W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer--Verlag Berlin Heidelberg, 1992. 44  Home/Search   Document Not in Database   Summary   Related Articles   Check

....property if , where denotes the strict subterm ordering. De nition 1. A west ordering is a well founded ordering on T (F) that ful ls the subterm property and that is total on T (F) it is called west after wellfounded, subterm and total) Every well founded ordering can be totalized [Wec91], and hence every wellfounded ordering satisfying the subterm property can be extended to a westordering. 3 B rewriting Now we de ne a special kind of rewrite relation, called blocked rewriting (Brewriting for short) which is always terminating. This relation restricts standard rewriting by ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1991. 40

....property if , where denotes the strict subterm ordering. Definition 1. A west ordering is a well founded ordering on T (F) that fulfils the subterm property and that is total on T (F) it is called west after wellfounded, subterm and total) Every well founded ordering can be totalized [Wec91], and hence every wellfounded ordering satisfying the subterm property can be extended to a westordering. 3 B rewriting Now we define a special kind of rewrite relation, called blocked rewriting (Brewriting for short) which is always terminating. This relation restricts standard rewriting by ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1991.

....of the editor s command set by adding functions to the ASF SDF specifications and adapting the SEAL script. The implementations of the ASF SDF Meta environment and SEAL allows us to do this fully interactively. Second, by extending the notion of a signature as known in the literature, see e.g. [Wec92], with flat lists our editor behaves more naturally when the focus is moved. Our approach has three limitations. First of all a textual view of the tree built is inevitable. Adding such a view requires the addition of a keyword skeleton for each function defined in the signature or extending the ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. SpringerVerlag, 1992.

....Structural Operational Semantics [15] and is closely related to Kahn s Natural Semantics [12] In this section, we introduce Relational Semantics through an example and give essential definitions for later sections. We assume familiarity with the basic concepts of Universal Algebras as in [20]. AppendixA contains a summary of the definitions used hereafter. The degree of formality of the following definitions is essential for later results. To illustrate Relational Semantics let us define a semantics of a simple functional language with local variables, which we call Fun. The first ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1992.

....system (TRS) to a term. Termination of the rewrite process can only be guaranteed if the TRS is confluent and weakly terminating. ASF is defined in [BHK89b] The theory of universal algebra, term rewriting, and other theoretical aspects of algebraic specifiation can be found in (among others) [EM85, Wec92, Wie91]. Implementation of (modular) algebraic specifications and term reduction systems is described in [Wal91, Hen91] 1.2.3 SDF SDF stands for Syntax Definition Formalism. An SDF specification defines the grammar or syntax of a language. A specification consists of a set of non terminals or sorts, a ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1992.

....the book of Padawitz [Pad92] but here the emphasis is more on logic programming than on functional programming. There are many papers on positive conditional specification. If no partiality is allowed then the initial model of the specification is considered to define the semantics of spec [Wec92]. The classical way to model partiality is to consider partial algebras, see e.g. BWP84] Here the problem arises how to define the equality appropriately. Both proposals, strong equality and existential equality, are unsatisfactory, as can be sen from Example 3. In [KM86] an equation t 1 = t 2 ....

W. Wechler. Universal algebra for computer scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1992.

....and total) Not all well founded orderings on terms can be extended to west orderings, even if they do not contradict the subterm property. For example, if a 1 f(b) and b 1 f(a) then, if is ( 1 [ we get a f(b) b f(a) a. But every well founded ordering can be totalized [Wec91], and hence every well founded ordering satisfying the subterm property can be extended to a west ordering. We also have the following: Lemma 2 Every reduction ordering r can be extended to a west ordering. Proof: Let rs be ( r [ Then rs is well founded: let s 1 rs s 2 rs : ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1991.

....and total) Not all well founded orderings on terms can be extended to west orderings, even if they do not contradict the subterm property. For example, if a 1 f(b) and b 1 f(a) then, if is ( 1 [ we get a f(b) b f(a) a. But every well founded ordering can be totalized [15], and hence every well founded ordering satisfying the subterm property can be extended to a west ordering. We also have the following: Lemma 2 Every reduction ordering r can be extended to a west ordering. Proof: Let rs be ( r [ Then rs is well founded. We will derive a ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1991.

.... P l;1 ; B 1 ; P l;n ; B n ; l; P l;1 = Phi 1 ; P l;n = Phi n ] where ( Phi 1 ; Phi n ) is the least fixpoint of Phi i = B[B i ] l 1; P l;1 = Phi 1 ; P l;n = Phi n ] 1 i n) Note, that with the Fixpoint theorem by Knaster and Tarski (cf. [Wec92]) Phi 1 ; Phi n ) G fS k Delta ( C ; C ) jj k 2 N 0 g with S Delta : C n (C n , defined by (S Delta ( Phi 0 1 ; Phi 0 n ) i : B[B i ] l 1; P l;1 = Phi 0 1 ; P l;n = Phi 0 n ] and C ( 0 ) undefined for all 0 2 S. The ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer--Verlag Berlin Heidelberg, 1992.

....and total) Not all well founded orderings on terms can be extended to west orderings, even if they do not contradict the subterm property. For example, if a 1 f(b) and b 1 f(a) then, if is ( 1 [ we get a f(b) b f(a) a. But every well founded ordering can be totalized [Wec91], and hence every well founded ordering satisfying the subterm property can be extended to a west ordering. We also have the following: Lemma 2 Every reduction ordering r can be extended to a west ordering. Proof: Let rs be ( r [ Then rs is well founded: let s 1 rs s 2 rs : ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1991.

....under f . The generability problem remains in P even with more than one function. More generally, it is the problem of deciding if, given a finite algebra, a subset of its universe and an element in it, this element is in the subalgebra generated by the given subset [31] See for example Wechler [54] for definitions. Actually, as we will see, generability is basically the same problem as nonemptiness of DTAs. In the following proof it is easily seen that all reductions can be carried out within logarithmic space, assuming reasonable representations of the problems, and we will not mention ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer Verlag, Berlin, 1992.

....i 1 ; v k ) need not be proper for u i ; v i . This is the case if the type S of f is not less than S i , the type of u i ; v i . Using the transitivity of we derive C[f(u 1 ; u k ) C[f(v 1 ; v k ) Hence f(u 1 ; u k ) f(v 1 ; v k ) cf. e.g. [Wec92], Sc. 2.1.2 for a similar type of argument. In general, the notion of contextual bisimilarity, as given in Definition 4.1, is not always a congruence. However, the main objection against the adaptation of Definition 4.1 as just proposed, is that it does not correctly generalize the usual notion ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer, 1992.

....P l;1 ; B 1 ; P l;n ; B n ; l; P l;1 = Phi 1 ; P l;n = Phi n ] where ( Phi 1 ; Phi n ) is the least solution of Phi i = B[B i ] l 1; P l;1 = Phi 1 ; P l;n = Phi n ] 1. 1) Note, that with the Fixpoint theorem of Knaster and Tarski (cf. [12]) we can give an alternative characterisation of (1.1) Phi 1 ; Phi n ) G fS k Delta ( C ; C ) jj k 2 N 0 g where we define the transformation S Delta : C n ( C n by (S Delta ( Phi 0 1 ; Phi 0 n ) i : B[B i ] l 1; P l;1 = Phi 0 1 ; ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer--Verlag, 1992.

....Any well founded or terminating relation suffices. This paper is a revised and extended version of the contribution in Bulletin of the EATCS 52, pp. 274 277, February 1994. This research was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D4 Projekt) Definition 1. 1 (cf. e.g. [Wec92]) Let R be a (binary) relation on a set A. ffl Let B be a non empty subset of A. An element b 2 B is said to be R minimal (or simply minimal) if, for all a 2 A, bRa implies a = 2 B. ffl R is called well founded (or Noetherian) if every non empty subset of A has a minimal element. 1 ffl R is ....

....sequence (a n ) n2IN such that a n Ra n 1 for all n 2 IN. Well foundedness and termination are equivalent notions. Well foundedness obviously implies termination, and the reverse direction is also easy but requires the Axiom of (Dependent) Choice (cf. e.g. Jec77] Theorem 1. 2 (cf. e.g. [Wec92]) A (binary) relation is well founded if and only if it is terminating. In practice, i.e. when trying to apply the general principle of proof by well founded induction, it often occurs that an appropiate well founded partial ordering either is not available or unknown, or if some partial ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1992.

....than on functional programming. There are many papers on Horn clause specification (i.e. only positive conditional equations are allowed) If in addition to this restriction no partiality is allowed then the initial model of the specification is considered to define the semantics of spec [Wec92]. If positive negative conditional equations are allowed, then there are two possibilities to evaluate these conditions for applying the equation: In [BG94] negation by failure is used to evaluate the conditions. The semantics are fixed by a reduction ordering , according to which the conditions ....

W. Wechler. Universal algebra for computer scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1992.

....over X, and the universal property means that any variable assignment ff : X jAj can be extended uniquely to a homomorphism ff : T Sigma (X) A such that the restriction of ff to X equals ff. That is, the universal property subsumes the principle of structural induction (algebraic recursion [Wec92]) More generally there exists for any inclusion SPEC 1 SPEC 2 of (conditional) equational specifications a functor F : Mod(SPEC 1 ) Mod(SPEC 2 ) left adjoint to the restricted forgetfull functor U : Mod(SPEC 2 ) Mod(SPEC 1 ) This result is essential in the area of algebraic ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin, 1992. This article was processed using the L A T E X macro package with LLNCS style

....Wong [15] 2 Kupert, Saake, and Wegner [14] gave three linear orderings on collection types in their study of duplicate detection and elimination. The ordering defined above coincides with one of them and is in fact a particular case of an order well known in universal algebra and combinatorics [13, 25]. An important feature of our technique of lifting linear orders is that the resulting linear orders are readily seen to be computable by our very limited language. Hence in the rest of the report, we assume that s , where s is not a base type, is a syntactic sugar as implemented in the theorem ....

W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monograph on Theoretical Computer Science. Springer-Verlag, Berlin, 1992.

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W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer--Verlag Berlin Heidelberg, 1992. 44

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W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer, 1992. 14

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