On Fixed Points of Strictly Causal Functions
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We ask whether strictly causal components form well defined systems when arranged in feedback configurations. The standard interpretation for such configurations induces a fixed-point constraint on the function modelling the component involved. We define strictly causal functions formally, and show that the corresponding fixed-point problem does not always have a well defined solution. We examine the relationship between these functions and the functions that are strictly contracting with respect to a generalized distance function on signals, and argue that these strictly contracting functions are actually the functions that one ought to be interested in. We prove a constructive fixed-point theorem for these functions, and introduce a corresponding induction principle.
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- 1.Zeigler, B.P.: Theory of Modeling and Simulation. John Wiley & Sons (1976)Google Scholar
- 2.Matsikoudis, E., Lee, E.A.: The fixed-point theory of strictly causal functions. Technical Report UCB/EECS-2013-122, EECS Department, University of California, Berkeley (June 2013)Google Scholar
- 6.Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002)Google Scholar
- 14.Lee, E.A., Varaiya, P.: Structure and Interpretation of Signals and Systems, 2nd edn. (2011), http://LeeVariaya.org
- 15.Enderton, H.B.: Elements of Set Theory. Academic Press (1977)Google Scholar
- 19.Scott, D.S., de Bakker, J.W.: A theory of programs. Unpublished notes, Seminar on Programming, IBM Research Center, Vienna, Austria (1969)Google Scholar
- 22.Roscoe, A.W.: Topology, computer science, and the mathematics of convergence. In: Reed, G.M., Roscoe, A.W., Wachter, R.F. (eds.) Topology and Category Theory in Computer Science, pp. 1–27. Oxford University Press, Inc., New York (1991)Google Scholar