# The Complexity of Counting Models of Linear-time Temporal Logic

Hazem Torfah and Martin Zimmermann

We determine the complexity of counting models of bounded size of specifications expressed in linear-time temporal logic. Counting word-models is \#P-complete, if the bound is given in unary, and as hard as counting accepting runs of nondeterministic polynomial space Turing machines, if the bound is given in binary. Counting tree-models is as hard as counting accepting runs of nondeterministic exponential time Turing machines, if the bound is given in unary. For a binary encoding of the bound, the problem is at least as hard as counting accepting runs of nondeterministic exponential space Turing machines, and not harder than counting accepting runs of nondeterministic doubly-exponential time Turing machines. Finally, counting arbitrary transition systems satisfying a formula is \#P-hard and not harder than counting accepting runs of nondeterministic polynomial time Turing machines with a PSPACE oracle, if the bound is given in unary. If the bound is given in binary, then counting arbitrary models is as hard as counting accepting runs of nondeterministic exponential time Turing machines.