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On the algebra of feedback and systems with boundary. (English) Zbl 1003.94051

Betti, Renato (ed.) et al., Categorical studies in Italy. Selected papers presented at the meeting “The Italian friends meet Bill Lawvere”, Perugia, Italy, May 1-3, 1997. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 64, 123-156 (2000).
From authors’ introduction: “The authors have introduced [P. Katis, N. Sabadini and R. F. C. Walters, J. Pure Appl. Algebra 115, 141-178 (1997; Zbl 0933.18008) and AMAST ‘97, Lect. Notes Comput. Sci. 1349, 307-321 (1997; Zbl 0885.18004)] two complimentary algebras for systems with boundaries: the first is the bicategory of circuits (or input-feedback-output systems) in the monoidal category ({Set}, \(\times)\) in which the natural feedback operation has delay; and the second is the compact closed bicategory of spans of graphs in which the natural feedback is instantaneous. The former provides a model of non-deterministic systems (such as asynchronous circuits), while the latter provides a model of deterministic systems (and also specifications of systems). These two bicategories are representatives of two broader classes of algebras, and, generally, there are homorphisms from the first type of algebra to the second.
“Both these classes of algebras may be studied in the context of bicategories-with-feedback, a notion the authors do not define in this paper. They do, however, consider specific bicategories-with-feedback and define the notion of a category-with-feedback, which includes traced symmetric monoidal categories [A. Joyal, R. Street and D. Verity, Math. Proc. Camb. Philos. Soc. 119, 447-468 (1996; Zbl 0845.18005)]. In this context we will consider functors between categories-with-feedback – functors which model and facilitate the interplay between systems and their behaviours that occur when specifying, building and analysing machines or programs.
To each of these algebras there is an associated geometry. That is, to an expression in the algebra there is an associated geometric presentation, which corresponds to the distributed nature of the system. (When considering categories of circuits, the usual circuit diagrams are obtained: in the context of algorithms, flow charts with feedback are obtained.) This relation between algebra and geometry is in the line of the Penrose diagrams [R. Penrose, Combinat. Math. Appl., Proc. Conf. Math. Inst., Oxford 1969, 221-244 (1971; Zbl 0216.43502)] for the tensor calculus, and the more recent developments in geometry and physics on braided tensor categories [A. Joyal and R. Street, Adv. Math. 102, 20-78 (1993; Zbl 0817.18007)], Frobenius algebras, cobordism, etc. This connection between algebra and geometry is being exploited to provide a graphical user interface for a computer program [R. Gates, P. Katis and R. F. C. Walters, A program for computing with the Cartesian bicategory Span(Graph) School of Mathematics and Statistics, University of Sydney (1998)] written by Robbi Gates for calculating in {Span(Graph)}. For our present purposes, we will only indicate how pictures of expressions can be drawn, deferring a formal treatment of this geometric presentation to another paper.”
It is the purpose of this note to give a brief introduction to and demonstrate the wide range of applicability of these algebras. After surveying the range of examples the authors concentrate on two new ones: in Section 4 they show that continuous linear systems are an example of the algebra which includes a model of RLC circuits, and they indicate how the model can be extended to incorporate more general continuous systems, and in Section 5, a formalization of double-entry accounting and other systems satisfying the continuity equation is given.
For the entire collection see [Zbl 0942.00025].

MSC:

94C15 Applications of graph theory to circuits and networks
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18B20 Categories of machines, automata
93A30 Mathematical modelling of systems (MSC2010)
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