Weighted Automata Theory and Applications

Weighted automata extend classical automata by assigning quantitative values to transitions and accepting runs, thus enabling the analysis of quantitative properties such as costs, probabilities and resource consumption. Fundamentally grounded in the algebraic framework of semirings, the theory unifies diverse models including min-plus and max-plus automata, probabilistic automata and cost-register machines. Core research has established hierarchies of expressiveness based on ambiguity, shown strict separation between classes of unambiguous, finitely-ambiguous and polynomially-ambiguous weighted automata, and explored closure properties under operations such as sum, product and homomorphism. Beyond words, tree-shaped inputs have given rise to weighted tree automata with equality and inequality constraints, broadening applicability to structured data in natural language processing and XML document analysis. On the logic side, weighted linear-time logics and weighted Büchi automata form a bridge between specification and verification of quantitative system properties, providing decidable fragments for model checking and specifying k-safe behaviours over infinite runs. Practical applications span performance analysis of embedded systems, speech recognition, natural language processing and verification of systems with quantitative resource constraints. Recent advances continue to deepen theoretical understanding while expanding algorithmic techniques for decision and optimisation problems in this richly interconnected field.

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Weighted Automata Theory and Applications publication trend

The graph below shows the total number of articles in weighted automata theory and applications across all publications each year (not limited to Nature Index journals).

Technical terms

Weighted automaton: A finite automaton in which transitions carry weights drawn from a semiring, and the weight of a run is computed by combining transition weights according to semiring operations.

Semiring: An algebraic structure with two binary operations—addition and multiplication—that provides a framework for combining quantitative values such as costs or probabilities.

Unambiguous automaton: A weighted automaton in which each accepted input has at most one accepting run, ensuring deterministic weight computation per input.

Tree automaton: An automaton that reads tree-structured inputs, assigning weights to nodes or branches and evaluating them via semiring operations to decide or quantify tree languages.

Weighted LTL: An extension of linear temporal logic that associates quantitative values with temporal formulae, interpreted over infinite runs via valuation monoids to capture cost or probability measures.

References

1. Pumping lemmas for weighted automata. Logical Methods in Computer Science (2021).
2. Weighted Tree Automata with Constraints. Theory of Computing Systems (2023).
3. Describing Weighted Safety with Weighted LTL over Product omega-valuation Monoids. Scientific Annals of Computer Science (2023).