The Eventual C-Element Theorem for Delay-Insensitive Asynchronous Circuits

In a seminal 1990 paper [9], Martin presents the C-element Theorem which implies, roughly, that the class of purely delay insensitive (DI) circuits is fundamentally limited in the set of functions it can implement. We provide circuit examples, both DI and not DI, that violate assumptions or results from [9], showing that the set of circuits considered by [9] is more limited than one might expect-and hence, the results from [9] are weaker than previously thought. In this paper we address this issue and amend matters. We use a model of circuits that naturally captures DI properties, and in particular admits all of our circuit examples. We prove the Eventual C-element Theorem for DI circuits. This is a variant of the C-element Theorem that similarly implies that the class of DI circuits is very limited. The proof of the new theorem uses novel techniques, and in particular makes use of a combinatorial property not present in previous work. With the new result, Martin's original insight and its profound implications are restored and provided with a new mathematical justification.