Iteration semirings are Conway semirings satisfying Conway's group identities. We show that the semirings $\N^{\rat}\llangle \Sigma^* \rrangle$ of rational power series with coefficients in the semiring $\N$ of natural numbers are the free partial iteration semirings. Moreover, we characterize the semirings $\N_\infty^{\rat}\llangle \Sigma^* \rrangle$ as the free semirings in the variety of iteration semirings defined by three additional simple identities, where $\N_\infty$ is the completion of $\N$ obtained by adding a point of infinity. We also show that this latter variety coincides with the variety generated by the complete, or continuous semirings. As a consequence of these results, we obtain that the semirings $\N_\infty^{\rat}\llangle \Sigma^* \rrangle$, equipped with the sum order, are free in the class of symmetric inductive $^*$-semirings. This characterization corresponds to Kozen's axiomatization of regular languages.
Deep Dive into Axiomatizing rational power series.
Iteration semirings are Conway semirings satisfying Conway’s group identities. We show that the semirings $\N^{\rat}\llangle \Sigma^* \rrangle$ of rational power series with coefficients in the semiring $\N$ of natural numbers are the free partial iteration semirings. Moreover, we characterize the semirings $\N_\infty^{\rat}\llangle \Sigma^* \rrangle$ as the free semirings in the variety of iteration semirings defined by three additional simple identities, where $\N_\infty$ is the completion of $\N$ obtained by adding a point of infinity. We also show that this latter variety coincides with the variety generated by the complete, or continuous semirings. As a consequence of these results, we obtain that the semirings $\N_\infty^{\rat}\llangle \Sigma^* \rrangle$, equipped with the sum order, are free in the class of symmetric inductive $^*$-semirings. This characterization corresponds to Kozen’s axiomatization of regular languages.
One of the most basic algebraic structures studied in Computer Science are the semirings Reg(Σ * ) of regular (or rational) languages over an alphabet Σ equipped with the star operation. Salomaa [32] has axiomatized these semirings of regular languages using a few simple identities and the unique fixed point rule asserting that if the regular language a does not contain the empty word then a * b is the unique solution of the fixed point equation x = ax + b. There are several ways of expressing the empty word property using a first-order language. Probably, the simplest way is by the inequality 1 + a = a. Using this, the unique fixed point rule can be formulated as the first-order axiom ∀a∀b∀x((1 + a = a ∧ ax + b = x) ⇒ x = a * b).
Salomaa’s result then amounts to the assertion that for any Σ, Reg(Σ * ) is freely generated in the class of * -semirings satisfying a finite number of (simple) identities and the above axiom. We have thus a finite first-order axiomatization of regular languages.
Because of the extra condition on a, the unique fixed point rule is not a quasi-identity. A finite axiomatization using only quasi-identities has been first obtained by Archangelsky and Gorshkov, cf. [2]. A second, and perhaps more serious concern is that several natural * -semirings which satisfy all identities of regular languages are not models of the unique fixed point rule. Examples of such semirings are semirings of binary relations with the reflexive-transitive closure operation as star, since for binary relations, the equation x = ax + b usually has several solutions, even if 1 + a = a (i.e., when a not reflexive). On the other hand, a * b is least among all solutions, so that
where a * b ≤ x may be viewed as abbreviation for a * b + x = x. And indeed, the semirings of regular languages can be characterized as the free algebras in a quasi-variety of semirings with a star operation axiomatized by a finite set of simple identities and the above least fixed point rule, or the least pre-fixed point rule
This result is due to Krob [25]. In [22,23], Kozen also required the dual of the least (pre-)fixed point rule ∀a∀b∀x(xa
and gave a simpler proof of completeness of this system. Several other finite axiomatizations are derivable from Krob’s and Kozen’s systems, see [12,13,9].
But the largest class of algebras in which the semirings of regular languages are free is of course a variety. This variety, the class of all semirings with a star operation satisfying all identities true of regular languages, is the same as the variety generated by all * -semirings of binary relations. The question whether this variety is finitely based was answered by Redko [30,31] and Conway [15], who showed that there is no finite (first-order or equational) axiomatization. The question of finding infinite equational bases was considered in [7,25]. The system given in Krob [25] consists of the Conway semiring identities, the identity 1 * = 1, and Conway’s group identities [15] associated with the finite (simple) groups. Conway semirings were first defined formally in [6,8].
Conway semirings are semirings equipped with a star operation satisfying (a+b) * = a * (ba * ) * and (ab) * = a(ba) * b + 1. Conway semirings satisfying the infinite collection of group identities are called iteration semirings, cf. [18]. The terminology is due to the fact that iteration semirings are exactly the semirings which are iteration algebras, i.e., satisfy the axioms of iteration theories [8] which capture the equational properties of the fixed point operation. Thus, Krob’s result characterizes the semirings of regular languages as the free iteration semirings satisfying 1 * = 1 (which implies that sum is idempotent). Another proof of this result using iteration theories can be obtained by combining the axiomatization of regular languages from [7] and the completeness (of certain generalizations of) the group identities for iteration theories, established in [18].
In this paper, we drop the idempotence of the sum operation and consider the semirings of rational power series N rat Σ * and N rat ∞ Σ * over the semiring N of natural numbers and its completion N ∞ with a point of infinity. The star operation in N rat Σ * is defined only on those proper power series having 0 as the coefficient of the empty word (the empty word property), whereas the star operation in N rat ∞ Σ * is totally defined. We prove that N rat ∞ Σ * is freely generated by Σ in the variety V of all iteration semirings satisfying the identities 1 * 1 * = 1 * , 1 * a = a1 * and 1 * (1 * a) * = 1 * a * . This result is also of interest because V coincides with the variety generated by those * -semirings that arise from (countably) complete or continuous semirings by defining a * as the sum n≥0 a n . Moreover, we prove that N rat Σ * is freely generated by Σ in the class of all partial iteration semirings. As a consequence of the equational axiomatizations, we show that N rat ∞ Σ * , equipped
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