Manfred Droste – Weighted automata and quantitative logics

Institute of Computer Science, Leipzig University, Germany

Quantitative models and quantitative analysis in Computer Science are
receiving increased attention. The goal of this talk is to investigate
quantitative automata and quantitative logics. Weighted automata on
finite words have already been investigated in seminal work of
Schützenberger (1961). They consist of classical finite automata in
which the transitions carry weights. These weights may model, e.g., the
cost, the consumption of resources, or the reliability or probability of
the successful execution of the transitions. This concept soon developed
a flourishing theory, as is exemplified and presented in several books
by Eilenberg, Salomaa-Soittola, Kuich-Salomaa,
Berstel-Reutenauer, Sakarovitch, and the "Handbook of Weighted Automata".
We investigate weighted automata and their relationship to weighted logics. For this, we present syntax and semantics of a quantitative logic; the semantics counts ‘how often’ a formula is true in a given word. Our main result (jointly with Paul Gastin), extending classical results of Büchi, Elgot and Trakhtenbrot (1961), shows that if the weights are taken from an arbitrary semiring, then weighted automata and a syntactically defined fragment of our weighted logic are expressively equivalent. A corresponding result holds for infinite words. Moreover, this extends to quantitative automata investigated by Henzinger et al. with (non-semiring) average-type behaviors, or with discounting or limit average objectives for infinite words. Finally, recall that by fundamental results of Schützenberger and McNaughton-Papert from the 1970s, the classes of first-order definable and aperiodic languages coincide. Very recently, this equivalence could be extended to weighted first order logic and weighed aperiodic automata.