The Complexity of Simulation and (Exotic) Matrix Multiplications

We study the complexity of computing the simulation preorder of finite transition systems, a crucial problem in model checking of temporal logic, showing that it is strongly related to some variants of matrix multiplication. We first show that any O(n)-time algorithm for n-states transition systems can be used to compute the product of two n × n boolean matrices in O(n) time. This reduction is the first evidence of the difficulty to get a truly subcubic combinatorial algorithm for the simulation preorder, and holds even restricting the problem to acyclic systems. For acyclic Kripke structures, we show an algorithm that employs fast (boolean) matrix multiplication and runs in n time (where ω < 2.4 is the exponent of matrix multiplication). Moreover, we exhibit O(n)-size canonical certificates that can be checked by verifying a constant number of n × n standard matrix multiplications over the integers, i.e., in O(n) (randomized) time. For cyclic structures, we give some evidence that the problem might possibly be harder. We define the max-semi-boolean matrix multiplication (MSBMM) as the matrix multiplication on the semi-ring (max,×) where one of the two matrices contains only 0’s and 1’s. We obtain O(n)-size canonical certificates for cyclic Kripke structures that can be checked by verifying a constant number of n × n MSBMMs. Then, we show that verifying a n × n MSBMM can be reduced to verifying the simulation preorder in a O(n log n)-states Kripke structure. Hence, for any α ≥ 2, if MSBMM can be verified in O(n) time, then the simulation preorder admits certificates that can be checked in O(n) time, otherwise the simulation preorder does not admit a O(n)-time algorithm. 1 ar X iv :1 60 5. 02 15 6v 1 [ cs .C C ] 7 M ay 2 01 6