Bounded Reordering in the Distributed Test Architecture

In the distributed test architecture, the system under test (SUT) interacts with its environment at multiple physically distributed ports and the local testers at these ports do not synchronize their actions. This presents many challenges and, in particular, apparently incorrect behaviors can be the consequence of an erroneous assumption about the exact order in which actions were performed at different ports. In previous work, we defined a conformance relation for the distributed test architecture. Essentially, the SUT is faulty if we observe a trace <inline-formula> <tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> such that no admissible reordering of the actions in <inline-formula><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> could have been produced by the specification. However, this notion can be weak if the compared traces might be <italic>too</italic> different. This paper introduces conformance relations where, for a given metric, a reordering is only considered if the distance between the two traces is at most a certain bound <inline-formula><tex-math notation="LaTeX">$k$</tex-math> </inline-formula>. We introduce two different metrics and provide algorithms to construct finite automata accepting these <italic>close</italic>, with respect to each metric, sequences. We also study the computational complexity of the two main problems associated with the new framework: deciding whether a trace is accepted by the new automaton and deciding whether one system conforms to a specification with respect to the new conformance relation.