# Infinite systems of equations over inverse limits and infinite synchronous concurrent algorithms

## Abstract

We consider the existence, uniqueness and effectiveness of solutions to infinite systems of equations in certain inverse limits of algebras. The notion of a guarded infinite system of equations is defined and used to establish existence and uniqueness results about the solutions. A domain structure associated with the inverse limit is used to prove the theorems. The use of infinite systems of equations is illustrated by the study of infinite synchronous concurrent algorithms (ISCAs). An ISCA is an infinite network of processors, operating in parallel, and synchronised by a global clock; the algorithm processes infinite streams of data. The algorithms are described by infinite systems of equations and, since the networks are deterministic, the equations are required to have unique solutions.

## Keywords

Inverse limits of algebras ultrametric algebras domains infinite systems of equations existence and uniqueness theorems synchronous concurrent algorithms streams infinitely parallel deterministic systems## Preview

Unable to display preview. Download preview PDF.

## References

- J P Crutchfield and K Kaneko, Phenomenology of spatio-temporal chaos, in H Bai-lin (ed.)
*Directions in Chaos*, World Scientific, 1987.Google Scholar - H Ehrig and B Mahr,
*Fundamentals of Algebraic Specifications 1 — Equations and initial semantics*, Springer-Verlag, 1985.Google Scholar - S M Eker, V Stavridou and J V Tucker, Verification of synchronous concurrent algorithms using OBJ3. A case study of the Pixel Planes architecture, In G Jones and M Sheeran (eds.)
*Designing Correct Circuits*, Springer-Verlag, 1991, pp. 231–252.Google Scholar - S M Eker and J V Tucker, Specification, derivation and verification of concurrent line drawing algorithms and architectures, in R A Earnshaw (ed.),
*Theoretical Foundations of Computer Graphics and CAD*, Springer-Verlag, 1988, pp. 449–516.Google Scholar - S M Eker and J V Tucker, Specification and verification of synchronous concurrent algorithms: a case study of the Pixel Planes architecture, in P M Dew, R A Earnshaw and T R Heywood (eds.),
*Parallel Processing for Computer Vision and Display*, Addison Wesley, 1989, pp.16–49.Google Scholar - F Fogelman Soulie, Y Robert, M Tchuente (eds.),
*Automata Networks in Computer Science*, Manchester University Press, 1986.Google Scholar - E Griffor, I Lindström and V Stoltenberg-Hansen,
*Mathematical Theory of Domains*, Cambridge Tracts in Theoretical Compiuter Science, to appear 1993.Google Scholar - J A Goguen, J W Thatcher, E G Wagner, and J B Wright, An initial algebra approach to the specification, correctness and implementation of abstract data types, in R T Yeh (ed.),
*Current Trends in Programming Methodology: IV Data Structuring*, Prentice Hall, 1978, pp. 80–149.Google Scholar - N A Harman and J V Tucker, Clocks, retimings, and the formal specification of a UART, in G Milne (ed.)
*The Fusion of Hardware Design and Verification*(Proceedings of IFIP Working Group 10.2 Working Conference), North-Holland, 1988, pp. 375–396.Google Scholar - N A Harman and J V Tucker, The formal specification of a digital correlator I: User specification process, in K McEvoy and J V Tucker [90], pp. 161–262.Google Scholar
- K M Hobley, B C Thompson, and J V Tucker, Specification and verification of synchronous concurrent algorithms: a case study of a convolution algorithm, in G Milne (ed.)
*The Fusion of Hardware Design and Verification*(Proceedings of IFIP Working Group 10.2 Working Conference), North-Holland, 1988, pp. 347–374.Google Scholar - A V Holden, J V Tucker and B C Thompson, The computational structure of neural systems, in A V Holden and V I Kryukov (eds.)
*Neurocomputers and Attention. I: Neurobiology, Synchronisation and Chaos*, Manchester University Press, 1990, pp. 223–240.Google Scholar - A V Holden, J V Tucker and B C Thompson, Can excitable media be considered as computational systems?
*Physica D*49 (1991) 240–246.Google Scholar - A V Holden, J V Tucker, M Poole and H Zhang, Coupled map lattices as computational systems,
*American Institute of Physics Chaos*2 (1992), 367–376.Google Scholar - K Kaneko (ed.),
*Coupled Map Lattices — Theory and Applications*, J Wiley, in press.Google Scholar - A R Martin and J V Tucker, The concurrent assignment representation of synchronous systems,
*Parallel Computing*9 (1988/89) 227–256.Google Scholar - K McEvoy and J V Tucker (eds.),
*Theoretical Foundations of VLSI Design*, Cambridge University Press, 1990.Google Scholar - B McConnell and J V Tucker, Infinite synchronous concurrent algorithms: the specifiation and verification of a hardware stack, in H Schwichtenberg (ed.)
*Logic and Algebra for Specification*, Springer-Verlag, 1993.Google Scholar - B McConnell and J V Tucker, Direct limits of algebras and the parameterisation of synchronous concurrent algorithms, Department of Computer Science, University College of Swansea, Report, in preparation.Google Scholar
- K Meinke and J V Tucker, Specification and representation of synchronous concurrent algorithms, in F H Vogt (ed.)
*Concurrency '88*, Lecture Notes in Computer Science 335, Springer-Verlag, 1988, pp. 163–180.Google Scholar - K Meinke and J V Tucker, Universal algebra, in S Abramsky, D Gabbay, T S E Maibaum (eds.)
*Handbook of Logic in Computer Science*, Oxford University Press, pp. 189–411.Google Scholar - B C Thompson, A mathematical theory of synchronous concurrent algorithms. PhD Thesis, School of Computer Studies, University of Leeds, 1987.Google Scholar
- B C Thompson and J V Tucker, Theoretical considerations in algorithm design, in R A Earnshaw (ed.),
*Fundamental Algorithms for Computer Graphics*, Springer-Verlag, 1985, pp. 855–878.Google Scholar - B C Thompson and J V Tucker, Equational specification of synchronous concurrent algorithms and architectures, Computer Science Division Research Report, University College of Swansea, 1991.Google Scholar
- V Stoltenberg-Hansen and J V Tucker, Complete local rings as domains,
*J. Symbolic Logic*53 (1988) 603–624.Google Scholar - V Stoltenberg-Hansen and J V Tucker, Algebraic and fixed point equations over inverse limits of algebras,
*Theoretical Computer Science*87 (1991) 1–24.Google Scholar - J V Tucker, Theory of computation and specification over abstract data types and its applications, in F L Bauer (ed), Proceedings of NATO Summer School 1989 at Marktoberdorf, in
*Logic, algebra and computation*, Springer, 1991, pp 1–39.Google Scholar - J V Tucker and J I Zucker, Theory of computation over stream algebras and applications, in I M Havel and V Koubek (eds),
*Mathematical Foundations of Computer Science 1992, 17th International Symposium, Prague*, Lecture Notes in Computer Science 629, Springer, Berlin, 1992, 62–80.Google Scholar - W Wechler,
*Universal Algebra for Computer Scientists*, EATCS Monographs, Springer Verlag, Berlin, 1991.Google Scholar - M Wirsing, Algebraic specification, in J van Leeuwen (ed)
*Handbook of Theoretical Computer Science. Volume B: Formal Models and Semantics*, Elsevier, 1990, pp. 675–788.Google Scholar