Getting Your Wires Crossed

Things should be as simple as possiblèand no simpler', so they say. This paper is about simplifying a construct in Ruby, to reduce the complexity of the algebra without sacriicing expressive power. Ruby, for which Mary Sheeran is largely responsible, is a language of relations and a collection of rules { equalities { for algebraic transformation of expressions in that language. The presentation here is by way of being a natural history of the combining forms that build state machines. It shows that over time the operators have evolved as we have been able to simplify calculation by chosing to name combining forms which have simple properties, rather than naming those forms which correspond to the original intuition about the application area, and this apparently without making it signiicantly harder to relate the calculations to reality. Whether there are any general lessons which can be drawn from this history, I leave to the reader. FP is a language of functions 6], from which Ruby has been developed. The in its name is the operator which constructs a state machine from its transition function, something traditionally denoted by. This operator constructs an automaton f; g] from two functions, f which delivers the output as a function of the present state and input, and g which delivers the next state also as a function of the present state and input. This is the characterisation of a state machine known as a Mealy machine, as opposed to Moore machines, which describe the same machine by a diierent pair of functions. An implementation is likely, for detailed operational reasons like propagation delay, to be more like a Mealy machine than the apparently simpler corresponding Moore machine which calculates its output as a function of its state. So it was the Mealy machine that was chosen as a named combining form to be added to Backus' FP 1]. It is an odd coincidence that Mealy, Moore, and Mary might all be being spelled with a capital. which is called C in reference 6]. Here the dot is (forward) composition of functions, the ones and twos are the FP notation for the projection functions which deliver the rst and second components of a pair, and f; g] is the function that returns the pair of results of applying each of f and g to its argument.