Defining concurrent processes constructively

Reviewer: Markus Wolf

Takayama presents a constructive type of theoretical system for reasoning about streams, where concurrent processes are viewed as stream transformers. The system is an extension of a first-order constructive logic with &lgr;&eegr;-calculus as term calculus. The computational streams correspond to the view of streams as lazy lists, whereas a logical stream I s is actually a function of the natural numbers to s . The situation is complex because the logic has to deal with the possible nontermination of a stream transformer. To get logical stream types to occur in specifications of concurrent systems and computational stream types in their implementation, a theorem is proven that states that logical and computational stream types are isomorphic in the sense that there is a compilation function between logical and computational streams. As a type theory, the system must provide methods for proving theorems over specifications of stream transformers. Takayama proposes a new proof rule, the MPST rule, that views a stream transformer as a system that at each “computation step” takes a finite initial segment of the input stream to produce an element of the output stream. This MPST rule is complex and difficult to understand. Takayama explains it at length, contrasting it with the usual co-induction rule. Takayama also shows how a realizability relation can be defined for the constructive system. Examples show that, although the MPST rule seems complex, it works quite well. The paper contains no major error s, although it is confusing at some points. The author assumes familiarity with his brand of constructivism, which derives from the PX system. The diamond operator, for example, is not explained sufficiently. Fortunately, the paper is detailed and contains explicit proofs. Takayama's explanations also help us to understand the formalities, which are sometimes presented too densely for my taste.