Probabilistic Weighted Automata

Nondeterministic weighted automata are finite automata with numerical weights on transitions. They define quantitative languages L that assign to each word w a real number L(w). The value of an infinite word w is computed as the maximal value of all runs over w, and the value of a run as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We introduce probabilistic weighted automata, in which the transitions are chosen in a randomized (rather than nondeterministic) fashion. Under almost-sure semantics (resp. positive semantics), the value of a word w is the largest real v such that the runs over w have value at least v with probability 1 (resp. positive probability). We study the classical questions of automata theory for probabilistic weighted automata: emptiness and universality, expressiveness, and closure under various operations on languages. For quantitative languages, emptiness and universality are defined as whether the value of some (resp. every) word exceeds a given threshold. We prove some of these questions to be decidable, and others undecidable. Regarding expressive power, we show that probabilities allow us to define a wide variety of new classes of quantitative languages, except for discounted-sum automata, where probabilistic choice is no more expressive than nondeterminism. Finally, we give an almost complete picture of the closure of various classes of probabilistic weighted automata for the following pointwise operations on quantitative languages: max, min, sum, and numerical complement.

💡 Research Summary

The paper introduces Probabilistic Weighted Automata (PWA), a natural extension of nondeterministic weighted automata (NWA) in which the choice of a transition is governed by a probability distribution rather than by nondeterminism. As in NWA, each transition carries a real‑valued weight and a run over an infinite word is evaluated by one of the classic quantitative criteria: maximum, lim sup, lim inf, limit‑average, or discounted sum. The novelty lies in the semantics used to lift run‑values to a word‑value. Two probabilistic semantics are defined:

• Almost‑sure semantics – the value of a word w is the greatest real v such that the set of runs whose value is at least v has probability 1.
• Positive semantics – the value of w is the greatest v such that runs of value at least v occur with positive probability.

With these definitions the authors formulate the standard decision problems of automata theory—emptiness (∃ w : L(w) > θ) and universality (∀ w : L(w) > θ)—for each combination of quantitative criterion and probabilistic semantics.

Decidability Landscape

The paper conducts a systematic decidability analysis for all ten meaningful combinations (five quantitative criteria × two semantics). The main findings are:

• Maximum automata: Both emptiness and universality are PSPACE‑complete under both semantics.
• Lim sup / lim inf automata: Under almost‑sure semantics, both problems are EXPTIME‑complete, while under positive semantics they become undecidable.
• Limit‑average automata: Almost‑sure emptiness is PSPACE‑complete and universality EXPTIME‑complete; under positive semantics both problems are undecidable.
• Discounted‑sum automata: All four decision problems remain PSPACE‑complete regardless of the semantics. Moreover, probabilistic choice does not increase expressive power beyond nondeterminism for this class.

These results delineate a clear boundary: the introduction of probability yields decidability gains for some criteria (maximum) but leads to undecidability for others (lim sup/lim inf, limit‑average) when the weaker positive semantics is used.

Expressive Power

The authors compare PWA with their nondeterministic counterparts. They prove that probabilistic choice strictly enlarges the class of definable quantitative languages for all criteria except discounted sum. For instance, an almost‑sure lim sup automaton can express the language “value = 1 iff the input contains infinitely many a’s”, which cannot be captured by any NWA with the same quantitative criterion. Conversely, for discounted‑sum automata, any probabilistic automaton can be simulated by a nondeterministic one, showing that probability adds no expressive benefit in this case.

Closure Properties

A major part of the work investigates closure under four pointwise operations on quantitative languages: max, min, sum, and numerical complement (1 − L). The results are summarized in a closure table; key observations include:

• Almost‑sure lim sup/lim inf are closed under max and sum but not under min.
• Positive limit‑average is closed under min and max, but not under sum or complement.
• Maximum automata are closed under max but not under sum or complement.
• Discounted‑sum automata are closed under sum and complement for both semantics, yet not under max or min.

These findings have practical implications: when designing specifications that combine quantitative requirements (e.g., taking the maximum of two safety scores), one must select a class of automata known to be closed under the intended operation; otherwise the combined specification may fall outside the original model’s expressive scope.

Significance and Applications

By integrating probabilistic transition selection, the paper opens a new dimension for quantitative language theory. The almost‑sure semantics captures “guaranteed” quantitative behaviour, while the positive semantics models “possible” behaviour with non‑zero likelihood. The decidability results give precise guidance on which quantitative criteria are amenable to algorithmic verification in a probabilistic setting. The expressive‑power analysis shows that probability can encode behaviours that nondeterminism alone cannot, except for discounted‑sum where the two notions coincide. Finally, the closure analysis equips researchers and practitioners with a toolbox for composing complex quantitative specifications without leaving the realm of automata that admit algorithmic analysis.

In summary, the work provides a comprehensive theoretical foundation for probabilistic weighted automata, mapping out their algorithmic limits, expressive advantages, and compositional properties. This foundation is poised to impact areas such as probabilistic model checking, quantitative verification of stochastic systems, and the synthesis of controllers where both quantitative performance and probabilistic uncertainty must be reasoned about simultaneously.