Better Complexity Bounds for Cost Register Automata

Cost register automata (CRAs) are one-way finite automata whose transitions have the side effect that a register is set to the result of applying a state-dependent semiring operation to a pair of registers. Here it is shown that CRAs over the tropical semiring \((\mathbb {N}\cup \{\infty \},\min ,+)\) can simulate polynomial time computation, proving along the way that a naturally defined width-k circuit value problem over the tropical semiring is \(\textsf {P}\)-complete. Then the copyless variant of the CRA, requiring that semiring operations be applied to distinct registers, is shown no more powerful than \(\textsf {NC}^{1}\) when the semiring is \((\mathbb {Z},+,\times )\) or \(({\Gamma }^{*}\cup \{\bot \},\max ,\text {concat})\). This relates questions left open in recent work on the complexity of CRA-computable functions to long-standing class separation conjectures in complexity theory, such as \(\textsf {NC}\) versus \(\textsf {P}\) and NC1 versus GapNC1.