On the Complexity of the Evaluation of Transient Extensions of Boolean Functions

TL;DR

This paper investigates the complexity of evaluating transient extensions of Boolean functions in hazard detection, introducing a class of functions that can be efficiently evaluated using short transients, and proves their properties.

Contribution

It proposes a method for evaluating general Boolean function extensions and identifies a class of functions that can be efficiently computed using short transients.

Findings

Deciding if the output transient exceeds a bound is NP-complete.

All three-variable functions have the property of efficient evaluation.

Certain other functions also have this property.

Abstract

Transient algebra is a multi-valued algebra for hazard detection in gate circuits. Sequences of alternating 0's and 1's, called transients, represent signal values, and gates are modeled by extensions of boolean functions to transients. Formulas for computing the output transient of a gate from the input transients are known for NOT, AND, OR} and XOR gates and their complements, but, in general, even the problem of deciding whether the length of the output transient exceeds a given bound is NP-complete. We propose a method of evaluating extensions of general boolean functions. We introduce and study a class of functions with the following property: Instead of evaluating an extension of a boolean function on a given set of transients, it is possible to get the same value by using transients derived from the given ones, but having length at most 3. We prove that all functions of three variables, as well as certain other functions, have this property, and can be efficiently evaluated.