# On Delay and Regret Determinization of Max-Plus Automata

## TL;DR

This paper investigates the decidability of determinizing weighted automata over a specific semiring by introducing k-delay and r-regret constraints, establishing complexity results and equivalences with the general problem.

## Contribution

It introduces k-delay and r-regret determinization notions for weighted automata and proves their equivalence to general determinization, along with complexity classifications.

## Key findings

- Decidability of determinization is linked to k-delay determinization.
- k-delay and r-regret problems are EXPtime-complete.
- Determining r-regret determinizability is in EXPtime.

## Abstract

Decidability of the determinization problem for weighted automata over the semiring $(\mathbb{Z} \cup {-\infty}, \max, +)$, WA for short, is a long-standing open question. We propose two ways of approaching it by constraining the search space of deterministic WA: k-delay and r-regret. A WA N is k-delay determinizable if there exists a deterministic automaton D that defines the same function as N and for all words {\alpha} in the language of N, the accepting run of D on {\alpha} is always at most k-away from a maximal accepting run of N on {\alpha}. That is, along all prefixes of the same length, the absolute difference between the running sums of weights of the two runs is at most k. A WA N is r-regret determinizable if for all words {\alpha} in its language, its non-determinism can be resolved on the fly to construct a run of N such that the absolute difference between its value and the value assigned to {\alpha} by N is at most r. We show that a WA is determinizable if and only if it is k-delay determinizable for some k. Hence deciding the existence of some k is as difficult as the general determinization problem. When k and r are given as input, the k-delay and r-regret determinization problems are shown to be EXPtime-complete. We also show that determining whether a WA is r-regret determinizable for some r is in EXPtime.