Totality, Definability and Boolean Ciruits

1 I n t r o d u c t i o n Adding parallel constants to a p rog ramming language strict ly increase the expressive power of the language, in general. For instance, extending Scot t ' s P C F with parallel-or, one can define any finite continuous funct ion [7]. However, it is an open problem whether parallelism adds expressive power, if we restrict our attent ion to total functions. Total i ty is a natura l not ion in domain theory: a g round object (such as an integer or a boolean), is total if it is defined (i.e. different f rom • and a function is tota l if it gives total values on tota l arguments . Hence tota l i ty is a logical predicate [6]. An equivalent definition of to ta l i ty may be given in terms of a logical (partial) equivalence relation: at ground types, x T y if x and y are equal and different f rom 2 ; at higher types, f ~ . r g if, whenever x T y at the appropr ia te type, then f (x ) T g(y). It turns out tha t f . J f if and only if f is total in the previously defined sense. Parallel-or is total , and it is ~T-equivalent to the str ict-or function, which is sequential (PCF-definable) . Our original mot ivat ion for this work was to explore the following conjecture, due to Berger [2]: For any total, parallel function f there exists a sequential function g such that f ,,~T g, where "parallel" means definable by P C F + ( P C F extended by parallel-or), "sequential" means PCF-def inable and the type frame we refer to is the Scott hierarchy of continuous functions over the flat domains of integer and boolean values 1. 1 Berger's conjecture is slightly complicated by the fact that, for infinite types, one has to take into account also the 3 functional.