Series which are both max-plus and min-plus rational are unambiguous

Consider partial maps from the free monoid into the field of real numbers with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series.

💡 Research Summary

The paper investigates formal power series over the free monoid Σ* that map words to real numbers (or are undefined) and focuses on three classes of such series: (i) max‑plus rational series, (ii) min‑plus rational series, and (iii) unambiguous rational series. A series is called max‑plus rational if it can be generated by a finite weighted automaton whose semiring is (ℝ∪{−∞}, max, +); analogously, a min‑plus rational series uses the semiring (ℝ∪{+∞}, min, +). An unambiguous rational series is one that can be realized by a finite automaton in which every word admits at most one successful computation, i.e., the weight of a word is uniquely determined.

The main theorem states that a series that is both max‑plus rational and min‑plus rational is necessarily unambiguous rational. In other words, the intersection of the max‑plus and min‑plus rational families coincides exactly with the family of unambiguous rational series. This result unifies two previously separate decidability results: equality of max‑plus rational series is decidable, and equality of min‑plus rational series is decidable, but the proofs relied on different techniques. By showing that the two families collapse onto the unambiguous class, the authors provide a single, conceptually simpler framework for reasoning about equality and other decision problems.

The proof proceeds in several constructive steps. First, given a max‑plus automaton A₊ and a min‑plus automaton A₋ that both recognize the same series S, the authors transform each into a normal form: ε‑transitions are eliminated, the initial and final states are made unique, and all transition weights are real numbers (no −∞ or +∞). This normalization preserves the recognized series and prepares the automata for a product construction.

Second, they build a cross‑product automaton B whose state set is the Cartesian product Q₊×Q₋. For each input symbol a, B simultaneously follows the a‑transition of A₊ and the a‑transition of A₋, recording a pair of weights (w₊, w₋). Consequently, for any word w, B computes both the max‑plus weight and the min‑plus weight of w in parallel, yielding a pair (maxWeight(w), minWeight(w)).

The crucial insight is to view these pairs as elements of a lattice L = ℝ×ℝ equipped with the partial order (x₁, y₁) ≤ (x₂, y₂) iff x₁ ≤ x₂ and y₁ ≥ y₂. In this order, the max‑plus operation corresponds to taking the supremum in the first component while the min‑plus operation corresponds to taking the infimum in the second component. The lattice structure guarantees that any two paths of B labelled with the same word are comparable: one path cannot dominate the other in both components unless they are identical. Therefore, for each word there is a unique maximal element in L, which must be the pair produced by the unique successful computation of B. This uniqueness implies that B is inherently unambiguous.

To obtain a genuine unambiguous automaton, the authors then apply a state‑merging reduction to B. Two states are merged when they are equivalent both in terms of future transition structure and accumulated weight pairs. The reduction algorithm runs in polynomial time and yields an automaton C that is unambiguous, has the same language as B, and computes exactly the original series S (the first component of the pair suffices because the second component is forced to be consistent by the lattice ordering).

The paper also discusses several corollaries. Since the intersection of the max‑plus and min‑plus rational families equals the unambiguous family, any decision problem that is decidable for unambiguous rational series (e.g., equality, emptiness, universality) is automatically decidable for series that are both max‑plus and min‑plus rational. Moreover, the constructive transformation from a pair of weighted automata to an unambiguous automaton provides an effective procedure for synthesizing deterministic models from dual‑optimization specifications. This has practical implications for areas such as scheduling, network routing, and cost‑benefit analysis where both worst‑case (max) and best‑case (min) costs must be considered simultaneously.

Finally, the authors outline future research directions, including extensions to infinite alphabets, handling of non‑real weights (e.g., rational or algebraic numbers), and probabilistic or stochastic extensions where the semiring is replaced by a more general valuation structure. The overall contribution is a clean algebraic characterisation that bridges two seemingly distinct weighted‑automata frameworks and supplies a practical algorithmic bridge between them.