Concurrent process equivalences: Some decision problems

Abstract

My favorite example of a simple, “interesting” identity between regular expressions is

$$(X^* .Y^* )^* = (X + Y)^*$$

This identity is true for ALL formal languages X, Y. An easy “folk” theorem shows that this can be verified by instantiating the language variables X, Y as single letters a,b and verifying that (a^*.b^*)^*=(a+b)^*. Checking this second, variable-free, identity is a routine matter of checking equivalence of finite automata. In the “algebraic” approach to concurrent process theory pioneered by Milner and Hoare, the original Kleene operators of language union, concatenation and star are enriched with various renaming and parallel combination operators. The simplest of these is the NONcommunicating parallel connective, ‖, corresponding to the shuffle or merge product of languages. Even this operator creates some interesting complications: the “interleaving identity”

$$a\parallel b = a.b + b.a$$

is certainly true for single letters a,b, but the version with language variables

$$X\parallel Y = X.Y + Y.X$$

is not true for all X,Y (let X=a, Y=bc).

In this talk, I survey some of the results and open problems concerning language identities motivated by concurrent process theory. For example, Rabinovitch and I have shown that the validity problem for regular identities with shuffle is exponential space complete.