Gate circuits in the algebra of transients

Issue		RAIRO-Theor. Inf. Appl. Volume 39, Number 1, January-March 2005 Imre Simon, the tropical computer scientist


Page(s)		67 - 91
DOI		10.1051/ita:2005004

Theoret. Informatics Appl. 39, 67-91 (2005)
DOI: 10.1051/ita:2005004

Gate circuits in the algebra of transients

Janusz Brzozowski¹ and Mihaela Gheorghiu²

¹ School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada; brzozo@uwaterloo.ca
² Department of Computer Science, University of Toronto, Toronto, ON, M5S 3G4, Canada; mg@cs.toronto.edu

Abstract

We study simulation of gate circuits in the infinite algebra of transients recently introduced by Brzozowski and Ésik. A transient is a word consisting of alternating 0s and 1s; it represents a changing signal. In the algebra of transients, gates process transients instead of 0s and 1s. Simulation in this algebra is capable of counting signal changes and detecting hazards. We study two simulation algorithms: a general one that works with any initial state, and a special one that applies only if the initial state is stable. We show that the two algorithms agree in the stable case. We also show that the general algorithm is insensitive to the removal of state variables that are not feedback variables. We prove the sufficiency of simulation: all signal changes occurring in binary analysis are predicted by the general algorithm. Finally, we show that simulation can be more pessimistic than binary analysis, if wire delays are not taken into account. We propose a circuit model that we conjecture to be sufficient for proving the equivalence of simulation and binary analysis for feedback-free circuits.

Mathematics Subject Classification. 06A06, 94C10, 94C12

Key words: Algebra -- digital circuit -- hazard detection -- signal changes -- simulation -- transient.

© EDP Sciences 2005

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