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ISBN-10: 0471974420

ISBN-13: 9780471974420

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2 —* F for all a 6 A, so called transitions having a label in A; 3. The empty frame 0 € F; 4. The binary frame operation ®, alternative frame composition or frame union. , 0)-(1 is considered a frame. The signature introduced above can be depicted as follows: . -2-> . (for all a £ A) The constant 0 denotes the empty frame, containing neither states nor transitions. The operation ® on frames gives the union of the states and transitions of both its arguments, and is defined in Table 1 (see also Bergstra and Ponse 1994).

Now g —» p has states S(g), each edge s -% t in g with t £ O(g) is replaced by an edge s -^-> p, and furthermore has the same entries as g and has as only exit p. 5. Initialisation operator. Let a process graph g be given, and let p £ V. p ~S> g has the same edges and exits as g. The set of entries becomes 1(9} ^ {P}- States are all states of g, plus in addition other entries than p (if they exist). 6. Exiting operator. Let a process graph g be given, and let p e V. g » p has the same edges and entries as g.

1984), 0 (zero process Baeten and Bergstra 1990). X + Y = Y +X (X + Y) + Z = X + (Y + Z) X +X =X (X + Y)-Z = X-Z + Y-Z (X • Y) • Z = X • (Y • Z) X +0= X a+6=a \$-X = 0 S-X = 6 X -% = X-t> a | 6 = 7(0, b) a\b = 6 if 7(0, 6) defined otherwise a |L (X + 6) = a • (X + a-X\b=(a\b)-X a | 6 • X = (a | 6) • X a-X\b-Y = ( a \ b ) - ( ( X + S ) \ \ ( Y + 6)) (X + Y)\Z = X \Z + Y\Z a) = a dH(a)=S if a £ H ifaGtf dH(X-Y)=dH(X)-dH(Y) TABLE 15. ACP0. TEFANESCU References Aceto, L , and M Hennessy 1992 Termination, deadlock, and divergence Journal of the ACM 39(1) 147-187 Baeten, J C M , and J A Bergstra 1990 Process algebra with a zero object In Proceedings CONCUR 90, Amsterdam, ed J C M Baeten and J W Klop, Lecture Notes in Computer Science, Vol 458, 83-98 Springer-Verlag Baeten, J C M , and J A Bergstra 1994 On sequential composition, action prefixes and process prefix Journal of Formal Aspects of Computing Science 6(3) 83-98 Baeten, J C M , and C Verhoef 1993 A congruence theorem for structured operational semantics with predicates In Proceedings CONCUR 93, Hildesheim, Germany, ed E Best, Lecture Notes in Computer Science, Vol 715, 0 - 0 Springer-Verlag Baeten, J C M , and C Verhoef 1995 Concrete Process Algebra In Handbook of Logic in Computer Science, ed S Abramsky, D M Gabbay, and T S E Maibaum Chap 2, 149-268 Oxford University Press To appear Baeten, J C M , and W P Weijland 1990 Process Algebra Cambridge Tracts in Theoretical Computer Science 18 Cambridge University Press Barendregt, H , H Wupper, and H Mulder 1994 Computable processes Report CSI-R9405 Computing Science Institute, Catholic University of Nijmegen Bartha, M 1992 An algebraic model of synchronous systems Information and Computation 97 79-131 Bergstra, J A 1989 Algebra of States and Transitions Report P8909 Programming Research Group, University of Amsterdam Bergstra, J A , I Bethke, and A Ponse 1994a Process algebra with combmators In Proceedings CSL '93, Swansea, ed E Borger, Y Gurevich, and K Memke, Lecture Notes in Computer Science, Vol 832, 36 65 Springer-Verlag Bergstra, J A I Bethke, and A Ponse 1994b Process algebra with iteration and nesting Computer Journal 37(4) 243-258 Bergstra, J A and J W Klop 1984 Process algebra for synchronous communication Information and Computation 60(1/3) 109-137 Bergstra, J A , and Gh §tefanescu 1994 Processes with multiple entries and exits modulo isomorphism and modulo bisimulation Report P9403 Programming Research Group, University of Amsterdam Bloom, S L , and Z Esik 1993 Iteration theories, the equatwnal logic of iterative processes Springer-Verlag Brookes, S D , C A R Hoare, and A W Roscoe 1984 A theory of communicating sequential processes Journal of the A CM 31(3) 560-599 Cazanescu, V E , and Gh §tefanescu 1990 Towards a new algebraic foundation of flowchart scheme theory Fundamenta Informaticae XIII 171-210 Cazanescu, V E , and Gh §tefanescu 1992 A general result of 1686 abstract flowchart schemes with applications Theoretical Computer Science 99 1-63 ExSpect-Team 1992 ExSpect user manual Eindhoven University of Technology PROCESS ALGEBRA WITH FEEDBACK / 37 Klusener, A S 1993. 