Minimization of Limit-Average Automata

LimAvg-automata are weighted automata over infinite words that aggregate weights along runs with the limit-average value function. In this paper, we study the minimization problem for (deterministic) LimAvg-automata. Our main contribution is an equivalence relation on words characterizing LimAvg-automata, i.e., the equivalence classes of this relation correspond to states of an equivalent LimAvg-automaton. In contrast to relations characterizing DFA, our relation depends not only on the function defined by the target automaton, but also on its structure. We show two applications of this relation. First, we present a minimization algorithm for LimAvg-automata, which returns a minimal LimAvg-automaton among those equivalent and structurally similar to the input one. Second, we present an extension of Angluin's L^*-algorithm with syntactic queries, which learns in polynomial time a LimAvg-automaton equivalent to the target one.

Pumping lemmas for weighted automata

We present pumping lemmas for five classes of functions definable by fragments of weighted automata over the min-plus semiring, the max-plus semiring and the semiring of natural numbers. As a corollary we show that the hierarchy of functions definable by unambiguous, finitely-ambiguous, polynomially-ambiguous weighted automata, and the full class of weighted automata is strict for the min-plus and max-plus semirings.

On Polynomial Recursive Sequences

We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is bn = n!. Our main result is that the sequence un = nn is not polynomial recursive.