Gate circuits in the algebra of transients

Janusz Brzozowski; Mihaela Gheorghiu

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2005)

  • Volume: 39, Issue: 1, page 67-91
  • ISSN: 0988-3754

Abstract

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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 0 s and 1 s; it represents a changing signal. In the algebra of transients, gates process transients instead of 0 s and 1 s. 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.

How to cite

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Brzozowski, Janusz, and Gheorghiu, Mihaela. "Gate circuits in the algebra of transients." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 39.1 (2005): 67-91. <http://eudml.org/doc/244623>.

@article{Brzozowski2005,
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 $0$s and $1$s; it represents a changing signal. In the algebra of transients, gates process transients instead of $0$s and $1$s. 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.},
author = {Brzozowski, Janusz, Gheorghiu, Mihaela},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {algebra; digital circuit; hazard detection; signal changes; simulation; transient; gate circuits; feedback-free circuits},
language = {eng},
number = {1},
pages = {67-91},
publisher = {EDP-Sciences},
title = {Gate circuits in the algebra of transients},
url = {http://eudml.org/doc/244623},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Brzozowski, Janusz
AU - Gheorghiu, Mihaela
TI - Gate circuits in the algebra of transients
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 1
SP - 67
EP - 91
AB - 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 $0$s and $1$s; it represents a changing signal. In the algebra of transients, gates process transients instead of $0$s and $1$s. 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.
LA - eng
KW - algebra; digital circuit; hazard detection; signal changes; simulation; transient; gate circuits; feedback-free circuits
UR - http://eudml.org/doc/244623
ER -

References

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  1. [1] Ann. Symp. Asynchronous Circuits and Systems, Proc. 9th (ASYNC ’03), IEEE Comp. Soc. (2003). 
  2. [2] J.A. Brzozowski and Z. Ésik, Hazard algebras. Formal Methods in System Design 23 (2003) 223–256. Zbl1073.68883
  3. [3] J.A. Brzozowski, Z. Ésik and Y. Iland, Algebras for hazard detection. Beyond Two: Theory and Applications of Multiple-Valued Logic, edited by M. Fitting and E. Orłowska. Physica-Verlag (2003) 3–24. Zbl1046.94021
  4. [4] J.A. Brzozowski and M. Gheorghiu, Simulation of gate circuits in the algebra of transients. Implementation and Application of Automata. Lect. Notes Comput. Sci. 2608 (2003) 57–66. Zbl1033.94568
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  8. [8] E.B. Eichelberger, Hazard detection in combinational and sequential circuits. IBM J. Res. Dev. 9 (1965) 90–99. Zbl0132.37106
  9. [9] J.D. Garside, AMULET3 revealed, in Proc. 5th Ann. Symp. Asynchronous Circuits and Systems (ASYNC ’99), IEEE Comp. Soc. (1999) 51–59. 
  10. [10] M. Gheorghiu, Circuit simulation using a hazard algebra. MMath Thesis, Department of Computer Science, University of Waterloo, Waterloo, ON, Canada (2001). 
  11. [11] M. Gheorghiu and J.A. Brzozowski, Simulation of feedback-free circuits in the algebra of transients. Int. J. Found. Comput. Sci. 14 (2003) 1033–1054. Zbl1101.68652
  12. [12] J. Kessels and P. Marston, Designing asynchronous standby circuits for a low-power pager, in Proc. 3rd Ann. Symp. Asynchronous Circuits and Systems (ASYNC ’97), IEEE Comp. Soc. (1997) 268–278. 
  13. [13] E.J. McCluskey, Transients in combinational logic circuits. Redundancy techniques for computing systems, edited by R.H. Wilcox and W.C. Mann. Spartan Books (1962) 9–46. 
  14. [14] D.E. Muller and W.C. Bartky, A theory of asynchronous circuits, in Proc. Int. Symp. on Theory of Switching, Annals of Comp. Lab. Harvard University 29 (1959) 204–243. Zbl0171.37902
  15. [15] C.-J.H. Seger and J.A. Brzozowski, Generalized ternary simulation of sequential circuits. Theor. Inf. Appl. 28 (1994) 159–186. Zbl0879.94040
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