Circuit Evaluation for Finite Semirings

TL;DR

This paper investigates the computational complexity of evaluating circuits over finite semirings, establishing a clear dichotomy based on algebraic properties that determines whether the problem is efficiently solvable or P-complete.

Contribution

It provides a complete classification of the circuit evaluation problem for finite semirings, identifying conditions that lead to efficient or hard computational complexity.

Findings

If the semiring's multiplicative semigroup is solvable and lacks a certain subsemiring, evaluation is in NC^2.

Otherwise, the circuit evaluation problem is P-complete.

The paper establishes a dichotomy based on algebraic properties of the semiring.

Abstract

The computational complexity of the circuit evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or 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 $1 \neq 0$, then the circuit evaluation problem for the semiring is in $\mathsf{DET} \subseteq \mathsf{NC}^2$. In all other cases, the circuit evaluation problem is $\mathsf{P}$-complete.