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Algebra of Flownomials; Part 1: Binary Flownomials; Basic Theory
"... ' morphism for connecting flowgraphs are used in [CaU82] and in all of our subsequent papers on flowchart schemes and flownomials, see [Ste87a, Ste87b, CaS88a, CaS90a, CaS92]. This chapter folows Chapter B, sec. 36 of [Ste91]. The main result is based on a series of papers dealing with the algebra ..."
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' morphism for connecting flowgraphs are used in [CaU82] and in all of our subsequent papers on flowchart schemes and flownomials, see [Ste87a, Ste87b, CaS88a, CaS90a, CaS92]. This chapter folows Chapter B, sec. 36 of [Ste91]. The main result is based on a series of papers dealing with the algebraization of flowchart schemes, including [CaU82, BlEs85, Ste86/90, Bar87a, CaS88a, CaS90b]. With different sets of operators various algebras for flowgraphs appear in [Mil79, Parr87, CaS90b, CaS88b]. In the classical algebraic calculus for regular languages it is often the case that certain abstract semirings are used instead of the Boolean f0; 1g semiring, e.g. by using formal series with such coefficients. 5 This property is similar to the universal property of the polynomials over a ring. Chapter 6 Graph isomorphism with various constants In this chapter we extend the axiomatistion for flowgraphs modulo isomorphism to the case where more constants for generating relations are present i...
Bisimulation is TwoWay Simulation
, 1994
"... We give here a simple proof of the fact that on transition systems bisimulation is the equivalence relation generated by simulation via functions. The proof entirely rests on simple rules of the calculus of relations. Keywords: theory of computation, concurrency, transition systems, equivalence Sim ..."
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We give here a simple proof of the fact that on transition systems bisimulation is the equivalence relation generated by simulation via functions. The proof entirely rests on simple rules of the calculus of relations. Keywords: theory of computation, concurrency, transition systems, equivalence Simulation is a standard notion of graph homomorphism that has been used in the study of flow diagram programs (see, e.g. [4, 3, 7]). Bisimulation is an equivalence on transition systems introduced by Park [6] in connection with Milner's work on concurrency [5]. In [2] we have shown that bisimulation is the equivalence relation generated by simulation via functions by using a translation between flowchart schemes and process graphs. We give here a simple proof of this fact using transition systems and simple rules of the calculus of relations, only. Remarks on notation: We use "ffi" to denote relational composition of two binary relations (i.e., R ffi T = f(x; z) j 9y : x R y and y T zg and Id...
Twisted Systems and the Logic of Imperative Programs
, 1998
"... Following Burstall, a flow diagram can be represented by a pair consisting of a graph and a functor from the free category to the category of sets and relations. A program is verified by incorporating the assertions of the FloydNaur proof method into a second functor and exhibiting a natural transf ..."
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Following Burstall, a flow diagram can be represented by a pair consisting of a graph and a functor from the free category to the category of sets and relations. A program is verified by incorporating the assertions of the FloydNaur proof method into a second functor and exhibiting a natural transformation to the program. A broader range of properties is obtained by substituting spans for relations and introducing oplaxness into both the functors representing programs and the natural transformations in the morphisms between programs. The apparent complexity of this generalization is overcome by the observation that an oplax functor J Sp(C) is essentially the same as a functor e J C where e J is the twisted arrow category of J. Thus, a program is a presheaf F (G) Set as are the properties of the program. By analogy with categorical models of firstorder logic, a program and the properties which pertain to it are subobjects of a suitably chosen base object. In this setting safety ...
Processes with Multiple Entries and Exits Modulo Isomorphism and Modulo Bisimulation
, 1994
"... . This paper proposes a framework for the integration of the algebra of communicating processes (ACP) and the algebra of flownomials (AF). Basically, this means to combine axiomatisations of parallel and looping operators. To this end a model of process graphs with multiple entries and exits is intr ..."
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. This paper proposes a framework for the integration of the algebra of communicating processes (ACP) and the algebra of flownomials (AF). Basically, this means to combine axiomatisations of parallel and looping operators. To this end a model of process graphs with multiple entries and exits is introduced. In this model the usual operations of both algebras are defined, e.g. alternative composition, sequential composition, feedback, parallel composition, left merge, communication merge, encapsulation, etc. The main results consist of correct and complete axiomatisations for process graphs modulo isomorphism and modulo bisimulation. 1