Allen Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....atoms supporting x. Thus we do indeed have the left to right implication in (14) Remark 4. 6 (Failure of the Axiom of Choice) In the literature, careful presentations of the de nition of capture avoiding substitution quite often make use of a choice function for picking out fresh variables: see [Sto88, Section 2], for example. The vague feeling that such concrete choices should be irrelevant crystallises here into the fact that such choice functions are inconsistent with FM set theory, because its axioms contradict the Axiom of Choice (AC) For example, the axiom (A Not Finite) implies that the set of co ....

Stoughton, A.: Substitution revisited. Theoretical Computer Science, 59:317-325, 1988.

....for CCS, we can view terms in T F (A) as given in a context A. The binding operator n is then an operation taking a term in a context with one fresh name, A 1, and binding the fresh name to obtain a term in A. We will now define substitution for terms, using normal forms for a equivalence, as in [Sto88]. Substitution We want to define a syntactic category from FCCS, and relate it to Proc CCS , by translation from the syntactic category to Proc CCS . We need to translate FCCS into CCS, which involves stripping off the n bindings of a term. Because bound names are then translated into free names, ....

....could be avoided if we were to adopt a suitable variable convention. The second rule is rather strong and would make it more difficult to prove properties of the transition relation; it should be a derived rule instead. We can avoid these complications by adapting the concept of a normal forms [Sto88], developed for T F in Section 4.2. Recall that we choose coproducts where old A (a) a for a 2 A, and write new A for the fresh name in A 1. Extend a function f : A B to a function Act( f ) Act(A) Act(B) as follows. We will usually just write f for Act( f ) when it is clear from ....

Allen Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....the possible applications of this presentation the proof of semantical properties of terms. In [KP93] bound and free variables are distinguished syntactically, which allows one to simplify the definition of substitution, and hence the proofs. Another point of view on substitution is presented in [Sto88], where a definition by structural recursion is presented; in this work, bound variables are systematically renamed in a substitution, while in our de Bruijn setting we always rename free variables. Miller has described in [Mil92] another interesting point of view onto calculus specification by ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....metaformula. The remaining two requirements say that the set MFrmL (X) is closed under applications of connectives and quantifiers. The requirement that v = fresh(X) guarantees that the resulting metaformulea are normalized wrt. a suitably defined notion of syntactic subsitution (in the sense of [12, 8]) Hence, we can avoid the complexity of working with equivalence classes (wrt. # conversion) of metaformulae. The syntax translation along metasignature morphisms is defined in a standard way. It is not di#cult to show that the construction indeed defines a functor from MSig to sDgm(Set) By the ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....containing them, just as the names of free variables do. This is done using a formulation suggested by Coquand [Coq91] based on syntactically distinguishing free from bound variables . Other work on formalization of binding and substitution using names includes [Coq96b, GM96, Owe95, Sat83, Sto88] but these do not work out any large examples using their binding notions. It would be interesting to compare our development with some similar example using the terms up to alpha conversion of [GM96] A presentation of type theory based on treating terms with named variables concretely is ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 17:317--325, 1988.

....for CCS, we can view terms in T F (A) as given in a context A. The binding operator n is then an operation taking a term in a context with one fresh name, A 1, and binding the fresh name to obtain a term in A. We will now define substitution for terms, using normal forms for a equivalence, as in [Sto88]. Substitution We want to define a syntactic category from FCCS, and relate it to Proc CCS , by translation from the syntactic category to Proc CCS . We need to translate FCCS into CCS, which involves stripping off the n bindings of a term. Because bound names are then translated into free names, ....

....could be avoided if we were to adopt a suitable variable convention. The second rule is rather strong and would make it more difficult to prove properties of the transition relation; it should be a derived rule instead. We can avoid these complications by adapting the concept of a normal forms [Sto88], developed for T F in Section 4.2. Recall that we choose coproducts A old A ## A 1 1 new A ## where old A (a) a for a 2 A, and write new A for the fresh name in A 1. Extend a function f : A B to a function Act( f ) Act(A) Act(B) as follows. We will usually just write f for Act( f ) ....

Allen Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....atoms supporting x. Thus we do indeed have the left to right implication in (14) Remark 4. 6 (Failure of the Axiom of Choice) In the literature, careful presentations of the de nition of capture avoiding substitution quite often make use of a choice function for picking out fresh variables: see [Sto88, Section 2], for example. The vague feeling that such concrete choices should be irrelevant crystallises here into the fact that such choice functions are inconsistent with FM set theory, because its axioms contradict the Axiom of Choice (AC) For example, the axiom (A Not Finite) implies that the set of co ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317-325, 1988.

....therefore tried to be as helpful as possible in choosing new variable names; a variable x will be renamed to x 0 or to x with some other subscript, if x 0 is not a fresh name. It would be even more helpful if we renamed variables only when necessary, but this makes the calculus less clear [22]) 3.2 Factoring standard reduction We factored standard reduction into two phases: computing all the filled evaluation contexts of a term, and taking the first indeed, the only one with a marked subterm reducible under name s Gamma Gamma Gamma Gamma . Of course, lazy evaluation means ....

Allen Stoughton. Substitution revisited. Theoretical Computer Science, Volume 59, pages 317--325, 1988.

....like [MPW92a, MPW92b, San93] all employ the notion of ff equivalence in the semantic definition, thus the following development of the semantic theory suffer from the complication caused by ff equivalence. In this paper, by using the notion of simultanous substitution introduced in [Sto88], we show that the semantics of calculus can be defined on the much simpler notion of syntactic identity instead of ff equivalence. This gives a considerable advantage in the development of the theory. The next section presents the basic algegbraic theory of the calculus. The section ....

....of y in P . It need to be clarified what this substitution exactly means, as there are possiblely bound names in a term, we have to be careful to avoid name clash when making such substitutions. In this paper we take the approach of simultaneous substitution introduced by Alan Stoughton [Sto88] for the lambda calculus. The definition below is a specialized version of that in [Sto88] in the sense that a name may only be substituted by another name in the calculus while a variable may be substituted by a term in the lambda calculus. We assume that there is a function fresh which for a ....

[Article contains additional citation context not shown here]

Allen Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....for all substitution rules that renaming is performed in parallel with the substitution. The renaming prevents any variable clashes that would cause substituted free variables to become bound; doing it in parallel allows substitution proofs to be correctly built as structurally inductive ones (see [8]) i) oe v: t ( ii) For each variable x occuring in t 0 , then (1) If x does not occur in v, then oe v: t (x) x, 2) else if x is bound and x = v then oe v: t (x) x, 3) else if x is unbound and x = v then oe v: t (x) t, 4) else if v = v 0 ; v 1 ) and x occurs in v i , ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....So far, all the properties were studied for processes without conflict of variables. In this section we show that the behaviour is preserved by ff conversion, which implies that our restriction to the subset of processes without conflict of variables is harmless. We base our studies on [Sto88] Definition 4.7 (Renaming of clocks in processes) Let v be a renaming. We extend the notion of renaming to terms in L according to the following recursive definition: v(stop) def = stop v(p q) def = v(p) v(q) v(a; p) def = a; v(p) v(fjCjg p) def = fjf (C)jg v[f ] p) v(OE7 7 p) ....

....q p j ff q p 0 j ff q 0 p p 0 j ff q q 0 f : C C 0 is bijective C 0 fv(fjCjg p) f ] p) j ff q fjCjg p j ff fjC 0 jg q If p j ff q then p and q are ff convertibles. 2 It can be proven that j ff is an equivalence, and hence it is a congruence by definition. We refer to [Sto88] for further studies in ff conversion. In the following we sate that for every term there is an ff conversion which does not have conflict of variables. Together with Theorem 4.10, we can state that for every term, there is another term which is timed bisimilar and does not have conflict of ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....definition of the substitution process . Indeed, there is a long history of erroneous definitions of substitution in the literature of logic and programming semantics. From a theoretical side, the problem of substitution and ff conversion are analysed in detail in A. Stoughton s paper [17], and one motivation behind the Preprint submitted to Elsevier Science 1 March calculus of explicit substitution [1] was to handle precisely these problems. One other attempt for making precise the substitution operation is the substitution calculus of P. Martin Lof, presented in the references ....

A. Stoughton. Substitution Revisited. Theoretical Computer Science 59 (1988), p. 317-325.

....OE ff[w] Delta q of p. A solution for this problem, which originates from the calculus, is to allow unrestricted substitution by renaming bound variables. In the sequel, process terms are considered modulo ff conversion, and when a substitution is applied, bound variables are renamed. Stoughton [12] presented a simple treatment of this technique. If for a substitution oe there are finitely many variables v 1 ; v n such that oe(v i ) 6= v i , then often we will use the standard notation p[oe(v 1 ) v 1 ; oe(v n ) v n ] for oe(p) 2.4 Operational semantics Table 1 contains an ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....for this problem, which originates from the calculus, is to allow unrestricted substitution by applying ff conversion, that is, by renaming bound variables. In the sequel, actual terms are considered modulo ff conversion, and when a substitution is applied, bound variables are renamed. Stoughton [38] presented a nice treatment of this technique. Remark 2.5 Bloom and Vaandrager [12] develop a framework for transition rules with types and a binding mechanism, in which they make a clear distinction between sorts for processes and sorts for data. We have chosen not to adopt this distinction, ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....These are arity preserving partial functions from agent identifiers to data closed terms, i.e. terms with no free data variables. We write [ F= X] for the substitution which maps X to F and is undefined elsewhere. A precise description of application of these substitutions may be found in [14]; we simply assume that they commute with data substitutions. We will have occasion to use a very specific form of data substitution called an environment. An environment is a substitution ffi : V ar V al which maps each variable to a designated value. We assume that each closed expression e or ....

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

....les canaux. Lors d une application d une substitution, il faudra s assurer qu aucun canal libre ne devient li e. La d efinition standard de la substitution est r ecursive; on ne peut pas la d efinir directement dans HOL. Une autre approche de substitution simultan e introduite par A Stoughton dans [31] nous donne une d efinition primitive r ecursive, qu on peut introduire dans HOL, en utilisant simplement la r egle d eriv e d introduction de nouvelles fonctions primitives r ecursives: newrecursivedefinition. Ce qui simplifie les preuves concernant la substitution en utilisant le principe ....

A Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

.... Language Denotational Semantics 6 Substitution and Translation We present here a brief discussion of proper substitution and expression translation; for more details, see [8] We define proper substitution on expressions using the technique of simultaneous substitutions, following Stoughton [14]. We represent substitutions by functions of type subst = var vexp. This describes an infinite family of single substitutions, all of which are considered to take place simultaneously. The normal single substitution operation of [v=x] may be defined as a special case: v=x] y: y = x = v ....

Stoughton, A.: Substitution Revisited. Theoretical Computer Science 59 (1988) 317--325

....terms to be equality up to ff equivalence of the formal parameters of bind expressions. This definition of equality of CoreDS terms makes Nd into a function. In practice, CoreDS terms can be compared by simultaneously substituting fresh variables for all the formal parameters of bind expressions [34]. 4.2 Unnaming CoreDS terms Unnaming a CoreDS term with Ud corresponds to performing the administrative reductions of the DS transformation. For each bind expression, we either unfold all the bindings, substituting each named term for its name, or we residualize the entire bind expression as a ....

Allen Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

No context found.

Allen Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

No context found.

Allen Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

No context found.

A. Stoughton. Substitution revisited. Theoretical Computer Science, 59:317--325, 1988.

No context found.

A. Stoughton, Substitution Revisited, Theoretical Computer Science 59 (1988) 317-325

No context found.

A. Stoughton, Substitution Revisited, Theoretical Computer Science 59 (1988) 317-325

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC