Real Time Models of the Asynchronous Circuits: The Delay Theory, in New Developments in Computer Science Research, Editor Susan Shannon, Nova Science Publishers, Inc., New York, 2005
Table of Contents
1. Introduction
2. Motivating Examples
2.1.Example 1 The Delay Circuit; 2.2.Example 2 Circuit with Feedback Using a Delay Circuit;
2.3.The Logical Gate NOT; 2.4.Circuit With Feedback Using a Logical Gate NOT; 2.5. First Conclusions
3. Preliminaries
3.1.The Boole Algebra with Two Elements; 3.2.Generalities on the R->{0,1} Functions; 3.3.Limits and
Derivatives. The Continuity and the Differentiability of the R->{0,1} Functions; 3.4.The Properties
of the Limits and of the Derivatives; 3.5 Conventions Concerning the Drawings of the Graphics of the R->{0,1} Functions
4. Signals
4.1.The Definition of the Signals; 4.2.Useful Lemmas
5. An Overview of the Delays: Informal Definitions
6. Delays
6.1.Stability. Rising and Falling Transmission Delays for Transitions; 6.2.Delays; 6.3.Determinism; 6.4.Order;
6.5.Time Invariance; 6.6.Constancy; 6.7.Rising-Falling Symmetry; 6.8.Serial Connection
7. Bounded Delays
7.1.The Consistency Condition; 7.2.Bounded Delays; 7.3.The Properties of the Bounded Delays; 7.4 Fixed and Inertial Delays
8. Absolute Inertial Delays
8.1.Absolute Inertia; 8.2.Absolute Inertial Delays; 8.3.The Consistency Condition; 8.4.Bounded Absolute Inertial Delays
9. Relative Inertial Delays
9.1.Relative Inertia; 9.2.Relative Inertial Delays; 9.3.The Consistency Condition; 9.4.Bounded Relative Inertial Delays;
9.5.Deterministic Bounded Relative Inertial Delays
10 Alternative Definitions. Symmetrical Deterministic Upper Bounded, Lower Unbounded Relative Inertial Delays
10.1.Alternative Definitions; 10.2.Symmetrical Deterministic Upper Bounded, Lower Unbounded Relative Inertial Delays
11. Other Examples and Applications
11.1.A Delay Line for the Falling Transitions Only; 11.2.Example of Circuit with Tranzient Oscillations; 11.3.Example of C Gate. Generalization
Digital electrical engineering is a non-formalized theory and one of the major causes of this situation
consists in the complexity of Mother Nature, things cannot be completely different from those in medicine,
for example. We are too restricted to finding quick solutions to the problems that arise in order to take
the time to strengthen a sound theoretical foundation of the reasoning that we do. Obviously, the political,
military, economical and technological importance of digital electrical engineering is itself an obstacle in
the spreading of consolidated theories. In fact, the reader of such literature can remark the existing
distance from the deductive theories, the way that the mathematicians use them.
The consequences of non-formalization are known. Many researchers do not give the right importance to the
scientific language and words like definition, theorem, proof are titles of descriptive paragraphs rather than
having the meaning that is accepted by the logicians. A fascinating job is, in this context, the translation
in a precise mathematical language of what is intuitively, imprecisely explained with natural language by the
engineers and this can be done in several ways. Our work has many such examples, let’s just mention here the
notion of inertia that is important and confusing at the same time.
The purpose of delay theory is that of writing systems of equations and inequalities with electrical signals,
that model the behavior of the asynchronous circuits.
The (electrical) signals are the functions f:R->{0,1} where R, the set of the real numbers, is the time set. We ask that they:
- be constant for t<0, with the variant that we have used elsewhere: be null for t<0, in other words 0 is the initial time instant
- be constant on intervals [t',t'') that are left closed and right open
- have a finite number of discontinuity points (i.e. a finite number of switches) in any bounded interval.
The asynchronous circuits (also called asynchronous systems, or asynchronous automata or timed automata in literature)
are these electrical devices that can be modeled by using signals.
The fundamental (asynchronous) circuit in delay theory is the delay circuit, also called delay buffer, the circuit that
computes the identity 1:{0,1}->{0,1} and the fundamental notion is that of delay condition, or shortly delay,
the real time model of the delay circuit. In our work we give first the definition of the delays. Second, the pure delays are defined.
Third, all the delays different from the pure delays are considered to be by definition inertial. Fourth, the serial connection of the
delays is their composition, as binary relations. The serial connection of the inertial delays results in an inertial delay, but the
type of inertia is likely to differ. The bounded delays have the nice property that, under the serial connection, the delays remain
bounded and thus the type of inertia remains the same; the absolute inertial delays are in the same situation. The relative inertial
delays are not closed under the serial connection.
The chapter of the book is organized in sections, numbered with 1, 2, 3, … the sections have several paragraphs:
2.1, 3.2, … and the paragraphs are usually organized in subparagraphs: 2.1.1, 4.5.2, … The important
equations and inequalities have been numbered, as well as all the figures and tables. The notation
3.2 (2) refers to equation or inequality (2) of paragraph 3.2 (that has no subparagraphs, in this case)
and the notation 4.1.2 (3) refers to equation or inequality (3) of the subparagraph 2 from the paragraph 4.1.
In Section 2 we give several examples of models for the sake of creating intuition and this is a presentation
of our intentions. The theory starting with section 3 is supposed to be self-contained. In Section 3 we fix
some fundamental concepts and notations on the R->{0,1} functions. Section 4 defines the signals and gives
some useful properties on them. In Section 5 we present the informal definitions of the delays, with long
quotations from several authors.
The sections that follow represent the core of this chapter. In Section 6 we define the delays,
as well as their determinism, order, time invariance, constancy, symmetry and serial connection.
Section 7 is dedicated to the bounded delays and in Sections 8, 9 we define and characterize the absolute and
the relative inertial conditions and delays. Section 10 shows some variants of the concepts from Sections
7, 8, 9 and introduces a special form of deterministic delays. Section 11 closes the chapter with new examples
and a generalization.
