A fragment of first order logic adequate for observation equivalence

We present a logical characterization of the Milner's notion of observation equivalence of processes (“at most one observable action at a time” variant) by using a restricted class of first order formulas. We use the game technique due to Ehrenfeucht as a means to achieve this characterization. First we extend the Ehrenfeucht game by introducing a pair of compatibility relations as a parameter to the game so that we can restrict the moves of the players on the basis of the previous moves. We then define the logic which corresponds to the extended game. Second we characterize the observation equivalence on a restricted class of labelled transition systems (with τ moves), called trace-unique labelled transition systems (t-τℓts's), as the equivalence induced by the games played on certain rcducts of the given t-τℓts's where the compatibility relations are defined in terms of bounded reachability in the t-τℓts's. Combining these two characterizations we get our main result.