Abstract
The complexity of software controlled systems is ever increasing, and so is the need for mathematical theories and tools that help with their design and verification . This need has not been answered by a single grand unified theory; on the contrary the great variety of programming languages and paradigms has led to an equally large number of different and, in most cases, incompatible formalisms. This diversity is of growing concern to academia and industry alike and we have reached the stage where the unification of existing theories should take priority over the invention of new ones.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. J. Aarts. Galois connections presented calculationally. Technical report, Eindhoven University of Technology, 1992. Available from ftp.win.tue.nl/pub/math.prog.construction/galois.dvi.Z.
R. C. Backhouse, T. S. Voermans, and J. van der Woude. A relational theory of datatypes. Available via world-wide web at http://www.win.tue.nl/win/cs/wp/papers, December 1992.
Ralf Behnke. Transformational program derivation in the framework of sequential and relational algebras. PhD thesis, Christian-Albrechts-Universität Kiel, 1998. In German.
Rudolf Berghammer, Peter Kempf, Gunther Schmidt, and Thomas Ströhlein. Relation algebra and logic of programs. In Algebraic Logic, volume 54 of Colloquia Mathematica Societatis János Bolyai. Budapest University, 1988.
Rudolf Berghammer and Burghard von Karger. Formal derivation of CSP programs from temporal specifications. In Bernhard Möller, editor, Mathematics of Program Construction, LNCS 947, pages 180–196. Springer-Verlag, 1995.
Rudolf Berghammer and Burghard von Karger. Towards a design calculus for CSP. Science of Computer Programming, 26:99–115, 1996.
Richard S. Bird and Oege de Moor. The Algebra of Programming. International Series in Computer Science. Prentice Hall, 1996.
H. Brandt. Über eine Verallgemeinerung des Gruppenbegriffs. Math. Ann., 96:360–366, 1926.
Stephen M. Brien. A time-interval calculus. In R. S. Bird, C. C. Morgan, and J. C. P. Woodcock, editors, Mathematics of Program Construction, LNCS 669. Springer-Verlag, 1992.
C. Brink, W. Kahl, and G. Schmidt, editors. Relational Methods in Computer Science. Advances in Computing Science. Springer, 1997.
Ronald Brown. From groups to groupoids. Bull. London Math. Soc., 19:113–134, 1987.
A. H. Clifford and G. B. Preston. The algebraic theory of semigroups. Number 7 in Mathematical Surveys. American Mathematical Society, 1961.
Edsger W. Dijkstra. The unification of three calculi. In Manfred Broy, editor, Program Design Calculi, pages 197–231. Springer-Verlag, 1993.
E. W. Dijkstra and C. S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, 1990.
Rutger Dijkstra. Computation calculus: Bridging a formalization gap. Science of Computer Programming, 37(1–3):3–36, 2000.
E. A. Emerson. Temporal and modal logic. In Jan van Leeuwen, editor, Formal Models and Semantics, volume B of Handbook of Theoretical Computer Science, chapter 16, pages 995–1072. Elsevier, 1990.
C. J. Everett. Closure operators and Galois theory in lattices. Trans. Amer. Math. Soc., 55:514–525, 1944.
Horst Herrlich and Miroslav Hušek. Galois connections. In Proceedings of MFPS, LNCS 239, pages 122–134. Springer-Verlag, 1985.
C. A. R. Hoare. Communicating Sequential Processes. Series in Computer Sciences. Prentice-Hall, 1985.
C. A. R. Hoare. Programs are predicates. In C. A. R. Hoare and J. C. Shepherdson, editors, Mathematical Logic and Programming Languages, pages 141–155. Prentice-Hall, 1985.
C. A. R. Hoare and He Jifeng. The weakest prespecification. Fundamenta Informaticae, 9:51–84, 217–252, 1986.
P. T. Johnstone. Stone Spaces. Cambridge University Press, 1992.
B. Jónsson. Varieties of relation algebras. Algebra Universalis, 15:273–298, 1982.
S. C. Kleene. Representation of events in nerve nets and finite automata. In Shannon and McCarthy, editors, Automata Studies, pages 3–42. Princeton University Press, 1956.
R. Maddux. The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica, L(3/4):421–456, 1991.
R. Maddux. A working relational model: The derivation of the Dijkstra-Scholten predicate transformer semantics from Tarski’s axioms of the Peirce-Schröder calculus of relations. Manuscript, 1992.
Zohar Manna and Amir Pnueli. The Temporal Logic of Reactive and Concurrent Systems—Specification. Springer-Verlag, 1991.
Antoni Mazurkiewicz. Traces, histories, graphs: Instances of a process monoid. In M. P. Chytil and V. Koubek, editors, Mathematical Foundations of Computer Science, LNCS 176, pages 115–133. Springer-Verlag, 1984.
Austin Melton, D. A. Schmidt, and George E. Strecker. Galois connections and computer science applications. In David Pitt, Samson Abramsky, Axel Poigné, and David Rydeheard, editors, Category Theory and Computer Programming, LNCS 240, pages 299–312. Springer-Verlag, 1986.
Austin Melton, Bernd S. W. Schröder, and George E. Strecker. Lagois connections—a counterpart to Galois connections. Theoretical Computer Science, 136(1):79–108, 1994.
A. Mili. A relational approach to the design of deterministic programs. Acta Informatica, 20:315–328, 1983.
Ben Moszkowski. A temporal logic for multi-level reasoning about hardware. IEEE Computer, 18(2):10–19, 1985.
Ben Moszkowski. Some very compositional temporal properties. In E. R. Olderog, editor, Proceedings of the IFIP TC2/WG2.1/WG2.2/WG2.3 Working Conference on Programming Concepts, Methods and Calculi (PROCOMET’ 94), San Miniato, Italy. IFIP Transactions A-56, North-Holland, 1994.
Oystein Ore. Galois connexions. Trans. Amer. Math. Soc., 55:493–513, 1944.
Amir Pnueli. The temporal logic of programs. In 18th Ann. IEEE Symp. on Foundations of Computer Science, pages 46–57, 1977.
Vaughan Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15:33–71, 1986.
W. Prenowitz. A contemporary approach to classical geometry. Amer. Math. Monthly, 68(1, Part II), 1968.
Gunther Schmidt and Thomas Ströhlein. Relations and Graphs. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1991.
Colin Stirling. Modal and temporal logics. In Samson Abramsky, Dov M. Gabbay, and Thomas S. E. Maibaum, editors, Background: Computational Structures, volume 2 of Handbook of Logic in Computer Science, pages 478–551. Clarendon Press, 1992.
A. Tarski and B. Jónsson. Boolean algebras with operators, parts I-II. Amer. J. Math., 73,74:891–939, 127–162, 1951/52.
Alfred Tarski. On the calculus of relations. Journal of Symbolic Logic, 6(3):73–89, 1941.
A. J. M. van Gasteren. On the Shape of Mathematical Argument, volume 445 of Lecture Notes in Computer Science. Springer, 1987.
Yde Venema. A modal logic for chopping intervals. J. Logic Computat., 1(4):453–476, 1991.
B. von Karger and R. Berghammer. A relational model for temporal logic. Logic Journal of the IGPL, 6(2):157–173, 1998. Available from http://www.oup.co.uk/igpl/Volume 06/Issue 02.
Burghard von Karger. Temporal Algebra. PhD thesis, Habilitationsschrift, Christian-Albrechts-Univ. Kiel, 1997. Available from http://www.informatik.uni-kiel.de/~bvk/.
Burghard von Karger. A calculational approach to reactive systems. Science of Computer Programming, 37(1–3):139–161, 2000.
Chaochen Zhou, C. A. R. Hoare, and Anders P. Ravn. A calculus of durations. Information Processing Letters, 40:269–276, 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
von Karger, B. (2002). Temporal Algebra. In: Backhouse, R., Crole, R., Gibbons, J. (eds) Algebraic and Coalgebraic Methods in the Mathematics of Program Construction. Lecture Notes in Computer Science, vol 2297. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47797-7_9
Download citation
DOI: https://doi.org/10.1007/3-540-47797-7_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43613-3
Online ISBN: 978-3-540-47797-6
eBook Packages: Springer Book Archive