Categories, Allegories and Circuit Design or the Picture of a Wiring Primitive (we Often Omit the Bounding Box) 3: Axioms for Pictures

Relational languages such as Ruby are used to derive hardware circuits from abstract speciications of their behaviour. Much reasoning is done informally in Ruby using pictorial representations of relational terms. We formalise this use of pictures in circuit design. We show that pictures naturally form a unitary pretabular allegory. Homomorphisms of pictures correspond to adding new wires or circuit components. Two pictures are mutually homomorphic if and only if they represent equal allegorical terms. We prove soundness and completeness results which guarantee that deriving circuits using pictures does not lead to errors. We illustrate the use of pictures by deriving the ripple adder implementation from a high level, behavioural speciication. Hardware circuit design involves translating abstract speciications of programs into eecient circuits which compute those programs. Pictures are widely used as an informal means of translating a speciication into an implementable circuit, and improving the layout of a circuit. A disadvantage of this informal approach is the lack of an independent means (apart from building and testing the circuit, which may be expensive and is not guaranteed to succeed) to verify that a picture indeed denotes the desired program, and that no errors have been introduced during the design process. In this paper we provide a relational semantics for pictures, together with an equivalence on pictures which shows how to transform one picture into another while preserving its semantics. The evident notion of homomorphism between pictures corresponds naturally to simple, relatively high level operations on pictures (adding new wires and components). Two pictures are provably equivalent if and only if they are mutually homomorphic, which is if and only if they denote the same relation for any interpretation of their basic components. These results lead us to a simple decision procedure for equivalence of circuits 3], which has been implemented 9]. Our results encourage the use of pictures in deriving circuits, by providing a formal foundation for that use. Pictures are easier and quicker to understand than syntactic terms, and so their use speeds up the process of circuit design. This paper illustrates these advantages in two ways, by illuminating the rather technical proof that the category of unitary pretabular allegories is isomorphic to the category of discrete cartesian bi-categories, and by presenting the derivation of a ripple adder from a high level behavioural speciication. Relations have been proposed as a paradigm for circuit development for several reasons. Relations provide …