Towards a Mathematical Theory of the Delays of the Asynchronous Circuits

Title: Towards a Mathematical Theory of the Delays of the Asynchronous Circuits

Abstract: This research aims to develop a mathematical theory for the delays found in asynchronous circuits. By using the pseudo-Boolean differential calculus, the authors attempt to write equations for the delays, considering bounded, fixed, and relative inertial delays. The theory is presented as a possible starting point for the semi-formalized reconstruction of digital electrical engineering, which is currently a non-formalized theory.

Main Research Question: Can a mathematical theory be developed for the delays of asynchronous circuits using the pseudo-Boolean differential calculus?

Methodology: The study uses the concepts of Boolean functions and delay elements, along with the language of differential equations and inequations on pseudo-Boolean functions. The authors consider various types of delays, such as bounded, fixed, and relative inertial delays, and present definitions and explanations for each.

Results: The research found that the definition of the inertial delay buffer was incorrectly defined in the literature, leading to apparent paradoxes. After a thorough definition of the inertial delay buffer, the concept did not offer the expected property of closure, as the serial connection of the inertial delay buffers is not an inertial delay buffer.

Implications: The study suggests that serious reasons exist for increasing the efforts of finding a good start in digital electrical engineering, as the delays and delay elements are related to the computation of the identity function on {0,1} and are the most simple circuits from electronics. The research also highlights the need for a more formalized theory in this field.

Link to Article: https://arxiv.org/abs/0402038v1 Authors: arXiv ID: 0402038v1