Circuits and Expressions over Finite Semirings

The computational complexity of the circuit and expression evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or a multiplicative identity. The following dichotomy is shown: If a finite semiring is such that (i) the multiplicative semigroup is solvable and (ii) it does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 not equal 0, then the circuit evaluation problem is in DET subset of NC2, and the expression evaluation problem for the semiring is in TC0. For all other finite semirings, the circuit evaluation problem is P-complete and the expression evaluation problem is NC1-complete. As an application, we determine the complexity of intersection non-emptiness problems for given context-free grammars (regular expressions) with a fixed regular language.