Courcelle B. "Infinite Trees in Normal Form and Recursive Equations Having a Unique Solution" in Mathematical Systems Theory 13, 131180. Springer-Verlag 1979. 23  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of labeled trees form a complete partial order under the partial ordering where t 1 t 2 , i# t 1 can be obtained from t 2 by replacing any number of subtrees with the singleton tree# In this setting, normal forms can be defined as unique limits of chains of approximations, as discussed in [3,4]. The singleton tree# is smaller than all other trees and corresponds to the type defined by Type T = T which we shall refer to as the vacuous type. Note that if two type expressions are equivalent, then their corresponding structural invariants must be equal. The equivalence is unique in ....

Courcelle B. "Infinite Trees in Normal Form and Recursive Equations Having a Unique Solution" in Mathematical Systems Theory 13, 131180. Springer-Verlag 1979. 23

....(lazy) values. The variety of different interpretations is investigated in [8] 4 2.1 Type Equivalence Several type expressions may be taken to denote the same type. These can be identified by an equivalence relation , which is defined as the identity of normal forms, using the techniques of [4,5]. To each type expression T we associate a unique normal form nf (T ) which is a possibly infinite labeled tree. Informally, the tree is obtained by repeatedly unfolding the type expression. Formally, we use the fact that the set of labeled trees form a complete partial order under the partial ....

....is enabled by the partial product aspect of the type ordering. There have been many suggestions for languages with a similar subtype polymorphism. Ours is unique in allowing truly imperative features such as assignments, subvariables, and variable parameters. Many systems rely on coercions [1,2,4,6] which have distinct disadvantages such as type loss and the update problem [2] We avoid these; for example, the procedure Proc Id(var skip end Id will be the identity on both the type and the value of any argument. The presence of variables or mutable types [1,2] have so far lead to unsafe ....

Courcelle, B. "Infinite Trees in Normal Form and Recursive Equations Having a Unique Solution" in Mathematical Systems Theory 13, 131-180. Springer-Verlag 1979.

....(lazy) values. The variety of different interpretations is investigated in [8] 4 2.1 Type Equivalence Several type expressions may be taken to denote the same type. These can be identified by an equivalence relation , which is defined as the identity of normal forms, using the techniques of [4,5]. To each type expression T we associate a unique normal form nf (T ) which is a possibly infinite labeled tree. Informally, the tree is obtained by repeatedly unfolding the type expression. Formally, we use the fact that the set of labeled trees form a complete partial order under the partial ....

....is enabled by the partial product aspect of the type ordering. There have been many suggestions for languages with a similar subtype polymorphism. Ours is unique in allowing truly imperative features such as assignments, subvariables, and variable parameters. Many systems rely on coercions [1,2,4,6] which have distinct disadvantages such as type loss and the update problem [2] We avoid these; for example, the procedure Proc Id(var x: Omega ) skip end Id will be the identity on both the type and the value of any argument. The presence of variables or mutable types [1,2] have so far ....

Courcelle, B. "Infinite Trees in Normal Form and Recursive Equations Having a Unique Solution" in Mathematical Systems Theory 13, 131-180. Springer-Verlag 1979.

.... Delta = e 0 i f f : ggg i2I We begin with the standard partial correctness property associated with transformation by equivalence : Theorem 8 (Partial Correctness) If e i = e 0 i for all i 2 I, then g i f i , i 2 I. This is the standard partial correctness result (see e.g. 17][5]) associated with e.g. unfold fold transformations. It follows easily from a least fixed point theorem for (the full details for this language can be found in [30] since the f are easily shown to be fixed points of the defining equations for functions g. Partial correctness is clearly not ....

B. Courcelle. Infinite trees in normal form and recursive equations having a unique solution. Mathematical Systems Theory, 13:131--180, 1979.

....with the extra complications of n ary programs in [10] and the issues we tackle here are complex enough without introducing others we already 1 The investigation of criteria for correct transformation is not a new one. In [11] and [12] for example, an algebraic approach was developed whilst in [5] general results based on term rewriting were developed. Both these strands have their intellectual roots in the work of Nivat (for example [13] Such analyses, however, do not uncover the intensional structure of transformations as putative proofs of equivalence or, more generally, as a ....

B. Courcelle. Infinite trees in normal form and recursive equations having a unique solution. Math. System Theory, 13:131--180, 1979.

....of the regular trees built from and using the operator. Thus, and are elements of D 0 and every other element x of D 0 is equal to x 0 x 00 for some x 0 ; x 00 2 D 0 . Furthermore, x 0 and x 00 are unique. It is well known that such a domain has contractive solutions [6]. Let x 0 y hold for all x; y 2 D 0 . Let x i 1 y hold iff x = or y = or x = x 0 x 00 and y = y 0 y 00 and x 00 i y 00 and y 0 i x 0 . Notice that i 1 i . Then x y holds iff x i y holds for all i 0 [4] First we check that has standard function types. x 0 ....

Bruno Courcelle. Infinite trees in normal form and recursive equations having a unique solution. Mathematical Systems Theory, 13:131--180, 1979.

....to non terminating rewrite systems: the use of approximations to the normal form and the use of transfinite reductions. The first of these approaches started with the work of Nivat [63] on the semantics of recursive program schemes and has been developed by many others like Boudol [12] Courcelle [16], Levy and Maibaum [56] Naoi and Inagaki [60, 61] and Raoult and Vuillemin [69] The whole approach is strongly linked with interpreting terms in the partial order defined by ADJ in [35] This order is defined using a special constant Omega which represents a computation that has not yet ....

Bruno Courcelle. Infinite trees in normal form and recursive equations having a unique solution. Mathematical Systems Theory, 13:131--180, 1979.

....of the regular trees built from and using the operator. Thus, and are elements of D 0 and every other element x of D 0 is equal to x 0 x 00 for some x 0 ; x 00 2 D 0 . Furthermore, x 0 and x 00 are unique. It is well known that such a domain has contractive solutions [Cou79]. Let x 0 y hold for all x; y 2 D 0 . Let x i 1 y hold iff x = or y = or x = x 0 x 00 and y = y 0 y 00 and x 00 i y 00 and y 0 i x 0 . Notice that i 1 i . Then x y holds iff x i y holds for all i 0 [AC93] First we check that has standard function ....

Bruno Courcelle. Infinite trees in normal form and recursive equations having a unique solution. Mathematical Systems Theory, 13:131--180, 1979.

....of subtrees, which is obtained as follows. Every subtree corresponds to a gap in the label L i . If the gap replaced the class C j , then the subtree is tree(C j ) if the gap replaced selfClass, then the subtree is tree(C i ) It is wellknown that such an equation system has a unique solution [19], which clearly is an L tree. If any part of a class is recursive, then the tree will be infinite. Quite often, recursive types are represented as regular trees with nodes labeled by type constructors [2] This is in fact what we have done, with the proviso that we consider every class as a ....

Bruno Courcelle. Infinite trees in normal form and recursive equations having a unique solution. Mathematical Systems Theory, 13:131--180, 1979.

....[25] and Erlang [3] A goal of our work is to support the design of recursive languages with concurrency, where the serial part is referentially transparent also in the precence of nondeterminism arising from the concurrency. Our results about folding and unfolding extend classical results [8, 10, 18] for deterministic systems. They are applicable to program transformation systems for program optimization [7, 13] and partial evaluation of nondeterministic languages. Semantics preserving partial evaluation of such languages is listed as the challenging problem no. 10.9 in [14] The only ....

....the results carry over more or less verbatim. Furthermore, the correctness of folding was proved under the simplifying condition that a rule would not be used to fold itself. In the classical theory for fold unfoldtranformations, conditions relaxing this restriction have been given by Courcelle [8] and Kott [18] We conjecture that these results carry over to our framework. ....

B. Courcelle. Infinite trees in normal form and recursive equations having a unique solution. Math. System. Theory, 13:131--180, 1979.

....instance roe of the right hand side) A position p in t at which a rewrite can take place is called a redex. We use jRj to denote the maximum depth of a left hand side of a system R. Since we are interested here primarily in sequences of rewrites issuing from finite terms t 0 (unlike [4, 10]) we will restrict our attention to that case: 3 Definition 7. A derivation of length ff, for ordinal ff, is a finite or transfinite sequence of (finite or infinite) terms t fi in T 1 , such that t 0 is a finite term in T and (t fi ) fi ff is an ff chain for R . In particular, if (t fi ) is ....

.... Delta Delta t 0 1 Clearly, lim n t 0 n = t 0 1 , which proves that t 0 R t 0 1 . 2 Neither hypotheses in this theorem suffices by itself. Example (4) is a non left linear, top terminating system, for which the theorem does not hold. Left linearity turns out to be crucial (as in [4], but cf. 11] and throughout this paper, we deal exclusively with left linear systems. Unfortunately, left linearity is insufficient. The following is an example (from 4 Our preliminary versions of this claim omitted the top termination requirement. 8 N. Dershowitz, S. Kaplan, D. A. Plaisted ....

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B. Courcelle, Infinite trees in normal form and recursive equations having a unique solution, Math. Systems Theory 13 (1979), 131--180.

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Courcelle, B. (1979). Infinite trees in normal form and recursive equations having a unique solution. Mathematical Systems Theory 13(1), 131--180.

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