I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....obvious, which also forms a natural basis for a concrete implementation. Consider the (one sorted) signature Sigma specified by Sigma 0 = fu; wg; Sigma 1 = fl; r; dg; Sigma 2 = fm; cg We write W ( Sigma; X) for the free Sigma algebra on a set of generators X . The free ordered Sigmaalgebra [GTWW77, Gue81] W ( Sigma; X) can be constructed as follows: the Sigma algebra structure is that of W ( Sigma; X[f g) where 62 X) and the order is generated by v x, subject to the condition that all operations are monotone. This is the match ordering: tvu just if u can be obtained from t by replacing ....

I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1981.

....identities. Next, consider the extension to infinite computations. That is reflexive and transitive is immediate. The fact that extends the prefix ordering is also clear, since fl is a prefix of ffi iff every finite prefix of fl is also a prefix of ffi. Now, by standard results (see, e.g. [11]) the ideal completion I of the set of finite computations, with respect to the preorder, is a Scott domain whose finite elements are exactly the principal ideals. We claim that the map h, taking each equivalence class [fl] to fffi : ffi finite; ffi flg (which is clearly an element of ....

I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes In Computer Science. Springer-Verlag, 1981.

....state of fl. Computations that are not finite are called infinite. A standard consequence of the ideal construction is that the set of all q computations of a CTS M , under inclusion order, is an algebraic directedcomplete partial order, whose isolated elements are exactly the finite computations [Gue81]. Define a subset fl of M (q; Delta) to be consistent if for every finite subset ffi of fl, the transition W ffi is not q . Proposition 7 Suppose fl is a consistent subset of M (q; Delta) Then there exists a least computation fl of M such that fl fl. 3 Semantics of Concurrent ....

I. Guessarian. Algebraic Semantics. Volume 99 of Lecture Notes in Computer Science, Springer Verlag, 1981.

.... induced by a denotational semantics given in terms of an algebraic domain [Plo81] As such preorders are completely determined by how they act on finite processes, we shall be interested in comparing them with the finitely observable , or finitary, part of the bisimulation in the sense of, e.g. [Gue81, Hen81]. The following definition is from [Abr91] Definition 2.3 Let R 2 Rel(P) The finitary part of R, R F is defined on any lts by sR F s 0 , 8t 2 ST(Lab) tRs ) tRs 0 : 2 6 An alternative method for using the functional F s to obtain a behavioural preorder is to apply it inductively as ....

....d =E j. 4 Preliminaries on Algebraic Semantics In this section, we review the basic notions of algebraic semantics and domain theory that will be needed in the remainder of this study. We assume that the reader is familiar with the basic notions of ordered and continuous algebras (see, e.g. [Gue81, Hen88a, AJ95]) however, in what follows we give a quick overview of the way a denotational semantics can be given to a recursive language like Trees following the standard lines of algebraic semantics [Gue81] The interested reader is invited to consult [Hen88a] for an explanation of the theory. In what ....

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I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer-Verlag, 1981.

....not hard to find. We shall describe perhaps the simplest and most obvious. Consider the (one sorted) signature Sigma specified by Sigma 0 = fu; wg; Sigma 1 = fl; r; dg; Sigma 2 = fm; cg We write W ( Sigma; X) for the free Sigma algebra on a set of generators X . The free ordered Sigmaalgebra [GTWW77, Gue81] W ( Sigma; X) can be constructed as follows: the Sigma algebra structure is that of W ( Sigma; X[f g) where 62 X) and the order is generated by v x, subject to the condition that all operations are monotone. This is the match ordering: tvu just if u can be obtained from t by replacing ....

I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1981.

....for future research. We assume that the reader is familiar with the basic notions of category theory. The necessary background can be found in [23, 2, 13] We also assume some familiarity with the theory of algebraic directed complete partial orders, as used in denotational semantics. Reference [11] provides background on this topic. 2 Concurrent Transition Systems In this section, we define concurrent transition systems and derive some of their basic properties. A graph is a tuple G = O; A; dom; cod) where O is a set of objects, A is a set of arrows, and dom; cod are functions from A to ....

....then every element of J is included in a maximal element of J . Proof (1) That the set of computations of (C; is an algebraic directed complete poset, with the finite computations as isolated elements, is a standard property of the completion by ideals of a partially ordered set (see e.g. [11]) 2) is obvious from the fact that ideals are defined by closure properties. 3) is just Zorn s Lemma applied to sets of computations. We conclude this section with a result that shows that our generalized definition of computations as ideals does not depart too radically from the more ....

I. Guessarian. Algebraic Semantics. Volume 99 of Lecture Notes in Computer Science, Springer Verlag, 1981.

....approaches. Recursive procedures seem to pose no particular difficulties; and we leave as future work the working out of the appropriate domain theory for constructing algebras with recursive methods and recursive data. Work on continuous algebras may be of use in solving such problems [16] 70] [20] [43] 74, Section 3.3.3] Another feature not in is block structure, more specifically local declarations of types and methods. The semantics could be modified to allow the declaration of types and methods in local blocks without conceptual difficulties; however, in doing so it is more ....

I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer-Verlag, New York, N.Y., 1981.

....system ST (Act) hST (Act) Act ; Gamma ; i, where ffl t if and only if Omega is included as a summand of t , and ffl t a i Gamma t i for each summand a i t i of t . Synchronization trees can be used to define a finitary preorder on processes of more general transition systems [Gue81] Definition 5.3 For a transition system hP ; Act ; Gamma ; i define the finitary preorder F P Theta P by p F q if and only if 8t 2 ST (Act) t B p ) t B q : Since finite synchronization trees are a model for finite processes, the finitary preorder can be ....

I. Guessarian. Algebraic Semantics. Volume 99 of Lecture Notes in Computer Science, Springer-Verlag, 1981.

....Next, consider the extension to infinite computations. That is reflexive and transitive is immediate. The fact that extends the prefix ordering is also clear, since fl is a prefix of ffi iff every finite prefix of fl is also a prefix of ffi. Now, by standard results (see, e.g. [12]) the ideal completion I of the set of finite computations, with respect to the preorder, is a Scott domain whose finite elements are exactly the principal ideals. We claim that the map h, taking each equivalence class [fl] to fffi : ffi finite; ffi flg (which is clearly an ....

I. Guessarian. Algebraic Semantics. Volume 99 of Lecture Notes in Computer Science, Springer Verlag, 1981.

....complex than the proofs for the other action notation combinators. Since unfolding A represents a family of action terms, whose size depends on the level of the unfolding, we must 7.7 Proof of Soundness 163 prove soundness for all such possible levels of unfolding. Using standard techniques[Gue81], we represent unfolding A as the union of all possible unfoldings A : Where A 1 A 2 means A 1 is operationally equivalent to A 2 , and A 1 A 2 means A 1 is defined as A 2 . For ACTRESS action notation, we can take A 1 is operationally equivalent to A 2 to mean ....

I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer-Verlag, 1981. Bibliography 191

....classes of terms, defined by a set of equations. However term rewriting techniques may be used to compute other relations than congruences. Particularly interesting are non symmetric relations like pre orders. For instance, logics of inequalities [7] rewriting logic [21] ordered algebras [8], subset logic [12, 24] unified algebras [2, 22] taxonomies [1, 23, 26] subtypes [5] refinement calculus [20] all them use some kind of pre order on expressions. In this paper we will show the applicability of rewriting techniques to monotonic pre order relations on first order terms ....

I. Guesarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer-Verlag, 1981.

....transition system ST (Act) hST (Act) Act ; Gamma ; i, where t if and only if Omega is included as a summand of t , and t a i Gamma t i for each summand a i t i of t . Synchronization trees can be used to define a finitary preorder on processes of more general transition systems [Gue81]. Definition 10. For a transition system hP ; Act ; Gamma ; i define the finitary preorder F P Theta P by p F q if and only if 8t 2 ST (Act) t B p ) t B q : Since finite synchronization trees are a model for finite processes, the finitary preorder can be ....

Guessarian, I.: Algebraic Semantics. Volume 99 of Lecture Notes in Computer Science, Springer-Verlag, 1981.

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I. Guessarian. Algebraic Semantics, volume 99 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1981.

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I. Guessarian, editor. Semantics of Systems of Concurrent Processes, volume 469 of Lecture Notes in Computer Science. Springer-Verlag, 1990.

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