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Compositional Relational Semantics for Indeterminate Dataflow Networks
, 1989
"... Given suitable categories T; C and functor F : T ! C, if X; Y are objects of T, then we define an (X; Y )relation in C to be a triple (R; r; ¯ r), where R is an object of C and r : R ! FX and ¯ r : R ! FY are morphisms of C. We define an algebra of relations in C, including operations of "rel ..."
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Cited by 17 (6 self)
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Given suitable categories T; C and functor F : T ! C, if X; Y are objects of T, then we define an (X; Y )relation in C to be a triple (R; r; ¯ r), where R is an object of C and r : R ! FX and ¯ r : R ! FY are morphisms of C. We define an algebra of relations in C, including operations of "relabeling," "sequential composition," "parallel composition," and "feedback," which correspond intuitively to ways in which processes can be composed into networks. Each of these operations is defined in terms of composition and limits in C, and we observe that any operations defined in this way are preserved under the mapping from relations in C to relations in C 0 induced by a continuous functor G : C ! C 0 . To apply the theory, we define a category Auto of concurrent automata, and we give an operational semantics of dataflowlike networks of processes with indeterminate behaviors, in which a network is modeled as a relation in Auto. We then define a category EvDom of "event domains," a (non...
Functional dynamics II: syntactic structure
 Physica D
, 2001
"... Functional dynamics, introduced in a previous paper, is analyzed, focusing on the formation of a hierarchical rule to determine the dynamics of the functional value. To study the periodic (or nonfixed) solution, the functional dynamics is separated into fixed and nonfixed parts. It is shown that t ..."
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Cited by 6 (1 self)
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Functional dynamics, introduced in a previous paper, is analyzed, focusing on the formation of a hierarchical rule to determine the dynamics of the functional value. To study the periodic (or nonfixed) solution, the functional dynamics is separated into fixed and nonfixed parts. It is shown that the fixed parts generate a 1dimensional map by which the dynamics of the functional values of some other parts are determined. Piecewiselinear maps with multiple branches are generally created, while an arbitrary onedimensional map can be embedded into this functional dynamics if the initial function coincides with the identity function over a finite interval. Next, the dynamics determined by the onedimensional map can again generate a ‘metamap’, which determines the dynamics of the generated map. This hierarchy of metarules can continue recursively. It is also shown that the dynamics can produce ‘metachaos ’ with an orbital instability that is stronger than exponential. The relevance of the generated hierarchy to biological and language systems is discussed, in relation with the formation of syntax of a network. 1
Functional dynamics I: Articulation process
 Physica D 138
, 2000
"... The articulation process of dynamical networks is studied with a functional map, a minimal model for the dynamic change of relationships through iteration. The model is a dynamical system of a function f, not of variables, having a selfreference term f ◦ f, introduced by recalling that operation in ..."
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Cited by 5 (1 self)
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The articulation process of dynamical networks is studied with a functional map, a minimal model for the dynamic change of relationships through iteration. The model is a dynamical system of a function f, not of variables, having a selfreference term f ◦ f, introduced by recalling that operation in a biological system is often applied to itself, as is typically seen in rules in the natural language or genes. Starting from an inarticulate network, two types of fixed points are formed as an invariant structure with iterations. The function is folded with time, until it has finite or infinite piecewiseflat segments of fixed points, regarded as articulation. For an initial logistic map, attracted functions are classified into step, folded step, fractal, and random phases, according to the degree of folding. Oscillatory dynamics are also found, where function values are mapped to several fixed points periodically. The significance of our results to prototype categorization in language is discussed. 1
Singular and Plural Nondeterministic Parameters
, 1997
"... : The article defines algebraic semantics of singular (calltimechoice) and plural (runtimechoice) nondeterministic parameter passing and presents a specification language in which operations with both kinds of parameters simultaneously can be defined. Sound and complete calculi for both semantic ..."
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Cited by 4 (1 self)
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: The article defines algebraic semantics of singular (calltimechoice) and plural (runtimechoice) nondeterministic parameter passing and presents a specification language in which operations with both kinds of parameters simultaneously can be defined. Sound and complete calculi for both semantics are introduced. We study the relations between the two semantics and point out that axioms for operations with plural arguments may be considered as axiom schemata for operations with singular arguments. Keywords: algebraic specification, manysorted algebra, nondeterminism, sequent calculus. AMS classifications: 68Q65, 68Q60, 68Q10, 68Q55, 03B60, 08A70. 1. Introduction The notion of nondeterminism arises naturally in describing concurrent systems. Various approaches to the theory and specification of such systems, for instance, CCS [16], CSP [9], process algebras [1], event structures [26], include the phenomenon of nondeterminism. But nondeterminism is also a natural concept in descr...
An extension theorem with an application to formal tree series
 BRICS Report Series
, 2002
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The Convex Powerdomain in a Category of Posets Realized By Cpos
 In Proc. Category Theory and Computer Science
, 1995
"... . We construct a powerdomain in a category whose objects are posets of data equipped with a cpo of "intensional" representations of the data, and whose morphisms are those monotonic functions between posets that are "realized" by continuous functions between the associated cpos. ..."
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. We construct a powerdomain in a category whose objects are posets of data equipped with a cpo of "intensional" representations of the data, and whose morphisms are those monotonic functions between posets that are "realized" by continuous functions between the associated cpos. The category of cpos is contained as a full subcategory that is preserved by lifting, sums, products and function spaces. The construction of the powerdomain uses a cpo of binary trees, these being intensional representations of nondeterministic computation. The powerdomain is characterized as the free semilattice in the category. In contrast to the other type constructors, the powerdomain does not preserve the subcategory of cpos. Indeed we show that the powerdomain has interesting computational properties that differ from those of the usual convex powerdomain on cpos. We end by considering the solution of recursive domain equations. The surprise here is that the limitcolimit coincidence fails. Nevertheless, ...