J.C.M. Baeten and J.A. Bergstra, On sequential composition, action prefixes and process prefix, Formal Aspects of Computing 6(3), 1994, pp. 250-268.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and assigned to a variable, which later on in the process expression can be used. A distinction can then be made between late and early instantiation of variables, a distinction which has consequences for the notion of equality and bisimulation of processes. Baeten and Bergstra have proposed [BB94] a general framework, Functional Prefix Algebra, which they use to come up with various algebraic specifications of late and early r(x) if x = 2 then P else Q r(x) 0 Q P Q 0 r(1) r(2) r(3) r(N) r(1) r(N) r(x) if x = 2 then P r(x) if x 6= 2 then Q 0 P 0 Q 0 Q r(1) r(2) r(3) r(N) r(1) ....

....all specifications given, show how the algebras work by executing the specifications using term rewriting, formulate the claims that should be proven in order to assess the correctness of all specifications, and propose directions for further research. This paper is to be read in combination with [BB94]. Before studying that paper, you might find it helpful to make the corrections we give in Appendix A. 1.1 Motivating Example The following example, taken from [MPW91, p.46] illustrates the differences between early and late input (here we are using CCS notation) R = r(x) if x = 2 then P else ....

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J.C.M. Baeten and J.A. Bergstra. On sequential composition, action prefixes and process prefix. Formal Aspects of Computing, 6:250--268, 1994.

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J.C.M. Baeten and J.A. Bergstra, On sequential composition, action prefixes and process prefix, Formal Aspects of Computing 6(3), 1994, pp. 250-268.

....of the above design goals. Finally, we had a few other design goals of lesser importance; we wanted that the existing smaller process algebras (without empty process, without time, or without either) could be embedded in the new process algebras in a simple way (i.e. in the terminology of [4] the old theories should be Subalgebras of a Reduced Model (SRMs) of the new theories) 1.4 Disclaimer This paper should be read in conjunction with [26] as the proofs of some of the theorems given here strongly lean on results and techniques described there. 7 2 Theories with Undelayable ....

J.C.M.BAETEN AND J. A. BERGSTRA. On sequential composition, action prefixes and process prefix. Formal Aspects of Computing, 6(3):250--268, 1994.

....as follows. We define new constants a by the equation a = a#, for each a # A. Then, we reduce the signature by deleting the prefix operators. The subalgebra of the initial algebra that is obtained by this reduced signature is now completely axiomatised by the theory BPA ## of [8, 7] Following [2], we call BPA ## an SRM specification (Subalgebra of Reduced Model Specification) of SPA. Then, we can reduce further by deleting #, or also #, and obtain the SRM specifications BPA # resp. BPA of [11, 8, 7] This means that all specifications and verifications of systems that have been obtained ....

J.C.M. Baeten and J.A. Bergstra. On sequential composition, action prefixes and process prefix. Formal Aspects of Computing, 6(3):250--268, 1994.

....with the defining equation P Q = P Delta (P Q) Q. The behaviour of P Q is zero or more repetitions of P followed by Q. early input and process prefixing We will additionally use early input action prefixing and the extension of this binding construct to process prefixing, both added to ACP in [19]. Early input action prefixing is defined by the equation er i (x) P = P d2D r i (d) Delta P [d=x] We use the extension to processes mainly to express parallel input: er 1 (x 1 ) k : k er n (x n ) P . We have: 4 (er 1 (x 1 ) k er 2 (x 2 ) P = X d12D r 1 (d 1 ) Delta (er 2 (x ....

....P [d 3 =x 3 ] etc. communication free merge We will not only use the merge operator (k) of ACP, but also the communication free merge operator (jjj) The communication free merge operator can be viewed as a special instance of the synchronisation merge operator k H of CSP, also added to ACP in [19], viz. the instance for H = It is defined by P jjj Q = P bbb Q Q bbb P , where bbb is defined as bb except that a Delta P bbb Q = a Delta (P jjj Q) Communication free merge can also be expressed in terms of parallel composition, encapsulation and renaming. 4 Basic network algebra BNA is ....

J.C.M. Baeten and J.A. Bergstra. On sequential composition, action prefixes and process prefix. Formal Aspects of Computing, 6:250--268, 1994.

....defining equation P Q = P Delta (P Q) Q. The behaviour of P Q is zero or more repetitions of P followed by Q. early input and process prefixing We will additionally use early input action prefixing and the extension of this binding construct to process prefixing, both added to ACP in [4]. Early input action prefixing is defined by the equation er i (x) P = P d2D r i (d) Delta P [d=x] We use the extension to processes mainly to express parallel input: er 1 (x 1 ) k : k er n (x n ) P . We have: er1(x1) k er2(x2) P = X d 1 2D r1(d1) Delta (er2(x2) P [d1 =x1 ....

....; P [d3=x3 ] etc. communication free merge We will not only use the merge operator (k) of ACP, but also the communication free merge operator (jjj) The communication free merge operator can be viewed as a special instance of the synchronisation merge operator k H of CSP, also added to ACP in [4], viz. the instance for H = It is defined by P jjj Q = P bbb Q Q bbb P , where bbb is defined as bb except that a Delta P bbb Q = a Delta (P jjj Q) Communication free merge can also be expressed in terms of parallel composition, encapsulation and renaming. priority The priority operator ....

J.C.M. Baeten and J.A. Bergstra. On sequential composition, action prefixes and process prefix. Formal Aspects of Computing, 6:250--268, 1994.

....of the above design goals. Finally, we had a few other design goals of lesser importance; we wanted that the existing smaller process algebras (without empty process, without time, or without either) could be embedded in the new process algebras in a simple way (i.e. in the terminology of [4] the old theories should be Subalgebras of a Reduced Model (SRMs) of the new theories) 1.4 Disclaimer This paper should be read in conjunction with [26] as the proofs of some of the theorems given here strongly lean on results and techniques described there. 2 Theories with Undelayable ....

J. C. M. BAETEN AND J. A. BERGSTRA. On sequential composition, action prefixes and process prefix. Formal Aspects of Computing, 6(3):250--268, 1994.

....x except those that are derivably equal to # (see Proposition 2.1.7) which implies that in a theory without immediate deadlock the law x cts(#) x will hold. The standard process algebra BPA # can be considered as an SRM specification (Subalgebra of Reduced Model, in the terminology of [3]) of BPA drt : consider the initial algebra of BPA drt , reduce the signature by omitting #, # rel , # rel , then BPA # is a complete axiomatization of the reduced model, under the interpretation of a, # (from BPA # ) by cts(a) cts(#) However, this is not the embedding of the time free ....

J. C. M. Baeten and J. A. Bergstra. On sequential composition, action prefixes and process prefix. Formal Aspects of Computing, 6(3):250--268, 1994.

....algebra of ACP pin . We claim that standard ACP is an axiomatisation of this algebra, again substituting a for p a p , and ffi for p ; Thus, ACP is a subalgebra of a reduced model specification (an SRM specification) of ACP pin . For more information on SRM specifications, we refer to [BaB94]. 9.2 ACP with ; Now extend the signature Sigma above with the extra constant ; We claim that the axioms in table 15 constitute an SRM specification for this signature (a; b 2 A [ fffig) 10 Conclusions We generalised process algebra to processes with multiple entries and exits, so called ....

J.C.M. Baeten and J.A. Bergstra, On sequential composition, action prefixes and process prefix, Formal Aspects of Computing 6(3), 1994, pp. 250-268.

....defining equation P Q = P Delta (P Q) Q. The behaviour of P Q is zero or more repetitions of P followed by Q. early input and process prefixing We will additionally use early input action prefixing and the extension of this binding construct to process prefixing, both added to ACP in [4]. Early input action prefixing is defined by the equation er i (x) P = P d2D r i (d) Delta P [d=x] We use the extension to processes mainly to express parallel input: er 1 (x 1 ) k : k er n (x n ) P . We have: er 1 (x 1 ) k er 2 (x 2 ) P = X d12D r 1 (d 1 ) Delta (er 2 (x 2 ....

....; P [d 3 =x 3 ] etc. communication free merge We will not only use the merge operator (k) of ACP, but also the communication free merge operator (jjj) The communication free merge operator can be viewed as a special instance of the synchronisation merge operator kH of CSP, also added to ACP in [4], viz. the instance for H = It is defined by P jjj Q = P bbb Q Q bbb P , where bbb is defined as bb except that a Delta P bbb Q = a Delta (P jjj Q) Communication free merge can also be expressed in terms of parallel composition, encapsulation and renaming. discrete time We need a ....

J.C.M. Baeten and J.A. Bergstra. On sequential composition, action prefixes and process prefix. Formal Aspects of Computing, 6:250--268, 1994.

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