On Determinism and Unambiguity of Weighted Two-Way Automata

This paper deals with one-way and two-way weighted automata. When the semiring of weights is commutative, we prove that unambiguous one-way automata, unambiguous two-way automata and deterministic two-way automata realize the same (rational) power series. If the semiring of weights is not commutative, unambiguous one-way automata and deterministic two-way automata realize the same rational power series, but unambiguous two-way automata may realize non rational power series.

Rationally Additive Semirings

We define rationally additive semirings that are a generalization of (omega-)complete and (omega-)continuous semirings. We prove that every rationally additive semiring is an iteration semiring. Moreover, we characterize the semirings of rational power series with coefficients in N_infty, the semiring of natural numbers equipped with a top element, as the free rationally additive semirings.

Inductive * -Semirings

One of the most well-known induction principles in computer science<br />is the fixed point induction rule, or least pre-fixed point rule. Inductive <br />*-semirings are partially ordered semirings equipped with a star operation<br />satisfying the fixed point equation and the fixed point induction rule for<br />linear terms. Inductive *-semirings are extensions of continuous semirings<br />and the Kleene algebras of Conway and Kozen.<br />We develop, in a systematic way, the rudiments of the theory of inductive<br />*-semirings in relation to automata, languages and power series.<br />In particular, we prove that if S is an inductive *-semiring, then so is<br />the semiring of matrices Sn*n, for any integer n >= 0, and that if S is<br />an inductive *-semiring, then so is any semiring of power series S((A*)).<br />As shown by Kozen, the dual of an inductive *-semiring may not be inductive. <br />In contrast, we show that the dual of an iteration semiring is<br />an iteration semiring. Kuich proved a general Kleene theorem for continuous<br /> semirings, and Bloom and Esik proved a Kleene theorem for all Conway <br />semirings. Since any inductive *-semiring is a Conway semiring<br />and an iteration semiring, as we show, there results a Kleene theorem <br />applicable to all inductive *-semirings. We also describe the structure<br />of the initial inductive *-semiring and conjecture that any free inductive<br />*-semiring may be given as a semiring of rational power series with <br />coefficients in the initial inductive *-semiring. We relate this conjecture to<br />recent axiomatization results on the equational theory of the regular sets.