Efficient Analysis of Unambiguous Automata Using Matrix Semigroup Techniques

Abstract

We introduce a novel technique to analyse unambiguous Büchi automata quantitatively, and apply this to the model checking problem. It is based on linear-algebra arguments that originate from the analysis of matrix semigroups with constant spectral radius. This method can replace a combinatorial procedure that dominates the computational complexity of the existing procedure by Baier et al. We analyse the complexity in detail, showing that, in terms of the set Q of states of the automaton, the new algorithm runs in time O(|Q| 4 ), improving on an efficient implementation of the combinatorial algorithm by a factor of |Q|.

Image-Binary Automata

“…As in [3] the algorithm will consist of solving a system of linear equations, split up in two parts. The first part is the basic linear system, which we will describe below, while the second part will add normalising equations in the form of cuts (or pseudo-cuts, if [18] is followed). Since the weights of the edges within SCCs are all on loops, and therefore all have weight 1, this part can be copied verbatim from Baier et al [3].…”