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15
Concrete Domains
 Theoretical Computer Science
, 1993
"... This paper introduces the theory of a particular kind of computation domains called concrete domains. The purpose of this theory is to find a satisfactory framework for the notions of coroutine computation and sequentiality of evaluation. Diagrams are emphasized because I believe that an important ..."
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Cited by 35 (1 self)
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This paper introduces the theory of a particular kind of computation domains called concrete domains. The purpose of this theory is to find a satisfactory framework for the notions of coroutine computation and sequentiality of evaluation. Diagrams are emphasized because I believe that an important part of learning lattice theory is the acquisition of skill in drawing diagrams. George Gratzer 1 Domains of computation In general, we follow Scott's approach [Sco70]. To every syntactic object one associates a semantic object which is found in an appropriate semantic domain. For technical details, we follow [Mil73] and [Plo78] rather than Scott. Definition 1.1 A partial order is a pair ! D; ? where D is a nonempty set and is a binary relation satisfying: i) 8x 2 D x x (reflexivity) ii) 8x; y 2 D x y; y x ) x = y (antisymmetry) iii) 8x; y; z 2 D x y; y z ) x z (transitivity) One writes x ! y when x y and x 6= y. Two elements x and y are comparable when either x y or y x. W...
Optimal Derivations in Weak Lambdacalculi and in Orthogonal Terms Rewriting Systems.
, 1991
"... We introduce the new framework of Labeled Terms Rewriting Systems (T l RS), a general framework to express sharing in Term Rewriting Systems (TRS). For Orthogonal T l RS, an important subclass of T l RS, we characterize optimal derivations. This result is applied to weak calculi, showing the ..."
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Cited by 34 (0 self)
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We introduce the new framework of Labeled Terms Rewriting Systems (T l RS), a general framework to express sharing in Term Rewriting Systems (TRS). For Orthogonal T l RS, an important subclass of T l RS, we characterize optimal derivations. This result is applied to weak calculi, showing the optimality of the lazy strategy, that is, the callbyname with sharing strategy. The result is also valid in the presence of ffi rules, as in PCF. Orthogonal T l RS is also useful as a calculus for proving syntactic properties of functional languages. 1 Compilation of the calculus Most compilers for functional languages translate their source language into some enriched calculus [17], and then, compile this intermediate language to a lowlevel language, such as mutually recursive supercombinators, as in LML [2, 10], or categorical combinators, as in CAML [4]. These lowlevel languages define different forms of weak fireduction. We now describe two of these lowlevel languages, superc...
An Abstract Standardisation Theorem
, 1992
"... The standardisation theorem is a key theorem in the calculus. It implies that any normal form can be reached by the normal order (leftmost outermost) strategy. The theorem states that any reduction may be rearranged in a topdown and lefttoright order. This also holds in orthogonal term rewriting ..."
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Cited by 28 (5 self)
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The standardisation theorem is a key theorem in the calculus. It implies that any normal form can be reached by the normal order (leftmost outermost) strategy. The theorem states that any reduction may be rearranged in a topdown and lefttoright order. This also holds in orthogonal term rewriting systems (TRS), although the lefttoright order is more subtle. We give a new presentation of the standardisation property by means of four axioms about the residual and nesting relations on redexes. This axiomatic approach provides a better understanding of standardisation, and makes it applicable in other settings, such as dags or interaction networks. We also treat conflicts between redexes (critical pairs in TRS). The axioms include Berry's stability, proving it to be a intrinsic notion of deterministic calculi. 1 Introduction The calculus has two main syntactic theorems. One is the ChurchRosser theorem, which induces uniqueness of normal forms. The second one is the standardisation...
HigherOrder Families
 In International Conference on Rewriting Techniques and Applications '96, LNCS
, 1996
"... A redex family is a set of redexes which are `created in the same way'. Families specify which redexes should be shared in any socalled optimal implementation of a rewriting system. We formalise the notion of family for orthogonal higherorder term rewriting systems (OHRSs). In order to comfort our ..."
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Cited by 14 (2 self)
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A redex family is a set of redexes which are `created in the same way'. Families specify which redexes should be shared in any socalled optimal implementation of a rewriting system. We formalise the notion of family for orthogonal higherorder term rewriting systems (OHRSs). In order to comfort our formalisation of the intuitive concept of family, we actually provide three conceptually different formalisations, via labelling, extraction and zigzag and show them to be equivalent. This generalises the results known from literature and gives a firm theoretical basis for the optimal implementation of OHRSs. 1. Introduction A computation of a result is optimal if its cost is minimal among all computations of the result. Taking rewrite steps as computational units the cost of a rewrite sequence is simply its length. Given a rewrite system the question then is: does an effective optimal strategy exist for it? In the case of lambda calculus, a discouraging result was obtained in [BBKV76]: th...
Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Cited by 13 (6 self)
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
Normalisation in Weakly Orthogonal Rewriting
, 1999
"... . A rewrite sequence is said to be outermostfair if every outermost redex occurrence is eventually eliminated. Outermostfair rewriting is known to be (head)normalising for almost orthogonal rewrite systems. In this paper we study (head)normalisation for the larger class of weakly orthogonal rewr ..."
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Cited by 8 (4 self)
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. A rewrite sequence is said to be outermostfair if every outermost redex occurrence is eventually eliminated. Outermostfair rewriting is known to be (head)normalising for almost orthogonal rewrite systems. In this paper we study (head)normalisation for the larger class of weakly orthogonal rewrite systems. Normalisation is established and a counterexample against headnormalisation is provided. Nevertheless, infinitary normalisation, which is usually obtained as a corollary of headnormalisation, is shown to hold. 1 Introduction The term f(a) in the term rewrite system fa ! a; f(x) ! bg can be rewritten to normal form b, but is also the starting point of the infinite rewrite sequence f(a) ! f(a) ! : : :. It is then of interest to design a normalising strategy, i.e. a restriction on rewriting which guarantees to reach a normal form if one can be reached. How to design a normalising strategy? Observe that in the example the normal form b was reached by contracting the redex closest...
Representation of Computations in Concurrent Automata by Dependence Orders
 THEORETICAL COMP. SCIENCE
, 1997
"... An automaton with concurrency relations A is a labeled transition system with a collection of binary relations indicating when two actions in a given state of the automaton can occur independently of each other. The concurrency relations induce a natural equivalence relation for finite computatio ..."
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Cited by 7 (2 self)
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An automaton with concurrency relations A is a labeled transition system with a collection of binary relations indicating when two actions in a given state of the automaton can occur independently of each other. The concurrency relations induce a natural equivalence relation for finite computation sequences. We investigate two graphtheoretic representations of the equivalence classes of computation sequences and obtain that under suitable assumptions on A they are isomorphic. Furthermore, the graphs are shown to carry a monoid operation reflecting precisely the composition of computations. This generalizes fundamental graphtheoretical representation results due to Mazurkiewicz in trace theory.
Axiomatic Rewriting Theory I  A Diagrammatic Standardization Theorem
, 2001
"... Machine translation ## calculus interpretation ## calculus Formally, the calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation. ..."
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Cited by 4 (0 self)
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Machine translation ## calculus interpretation ## calculus Formally, the calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation.
Axiomatic Rewriting Theory III  A factorisation theorem in Rewriting Theory
, 1997
"... . Some computations on a symbolic term M are more judicious than others, for instance the leftmost outermost derivations in the calculus. In order to characterise generically that kind of judicious computations, [M] introduces the notion of external derivations in its axiomatic description of Rewr ..."
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Cited by 2 (1 self)
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. Some computations on a symbolic term M are more judicious than others, for instance the leftmost outermost derivations in the calculus. In order to characterise generically that kind of judicious computations, [M] introduces the notion of external derivations in its axiomatic description of Rewriting Systems: a derivation e : M \Gamma! P is said to be external when the derivation e; f : M \Gamma! Q is standard whenever the derivation f : P \Gamma! Q is standard. In this article, we show that in every Axiomatic Rewriting System [M,1] every derivation d : M \Gamma! Q can be factorised as an external derivation e : M \Gamma! P followed by an internal derivation m : P \Gamma! Q. Moreover, this epimono factorisation is functorial (i.e there is a nice diagram) in the sense of Freyd and Kelly [FK]. Conceptually, the factorisation property means that the efficient part of a computation can always be separated from its junk. Technically, the property is the key step towards our illuminat...
Specifications, Algorithms, Axiomatisations and Proofs Commented Case Studies
 In the Coq Proof Assistant”, Summer School on Logic of Computation
, 1995
"... 1.1 An overview of the specification language Gallina.................... 5 ..."
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1.1 An overview of the specification language Gallina.................... 5