[Paper Review] Selected Topics in Asynchronous Automata

Serban E. Vlad|ArXiv.org|Oct 31, 2001

Formal Methods in Verification7 references3 citations

TL;DR

This paper presents a formal model for asynchronous circuits using {0,1}→R functions to represent electrical signals and asynchronous automata, solving their governing equations to analyze stability, fundamental mode operation, and semi-modularity. It establishes a connection between asynchronous automata and propositional temporal logic, offering a unified framework for semantics in continuous and discrete time.

ABSTRACT

The paper is concerned with defining the electrical signals and their models. The delays are discussed, the asynchronous automata - which are the models of the asynchronous circuits - and the examples of the clock generator and of the R-S latch are given. We write the equations of the asynchronous automata, which combine the pure delay model and the inertial delay model; the simple gate model and the complex gate model; the fixed, bounded and unbounded delay model. We give the solutions of these equations, which are written on R->{0,1} functions, where R is the time set. The connection between the real time and the discrete time is discussed. The stability, the fundamental mode of operation, the combinational automata, the semi-modularity are defined and characterized. Some connections are suggested with the linear time and the branching time temporal logic of the propositions.

Motivation & Objective

• To develop a formal mathematical model for asynchronous circuits based on {0,1}→R functions representing electrical signals.
• To formalize the equations of asynchronous automata and analyze their solutions.
• To characterize the fundamental mode of operation and stability in asynchronous systems.
• To explore semi-modularity and synchronous-like behavior in asynchronous automata.
• To connect the semantics of propositional temporal logic with systems theory through asynchronous automata.

Proposed method

• Models electrical signals as functions from time to {0,1}, using continuous and discrete time frameworks.
• Defines asynchronous automata via input and state functions with determinism relations (equations).
• Applies stability analysis to identify equilibrium points and transition behavior.
• Introduces the fundamental mode of operation as a condition for stable, hazard-free transitions.
• Uses the unbounded delay model to define semi-modular transitions and synchronous-like automata.
• Applies both linear and branching time temporal logic to describe system properties, with continuous and discrete time semantics.

Experimental results

Research questions

• RQ1 How can electrical signals in asynchronous circuits be formally modeled using {0,1}→R functions?
• RQ2 What conditions ensure stability and hazard-free operation in asynchronous automata?
• RQ3 How do semi-modular transitions enable synchronous-like behavior in inherently asynchronous systems?
• RQ4 How can linear and branching time temporal logic be interpreted within the framework of asynchronous automata?
• RQ5 What is the relationship between the fundamental mode of operation and the solvability of automaton equations?

Key findings

• The paper establishes that asynchronous automata can be formally described by systems of equations derived from input and state functions.
• Stability is characterized by equilibrium points where state functions remain constant under input changes.
• The fundamental mode of operation is identified as a necessary condition for safe, hazard-free transitions in asynchronous circuits.
• Semi-modular transitions enable the construction of synchronous-like autonomous automata even in the absence of a global clock.
• The semantics of linear and branching time temporal logic are formally defined in both continuous and discrete time within the automaton framework.
• The work reveals a lack of a unified mathematical theory for asynchronous automata, motivating the need for formal integration of logic and systems theory.