, such as ML and Haskell, and the ....

M. J. O'Donnell. Computing in Systems Described by Equations. Springer LNCS 58, 1977.

....of TRSs potentially suitable for functional logic computations have been extensively investigated. Figure 1 presents a containment diagram of some major classes. All the classes considered in the diagram are constructor based. Rewrite rules defining an operation with the constructor discipline [20] implicitly define a corresponding function over algebraic data types such as those of 6 WO OIS CB IS Fig. 1. Containment diagram of rewrite systems modeling functional logic programs. The outer area, labeled CB, represents the constructor based rewrite systems. The smallest darkest area, ....

....to functional logic computations. I briefly address these issues in this section. The focus, as in the rest of this paper, is on strategies. The classes of TRSs discussed earlier are all unconditional. The well known outermost fair rewrite strategy, which is normalizing for almost orthogonal TRSs [20], is also normalizing for conditional almost orthogonal TRSs [9] For the constructor based TRSs, the results presented earlier about evaluation strategies are extended to the conditional case with little effort. The strategies discussed in Section 3 are based, either directly or indirectly, on ....

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M. J. O'Donnell. Computing in Systems Described by Equations. Springer LNCS 58, 1977.

....in error reports. Examples like these will be presented in Section 5. For TRSs in areas different from programming language descriptions, there may be applications of origins as well. We will see that the notion of origins is an extension of the well known notion of residuals or descendants [O D77, HL79] Besides being an extension of residuals, origins are also defined for a wider class of TRSs, since we do not restrict ourselves to orthogonal TRSs, but allow ambiguous, not left linear, conditional TRSs. Residuals are typically used to study concepts like confluence, termination, or ....

....tracking in the preliminary implementation have been performed. In all benchmarks the run time overhead lies between 10 and 100 , excluding the costs of pre computing positional information. 6 Related Work In TRS theory, the notion of descendant [Klo91, Chapter 8] sometimes called residual [O D77, HL79] is well known. Descendants are very close to origins (see Section 3.3) but they are more limited, and are only defined for orthogonal (left linear and non overlapping) TRSs without conditions. The reason for introducing descendants is of a theoretical nature; they are used to study ....

M.J. O'Donnell. Computing in Systems Described by Equations, volume 58 of Lecture Notes in Computer Science. Springer-Verlag, 1977.

....it computes, and the ease of its implementation. The notion of an unavoidable step is well known for rewriting. Orthogonal systems have the property that in every term t not in normal form there exists a redex, called needed, that must eventually be reduced to compute the normal form of t [24, 30, 39]. Furthermore, repeated rewriting of needed redexes in a term suffices to compute its normal form, if it exists. Loosely speaking, only needed redexes really matter for rewriting in orthogonal systems. We extend this fact to narrowing in inductively sequential systems, a subclass of the orthogonal ....

....There are three research topics related to our work: 1) the concept of need as the foundation of laziness, 2) strategies for using narrowing in programming, and (3) implementations of narrowing in Prolog. 6. 1 Narrowing and need Seminal studies on the concept of need in rewriting appear in [24, 39]. Subsequent variations and extensions, e.g. 7, 21, 27, 30, 33, 40, 41, 45, 48] do not address narrowing, but limit the discussion to rewriting. We have introduced a concept of need for narrowing that extends a similar concept for rewriting. We have shown that the concept of need for narrowing ....

M. J. O'Donnell. Computing in Systems Described by Equations. Springer LNCS 58, 1977.

.... also to perform deduction when describing by inference rules a logic [GLT89] a theorem prover [JK86] or a constraint solver [JK91] It is of course central in systems making the notion of rule an explicit and rst class object, like expert systems, programming languages based on equational logic [O D77] algebraic speci cations (e.g. OBJ [GKK 87] functional programming (e.g. ML [Mil84] and transition systems (e.g. Murphi [DDHY92] It is hopeless to try to be exhaustive and the cases we have just mentioned show part of the huge diversity of the rewriting concept. When one wants to focus ....

M. J. O'Donnell. Computing in Systems Described by Equations, volume 58 of Lecture Notes in Computer Science. Springer-Verlag, 1977.

....An orthogonal TRS is left linear and nonoverlapping. If the non overlapping restriction is relaxed by allowing trivial overlays then we speak of almost orthogonal TRSs. A typical example of an almost orthogonal TRS that is not orthogonal is the two rule system f x ; x g. O Donnell [8] showed that the parallel outermost strategy which evaluates all outermost redexes in parallel is normalizing for all almost orthogonal TRSs. The question whether there exists a computable normalizing sequential reduction strategy for all (almost) orthogonal TRSs has received quite a bit of ....

....orthogonal TRS (from [4] a b c c f(x; b) d and the term t = f(c; a) The leftmost outermost strategy will select redex c in t and hence produce the in nite reduction sequence t t t . Nevertheless, parallel outermost succeeds in normalizing t: t f(c; b) d. O Donnell [8] showed that the leftmostoutermost strategy is normalizing for every left normal orthogonal TRS. 1 Left normality means that variables do not precede function symbols in the left hand sides of the rewrite rules. A typical example of a left normal orthogonal TRS is combinatory logic. 2) Huet ....

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M.J. O'Donnell, Computing in Systems Described by Equations, Lecture Notes in Computer Science 58, 1977.

....,R) Then, R(F ,R) is confluent if and only if R(F ,R) is outermost confluent. It is well known that outermost fair reductions are normalizing (i.e. for every term with a normal form, there is an outermost fair derivation leading to its normal form) for almost orthogonal systems (cf. O Donnell [4] and Klop [3] From this it follows that every weakly normalizing almost orthogonal system is weakly outermost normalizing and hence we have the following corollary. Corollary 1 Every weakly normalizing almost orthogonal system is outermost confluent. Proof. Follows from the above Theorem 2 ....

M.J. O`Donnell (1977): Computing in systems described by equations. Lecture Notes in computer Science, Vol. 58. Springer-Verlag

....a term of the form t 1 t 2 may be narrowed to normal form by narrowing either t 1 or t 2 , although we do not know of any criterion to make this choice without look ahead. To place our results in a context, we brie y review relevant results about rewriting strategies. O Donnell has shown [19] that the parallel outermost strategy is normalizing for almost orthogonal TRSs, hence for weakly orthogonal, constructor based TRSs. In general, some reductions performed by this strategy could be avoided. Huet and L evy have shown [11] that for the class of strongly sequential TRSs there is an ....

M. J. O'Donnell. Computing in Systems Described by Equations. Springer LNCS 58, 1977.

....ers it computes, and the ease of its implementation. The notion of an unavoidable step is well known for rewriting. Orthogonal systems have the property that in every term t not in normal form there exists a redex, called needed, that must eventually be reduced to compute the normal form of t [24, 30, 39]. Furthermore, repeated rewriting of needed redexes in a term suces to compute its normal form, if it exists. Loosely speaking, only needed redexes really matter for rewriting in orthogonal systems. We extend this fact to narrowing in inductively sequential systems, a subclass of the orthogonal ....

....There are three research topics related to our work: 1) the concept of need as the foundation of laziness, 2) strategies for using narrowing in programming, and (3) implementations of narrowing in Prolog. 6. 1 Narrowing and need Seminal studies on the concept of need in rewriting appear in [24, 39]. Subsequent variations and extensions, e.g. 7, 21, 27, 30, 33, 40, 41, 45, 48] do not address narrowing, but limit the discussion to rewriting. We have introduced a concept of need for narrowing that extends a similar concept for rewriting. We have shown that the concept of need for narrowing ....

M. J. O'Donnell. Computing in Systems Described by Equations. Springer LNCS 58, 1977.

....of) neededness when it comes to infinitary normalization. 1 Introduction In this paper we are concerned with reduction strategies for term rewriting systems. A reduction strategy is called normalizing if repeated contraction of the redexes selected by the strategy leads to normal form. O Donnell [13] showed that the parallel outermost strategy, which contracts all outermost redexes in parallel, is normalizing for orthogonal term rewriting systems. Parallel outermost is not an optimal reduction strategy since many of the redex contractions it performs are useless for obtaining normal forms. ....

....maximal non root stable subterms are never left undisturbed. Our notion of fairness, although much weaker than the one in Kennaway et al. 9, Definition 8.3] is sufficient for infinitary normalization. The fairness notion in [9] can be shown to be equivalent to O Donnell s outermost fairness [1, 13]. Definition 7.3 A reduction strategy S for a TRS is called infinitary fair normalizing if there are no perpetual fair S rewrite sequences starting from terms that admit an infinitary normalizing rewrite sequence. Theorem 7.4 Let R be a confluent TRS. Context free root normalizing reduction ....

M.J. O'Donnell, Computing in Systems Described by Equations, Lecture Notes in Computer Science 58, 1977.

....and has a normal form. We conclude by well foundedness that the strategy S is normalising. 3.5 Fair and hyper normalisation The reader interested in fair normalisation should observe that theorem 1 adapts easily to that setting. The idea appearing with the de nition of eventually outermost in [8] is to mix fairly ecient and unecient computations, and prove a normalisation result. Suppose for instance that a strategy S veri es for every term P that: 17 1. there exists n 2 N such that S n (P ) P Q is needed, 2. there exists n 2 N such that S n (P ) P Q is properly needed when ....

M. J. O'Donnell. \Computing in systems described by equations". Lecture Notes in Computer Science 58, Springer Verlag, 1977.

....i ; C j 0 , 8y: C j (y; a i ; C j 0 , 8y: T K j ( j; a i ; y) TK j 0 where 1 indicates divergence. Partial recursive rewrite systems are orthogonal (left linear and free of nontrivial critical pairs) As such they are confluent [17] and terminate iff they are innermost terminating [4] Moreover, they are constructor based (all but the outermost symbols on the left are constructors [0 or s] or variables) and complete (every non constructor ground term is reducible) We will refer to rewrite systems that are 1. left linear ....

Michael J. O'Donnell. Computing in systems described by equations, volume 58 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1977.

....there is a standard reduction from M to N . Proof: by Klop s method [4] See section 5 for a detailed proof. Now, a third axiom makes our definition of standard reduction more natural. Axiom 3 (Creation) t u; t[ u] t 0 ; 6 9v 0 : v 0 [ u] v) t 0 v This axiom, which appeared in [9], states that it is impossible to create a redex through another one. This is due to the fact that our reduction systems are conflict free (critical pairs for TRS) Graphically, u u t v v 1 The proposition below shows that standard is indeed outside in. Proposition 1 If a redex x ....

M. J. O'Donnell, Computing in Systems described by Equations. PhD thesis, Cornell, 1977.

....mandatory to prove the completeness of narrowing [EJ97] The confluence of a rewrite relation allows to evaluate expressions in a deterministic and efficient way by using rewriting strategies. Such strategies have been well investigated in the setting of finite and infinite orthogonal TRSs (e.g. O D77, HL91, KKSV95] In [Ant92] a strategy that computes outermost needed redexes based on definitional trees has been designed in the framework of orthogonal constructor based TRSs. In this article, we show that Antoy s strategy can be extended to orthogonal constructor based GRSs with the same ....

M. J. O'Donnell. Computing in Systems Described by Equations. Springer Verlag LNCS 58, 1977.

....Prolog interpreters and compilers, including commercial products. Rewriting [12, 29] is a computational paradigm that specifies which reductions can be performed but not where and when one should execute them during a computation. Computing requires some form of control referred to as a strategy [25, 29, 30, 42, 43]. Without an appropriate strategy a computation may fail to terminate even when termination is possible. We are concerned with a strategy, generally called lazy, that ensures that a computation terminates, if at all possible. Its foundations are based on the seminal work of Huet and L evy and ....

....an appropriate strategy a computation may fail to terminate even when termination is possible. We are concerned with a strategy, generally called lazy, that ensures that a computation terminates, if at all possible. Its foundations are based on the seminal work of Huet and L evy and O Donnell [25, 42, 43]. In a certain class of systems, one can find in every term t, not in normal form, a redex, called needed, that must be eventually reduced to compute the normal form of t. Furthermore, if each needed redex is eventually reduced, the computation of a term with a normal form terminates. The strategy ....

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Michael J. O'Donnell. Computing in Systems Described by Equations. Springer-Verlag, 1977. Lect. Notes in Comp. Sci., Vol. 58.

....two classes of TRSs for which we prove that the restricted strategies have normalizing properties, while the unrestricted ones do not. We define the strongly replacing independent TRSs for which every cs strategy is root normalizing. We define the left normal TRSs, a superset of left normal TRSs [13] for which the restricted leftmost outermost strategy is also root normalizing. We also extend these results to perform finite, efficient normalization and evaluation. In Section 2, we give some preliminary definitions. Section 3 recalls the basic concepts of csr and characterizes the strongly ....

....replacing independent iff, for all l 2 L(R) we have V ar com R (l) The TRS in Example 3.1 is strongly replacingindependent. The interesting property of these TRSs is that every replacing redex in non root stable terms is root needed, as we show below. Left normal TRSs are well known from [13] because the leftmost outermost reduction strategy is normalizing for this class of TRSs. A TRS is left normal if, in each lhs of the TRS, the function symbols precede the variable symbols when considering the ordering L . We generalize this notion. Definition 3.9 ( left normal TRS) Given a ....

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M.J. O'Donnell. Computing in Systems Described by Equations. LNCS 58, Springer-Verlag, Berlin, 1977.

.... full first order theories [Subrahmanyam 1981, Kapur 1980] This, of course, led to reasoning systems whose efficiency and 2 This observation extends to reactive systems as well, see [Kahn 1974] 3 simplicity could not match that of the existing systems for equational reasoning (see, e.g. O Donell 1977, Dershowitz 1990] for references) The difficulty with the set approach as sketched above is its inability to distinguish between the equivalence of terms and the equivalence of application of terms. This distinction is of no consequence in the deterministic case, because there every term ....

....domains of formal specifications. Thus the specifier need not be exposed to new (and possibly exotic) semantic concepts as long as he stays within the deterministic domain of discourse. 2) The existing tool base, created to cater for standard (equationally defined) algebras [Dershowitz 1990, O Donell 1977] can be retained. In 3 This may seem trivial, but was in fact hard enough to require the introduction of full first order logic in the early deduction systems [Kapur 1980, Subrahmanyam 1981] All classes of structures proposed as semantics of nondeterminism allow models in which such disjunctive ....

O'Donnell, M., "Computing in Systems Described by Equations", LNCS, vol. 58, Springer, 1977.

....(1991) F4.2, F4.1. Keywords Phrases: orthogonal term rewriting systems, perpetual reductions, strong normalization, reduction strategies. Note: Part of this work was completed during an enjoyable visit of the author at CWI in the summer of 1993. 1. Introduction It is shown in O Donnell [9] that the innermost strategy is perpetual for orthogonal term rewriting systems (OTRSs) That is, contraction of innermost redexes gives an infinite reduction of a given term whenever such a reduction exists. In fact, a strategy that only chooses redexes that do not erase any other redex is ....

....corresponding R reduction of P is also normalizing. Hence, by Lemma 2.3, t is strongly normalizable in R. Definition 2.4 (1) A TRS is called non erasing if left and right hand sides of each rule in it contain occurrences of the same variables. 2) A TRS is called weakly innermost normalizing [9] if each term has a normal form reachable by an innermost reduction. Corollary 2.2 (Church) Let R be a non erasing OTRS. Then R is weakly normalizing iff it is strongly normalizing. Corollary 2.3 (O Donnell [9] Let R be an OTRS. Then R is strongly normalizing iff it is weakly innermost ....

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O'Donnell M. J. Computing in systems described by equations. Springer LNCS 58, 1977.

....form c. A strategy can be seen as selecting one or more redex occurrences in a term that is not in normal form. A strategy is said to be normalising if repeatedly contracting redex occurrences selected by the strategy yields a normal form whenever the initial term has one. O Donnell shows in [24] that the parallel outermost strategy, which contracts all redexes that are outermost in a term in one step, is normalising for first order term rewriting systems that are left linear, and where trivial critical pairs are only allowed in a certain restricted form. A stronger result obtained in ....

....[24] that the parallel outermost strategy, which contracts all redexes that are outermost in a term in one step, is normalising for first order term rewriting systems that are left linear, and where trivial critical pairs are only allowed in a certain restricted form. A stronger result obtained in [24] is that outermost fair rewriting is normalising. A rewrite sequence is said to be outermost fair if every outermost redex occurrence is eliminated eventually. For example, in the rewriting system fa a; b c; f(c; x) f(b; x)g, the rewrite sequence f(b; a) f(c; a) f(b; a) is ....

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M.J. O'Donnell. Computing in Systems Described by Equations, volume 58 of Lecture Notes in Computer Science. Springer Verlag, 1977.

....OERSs the algorithm does not require actual transformation of the input term. As a corollary, we obtain an algorithm for computing the lengths of longest developments in OERSs. Keywords: Rewrite systems, calculus, perpetual reductions, strong normalization. 1 Introduction O Donnell [48] showed that the innermost strategy is perpetual for orthogonal Term Rewriting Systems (OTRSs) 13,39,3] This means that a repeated contraction of innermost redexes in a term yields an infinite reduction whenever the term has an infinite reduction. In fact, any strategy that contracts only the ....

....R reduction P starting from t is also normalizing. Hence by Lemma 14, t is strongly normalizing in R. Definition 21 1. A TRS is called non erasing if all variables in the lefthand side of any rule also occur in the corresponding right hand side. 2. A TRS is called weakly innermost normalizing [48] if each term has a normal form reachable by an innermost reduction. 13 Corollary 22 1. Church [9] Klop [39] 5 Let R be a non erasing OTRS. Then R is weakly normalizing iff it is strongly normalizing. 2. O Donnell [48] Let R be an OTRS. Then R is strongly normalizing iff it is weakly ....

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M.J. O'Donnell, Computing in Systems Described by Equations, Lecture Notes in Computer Science, Vol. 58 (Springer, Berlin, 1977).

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M. O'Donnell, Computing in Systems described by Equations, Springer LNCS 58, 1977.

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M.J. O'Donnell. Computing in Systems Described by Equations. LNCS 58, Springer-Verlag, Berlin, 1977.

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