Abstract
We introduce algebras capable of representing, detecting, identifying, and counting static and dynamic hazard pulses that can occur in the worst case on any wire in a gate circuit. These algebras also permit us to count the worst-case number of signal changes on any wire. This is of interest to logic designers for two reasons: each signal change consumes energy, and unnecessary multiple signal changes slow down the circuit operation. We describe efficient circuit simulation algorithms based on our algebras and illustrate them by several examples. Our method generalizes Eichelberger's ternary simulation and several other algebras designed for hazard detection.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.A. Breuer and L. Harrison, “Procedures for eliminating static and dynamic hazards in test generation,” IEEE Trans. on Computers, Vol. C-23, pp. 1069–1078, 1974.
J.A. Brzozowski, “A survey of regular expressions and their applications,” IRE Trans. on Electronic Computers, Vol. EC-11, No. 3, pp. 324–335, 1962.
J.A. Brzozowski, “Some applications of ternary algebras,” Publicationes Mathematicae (Debrecen), Vol. 54 (Suppl.), pp. 583–599, 1999.
J.A. Brzozowski, “De Morgan bisemilattices,” in Proc. 30th Int. Symp. on Multiple-Valued Logic, Portland, OR, IEEE Computer Society Press, Los Alamitos, CA, May 2000, pp. 173–178.
J.A. Brzozowski and Z. Ésik, “Hazard algebras (Extended Abstract),” in A Half-Century of Automata Theory, A. Salomaa, D. Wood, and S. Yu (Eds.), World Scientific, Singapore, 2001, pp. 1–19. To put a crossline over L
J.A. Brzozowski, Z. Ésik, and Y. Iland, “Algebras for hazard detection,” in Proc. 31st Int. Symp. on Multiple-Valued Logic, Warsaw, Poland, IEEE Computer Society Press, Los Alamitos, CA, May 2000, pp. 3–12. Also reprinted in M. Fitting and E. Orłowska (Eds.), Beyond Two: Theory and Applications of Multiple-Valued Logic, Physica-Verlag, 2003, pp. 3–24.
J.A. Brzozowski and M. Gheorghiu, “Simulation of gate circuits in the algebra of transients,” in 7th Int. Conf. on Implementation and Application of Automata, Tours, France, July 2002. Lecture Notes in Computer Science, vol. 2608, Springer 2003, pp. 57–66.
J.A. Brzozowski and C.-J.H. Seger, Asynchronous Circuits, Springer-Verlag, 1995.
S. Burris and H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, 1981.
S. Chakraborty and D.L. Dill, “More accurate polynomial-time min-max simulation,” in Proc. 3rd Int. Symp. on Advanced Research in Asynchronous Circuits and Systems, Eindhoven, The Netherlands, IEEE Computer Society Press, Los Alamitos, CA, April 1997, pp. 112–123.
B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1990.
E.B. Eichelberger, “Hazard detection in combinational and sequential switching circuits,” IBM J. Research and Development, Vol. 9, pp. 90–99, 1965.
G. Fantauzzi, “An algebraic model for the analysis of logic circuits,” IEEE Trans. on Computers, Vol. C-23, pp. 576–581, 1974.
M. Gheorghiu, “Circuit simulation using a hazard algebra,” MMath Thesis, Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1, Dec. 2001. http://maveric.uwaterloo.ca/publication.html.
M. Gheorghiu and J.A. Brzozowski, “Feedback-free circuits in the algebra of transients,” in 7th Int. Conf. on Implementation and Application of Automata, Tours, France, July 2002. Lecture Notes in Computer Science, vol. 2608, Springer 2003, pp. 106–116.
M. Goto, “Application of three-valued logic to construct the theory of relay networks” (in Japanese), in Proceedings of the Joint Meeting of IEE, IECE, and I.of Illumination E. of Japan, 1948.
G. Grätzer, Universal Algebra, 2nd ed., Springer-Verlag, 1979.
J.P. Hayes, “Digital simulation with multiple logic values,” IEEE Trans. on Computer-Aided Design, Vol.CAD-5, No. 2, 1986.
D.W. Lewis, “Hazard detection by a quinary simulation of logic devices with bounded propagation delays,” MSc Thesis, Electrical Engineering, Syracuse University, Syracuse, NY, Jan. 1972.
D.E. Muller, “A theory of asynchronous circuits,” Technical Report 66, Digital Computer Laboratory, University of Illinois, Urbana-Champaign, Illinois, USA, 1955.
D.E. Muller and W.S. Bartky, “A theory of asynchronous circuits,” in Proceedings of an International Symposium on the Theory of Switching, Annals of the Computation Laboratory of Harvard University, Harvard University Press, 1959, pp. 204–243.
A. Salomaa, Theory of Automata, Pergamon Press, Oxford, 1969.
S.H. Unger, Asynchronous Sequential Switching Circuits, Wiley-Interscience, 1969.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brzozowski, J., Ésik, Z. Hazard Algebras. Formal Methods in System Design 23, 223–256 (2003). https://doi.org/10.1023/A:1026218512171
Issue Date:
DOI: https://doi.org/10.1023/A:1026218512171