Stephen L. Bloom and Zolt'an ' Esik. Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algebra if it satisfies all identities satisfied by the continuous # algebras or, equivalently, the regular # algebras of [Tiu78] or the iterative # algebras of [Nel83,Tiu80] A morphism of iteration algebras is a morphism of # algebras. For axiomatic treatments of iteration algebras, see [BE93a] It is proved in [ Esi99] that an # algebra is an iteration algebra i# it is a Conway algebra satisfying certain group identities . When # is either clear from context or not important, we say only iteration algebra instead of # iteration algebra . The class of scalar iteration algebras is ....

.... group identities . When # is either clear from context or not important, we say only iteration algebra instead of # iteration algebra . The class of scalar iteration algebras is the class of all # iteration algebras, as # varies over all scalar signatures. The following is well known [BE93a] Proposition 2. If A is an iteration algebra, A satisfies the power identities: for each term t = t[x, y 1 , y p ] x.t # x.t n , n # 1, where t 1 = t and t k 1 : t[t k x] # 5 4 Scalar # iteration algebras We have already pointed out in Proposition 2 that any iteration ....

S.L. Bloom and Z. Esik. Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer--Verlag, 1993. 10

....algebra with the feedback operation. Our goal is to de ne a process algebra on which all operators of ACP [BeK84] are present as well as feedback, the key iteration construct of owchart theories. To this end we combine the results of [BaB95] BeS94] and various other results on owchart theories [Elg75, Ste87, Ste86, CaS90, BlE93, Ste94]. Like in owchart schemes [Ste86] feedback and alternative composition 3 suce to express all nite state systems. 3 is a mixture of disjoint sum , left composition with converses of functions and right composition with functions. In fact alternative composition and feedback allow ....

....to owchart schemes (see [CaS90, Ste94] for example) The present SIM axiom is related to the functoriality of functions. A weaker equational instance of the functoriality of functions is the key axiom of iteration theories and was used to get an axiomatisation of synchronization trees, cf. [BlE93]. 4 Conclusions We have presented a model of process graphs with multiple entries and exits. This model allows for a smooth integration of looping and parallel operators. Axiomatisations for such processes under graph isomorphism and strong bisimulation equivalences are given. An extension of ....

S.L. Bloom and Z. Esik. Iteration theories: the equational logic of iterative processes. EATCS Monographs in Theoretical Compputer Science, Springer Verlag, 1993.

....process algebra with the feedback operation. Our goal is to define a process algebra in which all operators of ACP [6] are present as well as feedback, the key iteration construct of flowchart theories. To this end we combine the results of [2] 8] and various other results on flowchart theories [11, 13, 14, 16, 22, 23]. Like in flowchart schemes [23] feedback and alternative composition 3 suffice to express all finite state systems. Actually, 3 is a mixture of disjoint sum 8 , left composition with converses of functions and right composition with functions. In fact alternative composition and ....

....of finite relations was widely used in various axiomatizations related to flowchart schemes (see [13] for example) The present SIM axiom is related to the functoriality of finite functions. A weaker equational instance of the functoriality of functions is the key axiom of iteration theories [11] and it has been used to get an axiomatisation of synchronization trees in [12] 4 Final Remarks We have presented a model of process graphs with multiple entries and exits. This model allows for a smooth integration of looping and parallel operators. Axiomatisations for such processes under ....

S.L. Bloom and Z. Esik. Iteration theories: the equational logic of iterative processes. EATCS Monographs in Theoretical Computer Science, Springer Verlag, 1993.

....and Corecursion Have the Same Equational Logic Lawrence S. Moss 1 Abstract This paper is concerned with the equational logic of corecursion, that is of definitions involving final coalgebra maps. The framework for our study is iteration theories (cf. e.g. Bloom and Esik [2, 3]) recently re introduced as models of the FLR 0 fragment of the Formal Language of Recursion [7, 8, 9] We present a new class of iteration theories derived from final coalgebras. This allows us to reason with a number of types of fixedpoint equations which heretofore seemed to require metric ....

....result in the subject. Section 8 discusses ideological aspects of the work, compares it to that of others, and lists some related projects as well. The reader may wish to consult it while reading the rest of the paper. 2 Background on iteration theories can be found in Bloom and Esik [2] or [3] on the Formal Language of Recursion in Hurkens et al. [7] on coalgebra in Rutten [14] and on our approach to corecursion in [11] This paper does make use of several results from these papers which of course we mention explicitly. Apart from these, the paper is self contained. 2 FLR 0 ....

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Stephen L. Bloom and Zolt'an ' Esik, Iteration Theories: The equational logic of iterative processes, EATCS Monographs on Theoretical Computer Science, Berlin: SpringerVerlag (1993).

....condition. However, this is also true of the functions [f ] from earlier. Another has to do with the formal properties of the operation f 7 f y given in Theorem 2.11. It can be shows that this association gives rise to an iterative 9 algebraic theory in the sense of Bloom and Esik [4]. This is equivalent to having a model of the FLR 0 fragment of Moschovakis formal language of recursion FLR 1 , a logical system studied in [5] The overall point is that the equational properties of f 7 f y are the same as those of the canonical form of recursive definitions, where f is a ....

Stephen L. Bloom and Zolt'an ' Esik, Iteration Theories: The equational logic of iterative processes, EATCS Monographs on Theoretical Computer Science, Berlin: SpringerVerlag (1993). 21

.... that we have a proper chain D # DB S (It would also be interesting to construct two mutually incomparable models strictly below S; perhaps this could be done by refining the current methods to construct models which do not satisfy some of the iteration theory axioms of Bloom and Esik[BE93] reading x.M as Y(#x.M ) The idea of the proof is to find a projection ## : D # # # D# such that ## (D # [ M ] #) D[ M ] #) 1) 22 for all L terms M . For this implies that EqTh L (D # ) # EqTh L (D) and so D # #DB . Equation 1 will be established by a logical ....

S. Bloom and Z. Esik, Iteration Theories: The Equational Logic of Iterative Processes, EATCS Monographs on Theoretical Computer Science, Berlin: Springer-Verlag, 1993.

.... mutual recursion as a primitive and is used and studied extensively in [12, 5] 7 The decidability of the class of standard identities is an old result, probably due to Courcelle, Kahn and Vuillemin [1] and an explicit axiomatization was probably first given (in categorical form) in Bloom Esik [2]. See [5] for a careful study of them and additional references, and also [11] for an extension of the decidability and axiomatization results to the full, Formal Language of Recursion FLR of [6] which allows functional recursion. Yiannis N. Moschovakis, to appear in Proceedings of CSL 97. ....

....several examples, to make sense; it is given in [9, Section 8] and again in [20, 12] for a finer model. Some care is also needed to make precise the notion of intensional model for FL 0 , and this, too, is done in the same papers, although some will prefer the categorical version in Bloom Esik [2]. When all this is said and done correctly, we have the following, basic result from [9, 12] C. Main Old Theorem. For each, countably complete poset D, the set P(D) of players with the implemented player functions, ipf composition and ipf recursion, form an intensional model of FL 0 which ....

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Stephen L. Bloom and Zoltan ' Esik. Iteration theories: the equational logic of iterative processes. EATCS Monographs in Theoretical Computer Science. SpringerVerlag, Berlin, 1993. Yiannis N. Moschovakis, to appear in Proceedings of CSL'97.

....there are many machine categories and many behaviours; the choice of machine category corresponds to a choice of communication, the choice of the behaviour corresponds to a choice of an abstraction level. The semantic categories are very often of the type considered in iteration theories [Elg75] [BE93]. The abstract mathematical setting (strictly speaking the theory of bicategories [B en67] CW87] CKW87] is not emphasized in this paper, rather the application to concurrency. A more precise analysis of the mathematical setting is begun in [KSW94] It has come as some surprise to the second ....

S.L. Bloom and Z. ' Esik. Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

....given in [B E94] together with an analogue of the variety theorem. They are specialized from work on iteration theories. These provide a general setting for the investigation of the equational logic of fixed point operators. We cannot do justice to the work here, but instead refer the reader to [B E93b] One can also look for axiomatisations which are sound in several intended models, as is done for example in [Koz94] for the expressions. In particular, axioms involving a guarded condition tend not to be valid in coarser models. Domain theoretic models of the expressions will be briefly ....

Stephen L. Bloom and Zoltan Esik. Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

....derived from work on iteration theories. Finally we should mention that various illuminating infinite but simple equational systems exist. Discussion of these may be found for example for language equivalence of star expressions in [Kro91] in the general setting of iteration theories in [BE93a, BE93b, BET93] and for bisimulation of expressions in [Sew] As to future work, several directions spring to mind. Firstly one could attempt to remove the restriction in the present work to axioms of order one. This requires an improved understanding of the interaction between fi reduction and the ....

Stephen L. Bloom and Zolt'an ' Esik. Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1993.

....a later paper. 2 1 INTRODUCTION Our goal is to define a process algebra on which all operators of ACP [BeK84] are present as well as feedback, the key iteration construct of flowchart theories. To this end we combine the results of [BaB94] BeS93] and various other results on flowchart theories [Gog74, Elg75, Ste87, Ste90, CaS90, BlE93]. Like in flowchart schemes [Ste90] feedback and alternative composition 3 suffice to express all finite state systems. 3 is a mixture of disjoint sum Phi , left composition with converses of functions and right composition with functions. In fact alternative composition and ....

....classes of relations was widely used in various axiomatizations related to flowchart schemes (see [CaS90] for example) The present SIM axiom is related to the functoriality of functions. A weaker equational instance of the functoriality of functions is the key axiom of iteration theories [BlE93] and was used to get an axiomatisation of synchronization trees in [BlET93] Theorem 12 The axioms of graph isomorphism, those for splitting (in Table 2) and SIM are correct and complete for process graphs modulo bisimulation. Proof: i) Completeness: By using the axioms in Table 2 it follows ....

S.L. Bloom and Z. Esik. Iteration theories: the equational logic of iterative processes. EATCS Monographs in Theoretical Compputer Science, Springer Verlag, 1993.

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Stephen L. Bloom and Zolt'an ' Esik. Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1993.

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Stephen L. Bloom and Zolt'an ' Esik. Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

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Stephen L. Bloom and Zoltan Esik. Iteration Theories: The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

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S.L. Bloom and Z. Esik (1991), Iteration Theories: the Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science 30, Springer-Verlag.

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Bloom, S.L. and Esik, Z. (1993). Iteration Theories: The Equational Logic of Iterative Processes. Springer-Verlag.

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S. L. Bloom and Z. Esik, Iteration Theories: The Equational Logic of Iterative Processes, EATCS Monographs on Theoretical Computer Science (W. Brauer, G. Rozenberg and A. Salomaa eds.), Springer-Verlag, 1993.

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S.L. Bloom and Z. Esik (1991), Iteration Theories: the Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science 30, Springer-Verlag.

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