Implementation Relations for Testing Through Asynchronous Channels

This paper concerns testing from an input–output transition system (IOTS) model of a system under test that interacts with its environment through asynchronous first in first out (FIFO) channels. It explores methods for analysing an IOTS without modelling the channels. If IOTS M produces sequence σ, then, since communications are asynchronous, output can be delayed and so a different sequence might be observed. Thus, M defines a language Tr(M) of sequences that can be observed when interacting with M through FIFO channels. We define implementation relations and equivalences in terms of Tr(M): an implementation relation says how IOTS N must relate to IOTS M in order for N to be a correct implementation of M. It is important to use an appropriate implementation relation since otherwise the verdict from a test run might be incorrect and also because it influences test generation. It transpires that it is undecidable whether IOTS N conforms to IOTS M and so also whether there is a test case that can distinguish between two IOTSs. We also investigate the situation in which we have a finite automaton P and either wish to know whether Tr(M) ∩ L(P) is empty or whether Tr(M) ∩ Tr(P) is empty and prove that these are undecidable. In addition, we give conditions under which conformance and intersection are decidable.