On Delay and Regret Determinization of Max-Plus Automata

Decidability of the determinization problem for weighted automata over the semiring (ℤ∪−∞,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 w in the language of N, the accepting run of D on w is always at most k-away from a maximal accepting run of N on w. 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 w 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 w 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.

(Note: It was recently shown that the r-regret determinization corresponds to a quantitative version of the notion of good-for-games.)

This is joint work with colleagues from ULB, Belgium.