# On the Complexity of Path Checking in Temporal Logics

Daniel Bundala ( University of Oxford )

- 11:30 22nd May 2013 ( week 5, Trinity Term 2013 )051

Given a formula in a temporal logic such as LTL or MTL, a
fundamental problem is the complexity of evaluating the formula on a
given trace. For LTL, the complexity of this task was recently
shown to be in NC. In this paper, we establish
a connection between LTL path checking and planar circuits, which
we then exploit to show that any further progress in determining the
precise complexity of LTL+Past path checking would immediately
entail a more efficient evaluation algorithm for a certain class of
planar circuits. Moreover, we show that the path-checking problem for
LTL extended with exclusive or is P-hard. We then present an
NC algorithm for MTL, a quantitative (or metric) extension of
LTL. Finally, we give an AC^1 algorithm for UTL, the
unary fragment of LTL. At the time of writing, this appears to be
the most expressive fragment of LTL with a more efficient
path-checking algorithm than for full LTL (subject to standard
complexity-theoretic assumptions).