Linear Temporal Logic over finite traces (LTLf) has proved itself to be an important and effective formalism in formal verification as well as in artificial intelligence. Pure past LTLf (pLTL) is the logic obtained from LTLf by replacing each (future) temporal operator by a corresponding past one, and is naturally interpreted at the end of a finite trace. It is known that each property definable in LTLf is also definable in pLTL, and vice versa. However, despite being extensively used in practice, to the best of our knowledge, there is no systematic study of their succinctness. In this paper, we investigate the succinctness of LTLf and pLTL. First, we prove that pLTL can be exponentially more succinct than LTLf by showing that there exists a property definable with a pLTL formula of size n such that the size of all LTLf formulas defining it is at least exponential in n. Then, we prove that LTLf can be exponentially more succinct than pLTL as well. This result shows that, although being expressively equivalent, LTLf and pLTL are incomparable when succinctness is concerned. In addition, we study the succinctness of Safety-LTL (the syntactic safety fragment of LTL over infinite traces) with respect to its canonical form G(pLTL), whose formulas are of the form G(α), G being the globally operator and α a pLTL formula. We prove that G(pLTL) can be exponentially more succinct than Safety-LTL, and that the same holds for the dual cosafety fragment. 2012 ACM Subject Classification Theory of computation → Modal and temporal logics; Theory of computation → Logic and verification
