# Theory of computation over stream algebras, and its applications

Invited Lectures

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## Abstract

The theory of computable functions on abstract data types is outlined. Methods for extending the theory to establish the scope and limits of computation on streams over abstract data types arc described. Applications of these methods to the theory of synchronous concurrent algorithms are discussed.

## Keywords

Computable Function Universal Algebra Systolic Array Computability Theory Springer Lecture Note
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## References

- J A Bergstra and J V Tucker, A characterisation of computable data types by means of a finite equational specification method, in J W de Bakker and J van Leeuwen (eds.)
*Automata, Languages and Programming, Seventh Colloquium*, Noordwijkerhout, 1980, Springer Lecture Notes in Computer Science 81, Springer-Verlag, Berlin, 1980, pp. 76–90.Google Scholar - J A Bergstra and J V Tucker, Algebraic specifications of computable and semicomputable data types,
*Theoretical Computer Science*, 50 (1987) 137–181.zbMATHMathSciNetCrossRefGoogle Scholar - J A Bergstra and J V Tucker, Initial and final algebra semantics for data type specifications: two characterisation theorems,
*Society for Industrial and Applied Mathematics (SIAM) J on Computing*, 12 (1983) 366–387.zbMATHMathSciNetGoogle Scholar - L Blum, M Shub and S Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines,
*Bulletin American Mathematical Society*21 (1989) 1–46.zbMATHMathSciNetCrossRefGoogle Scholar - B Courcelle, Recursive applicative program schemes, in J van Leeuwen (ed)
*Handbook of Theoretical Computer Science. Volume B: Formal Models and Semantics*, Elsevier, 1990, 459–492.Google Scholar - 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 - J E Fenstad,
*General Recursion Theory: An Axiomatic Approach*, Springer-Verlag, Berlin, 1980.zbMATHGoogle Scholar - M Fitting,
*Fundamentals of Generalised Recursion Theory*, North-Holland, Amsterdam, 1981.Google Scholar - F Fogelman Soulie, Y Robert, M Tchuente (eds.),
*Automata Networks in Computer Science*, Manchester University Press, 1986.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 - S Greibach,
*Theory of Program Structures: Schemes, Semantics, Verification*, Springer Lecture Notes in Computer Science 36, Berlin, 1975.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
- L A Harrington
*et al.*(eds.)*Harvey Friedman's Research on the Foundations of Mathematics*, North-Holland, 1985.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, 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.CrossRefGoogle Scholar - A V Holden, J V Tucker and H Zhang, Coupled map lattices as computational systems, Computer Science Division Research Report, University College of Swansea, 1992.Google Scholar
- G Kahn, The semantics of a simple language for parallel processing, in
*Proceedings IFIP Congress 74*, IFIP, 1974, 471–475Google Scholar - K Kaneko (ed.),
*Coupled Map Lattices — Theory and Applications*, J Wiley, in press.Google Scholar - D Kozen and J Tiuryn, Logics of programs, in J van Leeuwen (ed)
*Handbook of Theoretical Computer Science. Volume B: Formal Models and Semantics*, Elsevier, 1990, 789–840.Google Scholar - A R Martin and J V Tucker, The concurrent assignment representation of synchronous systems,
*Parallel Computing*9 (1988/89) 227–256.MathSciNetCrossRefGoogle 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, 1992.Google Scholar - K Meinke, Universal algrebra in higher types,
*Theoretical Computer Science*, to appear.Google Scholar - K Meinke and J V Tucker, Specification and representation of synchronous concurrent algorithms, in F H Vogt (ed.)
*Concurrency '88*, Springer 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 - J C Shepherdson, Algorithmic procedures, generalised Turing algorithms, and elementary recursion theory, in Harrington
*et al.*[1985], pp. 285–308.Google Scholar - R Stephens and B C Thompson, Composition of cartesian stream transformers, in preparation.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, Algebraic specification of synchronous concurrent algorithms and architectures, Computer Science Division Research Report, University College of Swansea, 1991.Google Scholar
- J V Tucker, Computing in algebraic systems, in F.R. Drake and S.S. Wainer (eds.)
*Recursion Theory, its Generalisations and Applications*, London Mathematical Society Lecture Note Series 45, Cambridge University Press, Cambridge, 1980, pp. 215–235.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,
*Program Correctness over Abstract Data Types, with Error State Semantics*, North Holland, 1988.Google Scholar - J V Tucker and J I Zucker, Horn programs and semicomputable relations on abstract structures,
*Automata, Languages and Programming 1989, Proceedings of the Sixteenth Colloquium*,*Stresa*, Springer Lecture Notes in Computer Science 372, Springer-Verlag, 1989, pp.745–760Google Scholar - J V Tucker, S S Wainer and J I Zucker, Provably computable functions on abstract data types,
*Automata, Languages and Programming 1990, Proceedings of the Seventeenth Colloquium*,*Coventry*, Springer Lecture Notes in Computer Science, Springer-Verlag, 1990.Google Scholar - J V Tucker and J I Zucker, Projections of semicomputable relations on abstract data types,
*International J of Foundations of Computer Science*2 (1991) 267–296.zbMATHMathSciNetCrossRefGoogle Scholar - J V Tucker and J I Zucker, Deterministic and nondeterministic computation, and Horn programs, on abstract data types,
*Journal of Logic Programming*, 13 (1992) 23–55. a.zbMATHMathSciNetCrossRefGoogle Scholar - J V Tucker and J I Zucker, Examples of semicomputable sets of real and complex numbers, in J P Myers and M J O'Donnell (eds.),
*Constructivity in Computer Science*, Springer Lecture Notes in Computer Science, Berlin. b.Google Scholar - J V Tucker and J I Zucker, Provable computable selection functions on abstract structures, in P Aczel, H Simmons and S S Wainer (cds.)
*Proof Theory*, Cambridge University Press, 1992, in press. c.Google Scholar - J V Tucker and J I Zucker, Algebraic specifications of generalised computable functions over abstract data types, in preparation.Google Scholar
- J V Tucker, J I Zucker, and K Meinke, Computability on abstract structures, in S. Abramsky, D. Gabbay and T Maibaum (eds.)
*Handbook of Logic for Computer Science*, Oxford University Press, in preparation.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

## Copyright information

© Springer-Verlag Berlin Heidelberg 1992