Two variable and linear temporal logic in model checking and games

Model checking linear-time properties expressed in first-order logic has nonelementary complexity, and thus various restricted logical languages are employed. In this paper we consider two such restricted specification logics, linear temporal logic (LTL) and two-variable first-order logic (FO). LTL is more expressive but FO can be more succinct, and hence it is not clear which should be easier to verify. We take a comprehensive look at the issue, giving a comparison of verification problems for FO, LTL, and various sublogics thereof across a wide range of models. In particular, we look at unary temporal logic (UTL), a subset of LTL that is expressively equivalent to FO; we also consider the stutterfree fragment of FO, obtained by omitting the successor relation, and the expressively equivalent fragment of UTL, obtained by omitting the next and previous connectives. We give three logic-to-automata translations which can be used to give upper bounds for FO and UTL and various sub-logics. We apply these to get new bounds for both nondeterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). We couple these with matching lower-bound arguments. Next, we look at combining FO verification techniques with those for LTL. We present here a language that subsumes both FO and LTL, and inherits the model checking properties of both languages. Our results give both a unified approach to understanding the behaviour of FO and LTL, along with a nearly comprehensive picture of the complexity of verification for these logics and their sublogics.