J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we also need to use second order quantification to deal with iteration and recursive procedures. It is not at all obvious that such complex definitions are well behaved or even consistent. Of course ours is not the first formal model of concurrency. In fact, well developed approaches are available [17, 25, 4, 39] 1 and our work inherits many of the intuitions behind them. However, it is distinguished from these in at least two fundamental ways. First, it allows incomplete information about the environment surrounding the program. In contrast to typical computer programs, the initial state of a ConGolog ....

J. De Bakker and E. De Vink. Control Flow Semantics. MIT Press, 1996.

.... processes as concrete objects, such as failure traces [Phi87] traces decorated with actions that cannot be executed at certain moments, Mazurkiewicz traces [Maz88] which contain an explicit indication of parallelism, event structures [Win87] Petri nets [Rei85, Jen92] objects in metric spaces [dBdV96] etc. etc. A partial overview of process models is given in [vG90, vG93] A very useful perspective, which we employ for illustrations, is the view of a process as an automaton of which the transitions are labelled with actions. Each traversal through the automaton is a run of the process. This ....

J.W. de Bakker and E. de Vink. Control Flow Semantics. MIT Press, 1996. 61

....(3) some propositions that we wish to prove, and possibly, 4) some details of the actual proof. See the above diagram in which these four points can be recognised. This project makes heavy use of traditional results and techniques from the semantics of programming languages, see e.g. [5, 10]. In a nutshell, traditional reasoning about programs in a language L proceeds as follows. First, a suitably rich mathematical structure D is identified, which can serve as semantic domain for L, and as domain of reasoning. Then, an interpretation function [ Gamma ] L D is written out, ....

J.W. de Bakker and E. Vink. Control Flow Semantics. The MIT Press, Cambridge, MA, 1996.

....For hA; di with A 6= a com21 plete metric space and f : A Gamma A a contracting function on hA; di we have (1) f has a unique fixed point, say x, and (2) any sequence (x n ) such that x i 1 = f(x i ) has limit x. 5. 2 Denotational semantics We only give a brief account of our approach; see [35,10,6,11] for more information on the use of metrics for denotational semantics. The semantic domain S in our case a suitable variant of TES for PA is equipped with a set Op 0 of operators that reflect the operators Op of Expr. For any fixed declaration decl, the function P 7 M(decl; P ) for P ....

....Traces(E 1) f Theorem 26 g S t 0 Traces( E 1) t) f Lemma 23 g 25 S t 0 Traces(E t) f Theorem 26 g Traces(E) The equivalence intended above is now defined by E 1 E 2 if and only if E 1 1 = E 2 1. It is quite standard to abstract from event identities in metric semantics [11,29], i.e. to deal with isomorphism classes of semantic structures. The event identities are only needed for technical reasons but they are meaningless for the semantics of an expression. The following definition is the usual notion of isomorphism with the only exception that the bijection is defined ....

J.W. de Bakker and E.P. de Vink. Control Flow Semantics. MIT Press, 1996.

....is indeed mathematically well behaved. The paper is self contained in terms of definitions and theorems; motivation and examples, however, are to be found in the companion paper. Of course ours is not the first formal model of concurrency. In fact, well developed approaches are available [13, 17, 3, 24] 1 and our work inherits many of the intuitions behind them. However, it is distinguished from these in at least two fundamental ways. First, it allows incomplete information about the environment surrounding the program. In contrast to typical computer programs, the initial state of a ConGolog ....

J. De Bakker and E. De Vink. Control Flow Semantics. MIT Press, 1996.

....The relevance of 2 Stone spaces, i.e. Partially supported by EC HCM project Lambda Calcul Typ e, CHRX CT92.0046. countably based, totally disconnected compact Hausdorff spaces, arises from the fact that compact ultrametric spaces , a category of spaces widely used in metric semantics (see [8]) are 2 Stone spaces. A natural partialization of a 2 Stone space hX; i by a Scott domain can be immediately obtained as the ideal completion of the collection K Omega ne (X) of non empty compact open subsets of X , ordered by reverse inclusion D X 1 = Idl(K Omega ne (X) Such domains ....

J.W. de Bakker and E. de Vink. Control Flow Semantics. MIT Press, 1996.

....language is that it allows us to conveniently formulate agent controllers that pursue goal oriented tasks while concurrently monitoring and reacting to conditions in their environment. Of course ours is not the first formal model of concurrency. In fact, well developed approaches are available [16, 23, 4, 35] 1 and our work inherits many of the intuitions behind them. However, it is distinguished from these in at least two fundamental ways. First, it allows incomplete information about the environment surrounding the program. In contrast to typical computer programs, the initial state of a ConGolog ....

J. De Bakker and E. De Vink. Control Flow Semantics. MIT Press, 1996.

....guidelines. This implied that the formal description had to be on a level of abstraction that: ffl hides implementation details, but ffl is still easy to derive implementations from. A first consequence of this is the choice for an operational semantics as opposed to denotational semantics [1]. Using an operational semantics has the advantage over an actual implementation, that we are free to abstract from many implementation details that are of less importance, or that we simply do not want to fill in yet. Still, we can choose a level of abstraction that is still sufficiently close to ....

J. d. Bakker and E. d. Vink. Control Flow Semantics. The MIT Press, Cambridge, Massachusetts, 1996.

....1) let E 1 t E 2 , E 1 [t;t] E 2 . By straightforward proof one can establish that TES is closed under the operators a I : nA, jj A , and t . 5 A metric denotational semantics The approach. We only give a brief account of our approach; see [2] for a full treatment, and [15, 5, 6] for more information on the use of metrics for denotational semantics. The semantic domain S for PA is equipped with a set Op 0 of operators that reflect the operators Op of Expr. For any fixed declaration decl, the function P 7 M(hdecl; P i) is a homomorphism from (Expr; Op) to (S; Op 0 ) ....

J.W. de Bakker and E.P. de Vink. Control Flow Semantics. MIT Press, 1996.

....of concurrency. The importance of SFP domains is unquestionable (see [16] The relevance of 2 Stone spaces, i.e. countably based, totally disconnected compact Hausdorff spaces, arises from the fact that compact ultrametric spaces , a category of spaces widely used in metric semantics (see [8]) are 2 Stone spaces. A natural partialization of a 2 Stone space hX; i by a Scott domain can be immediately obtained as the ideal completion of the collection K Omega ne (X) of non empty compact open subsets of X , ordered by reverse inclusion D X 1 = Idl(K Omega ne (X) Such domains ....

J.W. de Bakker and E. de Vink. Control Flow Semantics. MIT Press, 1996.

....may be described as construction versus observation. It may be found in process theory [50] data type theory [21, 25, 5, 40] including the theory of classes and objects in objectoriented programming [61, 30, 37, 35] semantics of programming languages [46] denotational versus operational [64, 67, 6]) and of lambda calculi [58, 59, 20, 31] automata theory [53] system theory [65, 34] natural language theory [10, 62] and many other fields. We assume that the reader is familiar with definitions and proofs by (ordinary) induction. As a typical example, consider for a fixed data set A, the set ....

J.W. de Bakker and E. Vink. Control Flow Semantics. The MIT Press, Cambridge, MA, 1996.

....= 0 for x = 2 fx 1 ; xn g. D(X) denotes the collection of all simple probability distributions on X . For E X , E] is short for P x2E (x) This way, a simple probability distribution corresponds to a convex linear combination of Dirac measures. Metric spaces (See, e.g. the monograph [BV96]. A pair (M; d) with M a nonempty set and d: M 2 [0; 1] is called an ultrametric space if, for all x; y; z 2 M : d(x; y) d(y; x) d(x; y) 0 , x = y, and d(x; z) maxfd(x; y) d(y; z)g. The last expression is referred to as the strong triangle inequality. For metric spaces M 1 ; M 2 , a ....

....a topology as is the case if the set of processes is endowed with an order or a metric structure the obvious choice for this oe algebra is the Borel oe algebra, i.e. the least oe algebra containing all the open sets. As mentioned in the introduction, we prefer the use of ultrametric (cf. [BV96]) above order, because of a combination of the following two reasons: 1) the technical advantage of a close relationship between standard measure theory and metric topology, and (2) the availability of a final coalgebra theorem in the metric setting, leading to a fully abstract domain for general ....

J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, 1996.

.... is called the support of . D(X) denotes the collection of all simple probability distributions on X. For E # X, E] is short for P x#E (x) This way, a simple probability distribution corresponds to a convex linear combination of Dirac measures. Metric spaces (See, e.g. the monograph [BV96]. A pair (M, d) with M a nonempty set and d: M 2 # [0, 1] is called an ultrametric space if, for all x, y, z # M : d(x, y) d(y, x) d(x, y) 0 # x = y, and d(x, z) # max d(x, y) d(y, z) The last expression is referred to as the strong triangle inequality. For metric spaces M 1 , M ....

....: O # X = # ## O # Y = # . We will employ the following result: If M is a complete ultrametric space, then the collection P co (M) of all compact subsets of M supplied with the Hausdor# distance is also a complete metric space. For a proof of this fact the reader may consult, e.g. [BV96], Appendix A. For a Borel probability measure with compact support we define its support spt( by spt( T K # M K compact, vanishes outside K . For any open O # O(M) we then have the equivalence (O) 0 ## spt( # O = #. See, e.g. Rud66, p57] Lemma 5.10 (a) If M ....

J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, 1996.

....function space. By Banach s Theorem they have a unique fixed point, which is the object specified by the equations of the original definitions. The nice point of using metric spaces as underlying mathematical structure is the close connection between measure theory and metric topology. We refer to [15, 3] for a further discussion on the use of complete metric spaces and Banach s Theorem in the area of programming language semantics. The advantage of using metric spaces is also reflected in a number of semantical investigations related to probability. See [6, 16, 4] for example. The mathematical ....

....ff 0 j 8x 2 S 1 9y 2 S 2 [d(x; y) ff] 8y 2 S 2 9x 2 S 1 [d(x; y) ff] g : It is well known that the collection of nonempty compacta with the Hausdorff distance is complete whenever the underlying metric space is complete. Thus, in our situation, SPMS is a complete metric space. See, e.g. [3] for more details on dB and dH . The Dirac measure for the string w is denoted by 1 w . So 1 w (B) 1 if w 2 B, and 1 w (B) 0 otherwise, for any Borel set B A 1 ffi . Dirac measures are Borel probability measures of compact support, viz. the singletons fwg. Below we encounter 1 ffl , 1 ....

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J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, 1996.

....noted that in this paper we deal with a process language without synchronization. Operational and denotational semantics for a language with synchronization are feasible as well. One then has to resort to a branching time domain for the construction of the denotational semantics (see, for example, [BV96, Har96]) It is a topic of further investigations to see whether, e.g. failure sets can be employed to obtain a fully abstract model for such a language. We now discuss some related investigations dealing with full abstractness for action refinement. An early result concerning full abstraction and ....

....constructed employing the notion of a semantic refinement. Section 5 treats the comparison of the operational and denotational model and presents the full abstractness result. 2 Mathematical preliminaries We refer to [Dug76, Eng89] standard textbooks in general topology, and to the monograph [BV96] or the overview [Smy92] which have a theoretical computer science perspective, for the various basic definitions and facts from elementary metric topology: metric space, ultrametric, 1 bounded metric, completeness, compact subset. Here we discuss the metric spaces, constructions and specific ....

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J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, 1996.

....subsets of PMS. We assume SPMS to come equipped with the Hausdorff distance for which d(S 1 ; S 2 ) 2 Gamman if S 1 ; S 2 intersect the same balls B 2 Gamman ( 2 PMS, but not the same of ball of radius 2 Gamma(n 1) In our set up, SPMS is a complete metric space. See, e.g. [2] for more details on the Baire distance and the Hausdorff distance and [24] for the distance on measures. The Dirac measure for the string w is denoted by 1 w . So 1 w (B) 1 if w 2 B, and 1 w (B) 0 otherwise, for any Borel set B A 1 ffi . Below we encounter 1 ffl , 1 ffi and 1 a (and a ....

....that (i;j) 0 only if (i; j) 2 C and P m j=1 (i 0 ;j) ae i 0 and P n i=1 (i;j 0 ) oe j 0 . Proof This lemma is a consequence of the max flow min cut theorem (see, e.g. 4] 2 We can show the following properties of the metric d on SPMS. The first two properties are standard (cf. [2] for example) The third is easy to check using the definition of the lifted Phi. The last inequality is based on the Splitting Lemma 2.1 above. Lemma 2.2 (a) d(a;S; a;S 0 ) 1 2 d(S; S 0 ) b) d( S f S i j i 2 I g; S f S 0 i j i 2 I g) maxf d(S i ; S 0 i ) j i 2 I g) c) ....

J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, 1996.

.... (hd; S 1 2S 2 i) maxfwgt 0 (hd; S 1 i) wgt 0 (hd; S 2 i)g 1; The function wgt 0 is well defined for each pair hd; Si 2 L 0 as can be easily seen by induction on the syntactic complexity first of guarded statements and then of general statements (for more on the use of the weight function see [BV95]) We are now ready for the justification of wp 0 . It is based on a mapping Psi 0 : Sem 0 Sem 0 where (F 2) Sem 0 = L 0 Pi 0 . Pivotal is the clause Psi 0 (F ) S 1 ;S 2 ) Psi 0 (F ) S 1 ) OE(S 2 ) for the sequential command. Note that F and not Psi 0 (F ) is applied to the second ....

....then needs to be adapted as well. Conjunctivity will replace the multiplicativity condition. The dual domains of state transformers need to be extended with a special element to cater for a denotation for abort. Locality can be dealt with, using techniques developed in the metric setting. cf. [BV95]. It is an open problem, however, how to deal with angelic nondeterminacy here. Complementary to this is the issue of full abstractness or, more restrictedly, of the adequacy of linear models for L 2 and L 3 . Since no such feature as message passing is present in the languages we may expect ....

J.W. de Bakker and E. de Vink. Control Flow Semantics. MIT Press, 1995. To appear.

....the synchronous execution of c s and c x is modelled by passing the denotation of s and storing it in x in the semantic store. For a large variety of languages denotational semantics based on ultrametric spaces have been developed (numerous examples are provided by De Bakker and De Vink in [BV95]) Higher order notions have been modelled denotationally by, e.g. Hennessy [Hen94] Jagadeesan and Panangaden [JP90] and Thomson [Tho90] To link the operational and denotational semantics we introduce an intermediate semantics. Like the operational semantics the intermediate semantics is ....

J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, Cambridge, 1995. To appear.

....function space. By Banach s Theorem they have a unique fixed point, which is the object specified by the equations of the original definitions. The nice point of using metric spaces as underlying mathematical structure is the close connection between measure theory and metric topology. We refer to [14, 3] for a further discussion on the use of complete metric spaces and Banach s Theorem in the area of programming language semantics. The advantage of using metric spaces is also reflected in a number of semantical investigations related to probability. See [5, 15, 4] for example. The mathematical ....

....used. Though straightforward, the proof is cumbersome and omitted here. However, it is here that we benefit from using the metric approach to programming language semantics, where Banach s Fixed Point Theorem is instrumental in the justification of circular definitions. We mention the references [14, 3, 8] for more details of the application of this technique. Lemma 3.6 (a) T 0 is finitely branching. b) The transition system T 0 has no internal divergence, i.e. there are no statements s 0 ; s 1 ; s 2 ; and 1 ; 2 ; in fg [ 0; 1) such that s 0 1 s 1 2 s 2 Delta Delta ....

J.W. de Bakker and E.P. de Vink. Control Flow Semantics. The MIT Press, 1996.

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J.de Bakker, E.de Vink. Control Flow Semantics, MIT Press, 1996.

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BV96. J.de Bakker, E.de Vink. Control Flow Semantics, MIT Press, 1996.

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J. De Bakker and E. De Vink. Control Flow Semantics. MIT Press, 1996.

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