A Conway semiring is a semiring $S$ equipped with a unary operation $^*:S \to S$, always called 'star', satisfying the sum star and product star identities. It is known that these identities imply a Kleene type theorem. Some computationally important semirings, such as $N$ or $N^{\rat}\llangle \Sigma^* \rrangle$ of rational power series of words on $\Sigma$ with coefficients in $N$, cannot have a total star operation satisfying the Conway identities. We introduce here partial Conway semirings, which are semirings $S$ which have a star operation defined only on an ideal of $S$; when the arguments are appropriate, the operation satisfies the above identities. We develop the general theory of partial Conway semirings and prove a Kleene theorem for this generalization.
Deep Dive into Partial Conway and iteration semirings.
A Conway semiring is a semiring $S$ equipped with a unary operation $^*:S \to S$, always called ‘star’, satisfying the sum star and product star identities. It is known that these identities imply a Kleene type theorem. Some computationally important semirings, such as $N$ or $N^{\rat}\llangle \Sigma^* \rrangle$ of rational power series of words on $\Sigma$ with coefficients in $N$, cannot have a total star operation satisfying the Conway identities. We introduce here partial Conway semirings, which are semirings $S$ which have a star operation defined only on an ideal of $S$; when the arguments are appropriate, the operation satisfies the above identities. We develop the general theory of partial Conway semirings and prove a Kleene theorem for this generalization.
It is well-known that there exists no finite base of identities for the regular languages equipped with the operations of union +, product • and (Kleene) star * ; cf. [6,18,19]. The notion of Conway semirings involves two important identities for the star operation: the sum star and the product star identities, (a + b) * = a * (ba * ) * (ab) * = 1 + a(ba) * b.
It has been shown that Kleene’s theorem for languages and automata, as well as its generalization to weighted automata, are consequences of these identities. Thus, it is possible to derive Kleene’s theorem by purely equational reasoning from the axioms of Conway semirings, cf. [6,3,13]. Important examples of Conway semirings are
• the boolean semiring B;
• the semirings B rat Σ * of rational power series with coefficients in B, which are isomorphic copies of the semirings of regular languages,
• the continuous or complete semirings [6,8,3].
However, many computationally important semirings do not have a totally defined star operation satisfying the Conway identities. Some examples of such semirings are the semirings S rat Σ * of rational power series over Σ with coefficients in the semiring S, where S is either the semiring N of natural numbers or a nontrivial ring (if 1 * = 1 * • 1 + 1, then 0 = 1). The semiring N can be embedded into a Conway semiring, namely the semiring N ∞ obtained by adding a point ∞. By means of this embedding, Kleene’s theorem for Conway semirings becomes indirectly applicable to weighted finite automata over N. On the other hand, such an embedding does not exist for all semirings, so that Kleene’s theorem for Conway semirings does not cover weighted finite automata over such semirings.
In this paper, we introduce partial Conway semirings as a generalization of Conway semirings. In a partial Conway semiring S, the domain D(S) of the star operation is an ideal of the semiring; further, when restricted to this domain, the sum star and product star identities hold. We prove a Kleene theorem for partial Conway semirings, and thus obtain a single unified result which is directly applicable in all of the above situations. We also outline the general theory of partial Conway semirings which parallels with the theory of Conway semirings. This general theory provides the background for the Kleene theorem mentioned above.
Moreover, we also introduce partial iteration semirings, which are partial Conway semirings satisfying Conway’s group identities, cf. [6,16]. We define partial iterative semirings as star semirings in which certain linear equations have unique solutions. We prove that partial iterative semirings are partial iteration semirings. As an application of this result, we show that for any semiring S and set Σ, the power series semiring S Σ * is a partial iterative semiring and thus a partial iteration semiring.
The results of this paper are used in [4], where the semirings N rat Σ * are characterized as the free partial iteration semirings, and the semirings N rat ∞ Σ * as the free algebras in a subvariety of iteration semirings satisfying three additional simple identities.
A semiring [14] is an algebra S = (S, +, •, 0, 1) such that (S, +, 0) is a commutative monoid, (S, •, 1) is a monoid, moreover 0 is an absorbing element with respect to multiplication and product distributes over sum:
for all a, b, c ∈ S. The operation + is called sum or addition, and the operation • is called product or multiplication. A semiring S is called idempotent if a + a = a for all a ∈ S. A morphism of semirings preserves the sum and product operations and the constants 0 and 1. Since semirings are defined by identities, the class of all semirings is a variety (see e.g., [15]) as is the class of all idempotent semirings.
An important example of a semiring is the semiring N = (N, +, •, 0, 1) of natural numbers equipped with the usual sum and product operations, and an important example of an idempotent semiring is the boolean semiring B whose underlying set is {0, 1} and whose sum and product operations are the operations ∨ and ∧, i.e., disjunction and conjunction. Actually N and B are respectively the initial semiring and the initial idempotent semiring.
We end this section by describing three constructions on semirings. For more information on semirings, the reader is referred to Golan’s book [14].
Suppose that S is a semiring and Σ is a set. Let Σ * denote the free monoid of all words over Σ including the empty word ǫ. A formal power series, or just power series over S in the (noncommuting) letters in Σ is a function s : Σ * → S. It is a common practice to represent a power series s as a formal sum w∈Σ * (s, w)w, where the coefficient (s, w) is ws, the value of s on the word w. The support of a series s is the set supp(s) = {w : (s, w) = 0}. When supp(s) is finite, s is called a polynomial. We let S Σ * and S Σ * respectively denote the collection of all power series and polynomials over S in the letters Σ.
We define the sum s + s
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