Bisimulation and Language Equivalence

One way to understand an interactive system is firmly rooted in language theory, that a system is its set of runs (or words). Properties of systems are described in a linear time temporal logic. Relationships between automata, language theory and logic are then utilised, such as the theory of ω-regular languages and Büchi automata. An alternative viewpoint is that an interactive system should be understood as its capability for interacting with other systems. Language and automata theory then have less relevance because a more intensional account of system behaviour is needed than that given by sets of words. Bisimulation equivalence has a pivotal role within this approach. Bisimulation is a rich concept which appears in various areas of theoretical computer science. Besides its origin for understanding concurrency, it was independently developed in the context of modal logic. In this paper we make some contrasts between bisimulation equivalence and language equivalence. There are two threads. First is that because bisimulation is more intensional, results in language and automata theory can be recast for bisimulation. The second thread is the contrast between definability of language equivalence and bisimulation equivalence. Bisimulation equivalence is definable as a “simple” formula in first-order logic with fixed points. Language equivalence is not definable as an unconditional projection of simple least fixed point. This should be contrasted with a known normal form result for least fixed point logic: any least fixed point definable relation is definable as a projection of a simple least fixed point under equality conditions on its components. It should be noted that undefinability of language equivalence in least fixed point logic per se would actually imply P 6= NP. In section 2 we consider the two origins of bisimulation. In section 3 we describe some results which contrast bisimulation equivalence and language equivalence on automata. The final two sections discuss logics and the undefinability result.