Diagrammatic Reasoning for Delay-Insensitive Asynchronous Circuits

Abstract

Abstract. In this paper we construct a new trace model of delay-insensitive asynchronous circuits inspired by Ebergen's model in such a way that it satisfies the compositional properties of a category, with additional monoidal structure and further algebraic properties. These properties taken together lay a solid mathematical foundation for a diagrammatic approach to reasoning about asynchronous circuits, which represents a formalisation of common intuitions about asynchronous circuits and their properties. Asynchronous circuitsIn the last few decades interest in asynchronous digital design ebbed and flowed. On the one hand, many studies have identified a great promise in asynchronous circuits, in particular low power consumption and modularity. On the other hand problems such as large silicone footprint and difficulties of fabrication hampered the adoption of asynchronous technology into the mainstream. These are just some of the well known advantages and disadvantages of the technology [1].Another challenge raised by asynchronous design is that of reasoning about the correctness of circuits, and it has attracted a great deal of research interest. Several models of asynchronous circuits exists, such as Huffman [2] and burstmode circuits [3], which fall in the broader category of bounded-delay circuits, and delay-insensitive circuits, of which a notable version are the so-called micropipelines [4].The bounded-delay model takes explicitly into account the precise propagation delays of signals along circuit paths, or at least bounds on these delays. This is a fairly obvious model, but it has serious disadvantages. The first one is that computing delays is complicated, as propagation delays in a circuit can be data-dependent. The second one is that reasoning needs to be "geometric" rather than "topological", as wire lengths are highly relevant. This means that accurate reasoning can only be made after a circuit is placed and routed. Because one logical design can have a large number of concrete instantiations (layouts) this low-level way of reasoning is highly undersirable.Far more attractive is the delay-insensitive model, which aims to design circuits that behave well no matter what the delays in the circuit. This is the model we will focus on. Typically, delay insensitive circuits are constructed out of a fixed set of primitive gates. Some of the most common are: CThe Muller C-element is the typical synchronisation gate. It produces an output if it receives signals on both inputs.

Confluence of Graph Rewriting with Interfaces

“…An analogous result (Corollary 41) holds for those SMTs that do not include a special Frobenius structure, but whose set of rules satisfies the left-connected conditions. Hence it applies to a variety of other nonFrobenius theories, such as those in [35,26,21]. In both cases, these decision procedures are amenable to implementation in string diagram rewriting tools like Quantomatic [32] (via an encoding of hypergraphs) or directly in hypergraph-based rewriting tools.…”

“…The rules of these systems need to satisfy a simple condition that we call left-connectedness. Many rewrite systems arising from SMTs (e.g., [35,26,21]), including aforementioned Yang-Baxter rule, enjoy this property. Amongst these, we consider a rewriting system for non-commutative bimonoids that has been shown to be terminating in [4].…”

Towards Compositional Graph Theory

“…Note also that the definition of monoidal width is not tied to ABUV and makes sense in any monoidal theory, which solves the problem of generality outlined in the Introduction. Our claims are that (i) ABUV is a canonical, compositional algebra of simple graphs, and (ii) monoidal width is a robust concept that we believe will be useful in a number of different applications areas [1,4,7,10,19] where monoidal theories are used.…”

“…Walters and collaborators [11,12]. Although we do not concentrate on applications in this paper, we believe that our theory will be relevant in several settings as there has been a recent surge in the applications of symmetric monoidal theories: amongst other works we mention signal flow graphs [1,4], Petri nets [19][20][21], asynchronous circuits [10] and quantum circuits [5,7].…”

Interacting Hopf algebras

“…We now have all the ingredients to state the soundness of kernel computation for an arbitrary R-matrix of Mat R. By Proposition 5.13, the columns of the matrix U r : r → n yield a basis for the kernel of A. Thus U r : r → n together with ¡ : r → 0 is also a pullback span in (7) and since S HA R (¡ : r → 0) = r we know by Lemma 5.9 that n U r r ? = ?…”