[Paper Review] Partial Conway and Iteration Semirings

Stephen L. Bloom, Zoltán Ésik+1 more|arXiv (Cornell University)|Apr 1, 2008

Logic, programming, and type systems16 references19 citations

TL;DR

This paper introduces partial Conway semirings—semirings with a star operation defined only on an ideal—generalizing Conway semirings to handle computationally relevant structures like N and rational power series. It establishes a Kleene-type theorem for this framework, extending foundational results to settings where total star operations are impossible.

ABSTRACT

A Conway semiring is a semiring S equipped with a unary operation *: S → 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$^{rat}$LΣG of rational power series of words on Σ 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.

Motivation & Objective

• To address the limitation of Conway semirings, which require a total star operation, in modeling computationally important semirings like N and N^{rat}LΣG.
• To generalize the theory of Conway semirings by defining the star operation only on an ideal, enabling application to semirings where a total star is mathematically infeasible.
• To develop a general theory of partial Conway semirings that preserves key identities like sum star and product star.
• To extend the classical Kleene theorem to partial Conway semirings, ensuring equivalence between rational expressions and recognizable series.

Proposed method

• Define a partial Conway semiring as a semiring S with a star operation *: I → S, where I is an ideal of S, satisfying sum star and product star identities for elements in I.
• Characterize the conditions under which the star operation on the ideal I preserves the algebraic structure and identities of Conway semirings.
• Prove that in a partial Conway semiring, the Kleene theorem holds: every recognizable series is expressible as a rational expression.
• Use ideal-based star operations to model rational power series over N, where a total star operation cannot exist.
• Establish closure properties of the ideal I under the semiring operations and the star operation.
• Leverage the theory of rational series and formal power series to validate the generalization in concrete computational settings.

Experimental results

Research questions

• RQ1 Can the theory of Conway semirings be extended to semirings where a total star operation is impossible, such as N or N^{rat}LΣG?
• RQ2 What algebraic conditions must be satisfied by a partial star operation on an ideal to preserve the key identities of Conway semirings?
• RQ3 Does a Kleene-type theorem hold in the context of partial Conway semirings, ensuring equivalence between rational expressions and recognizable series?
• RQ4 How can the partial star operation be systematically defined and characterized in computationally relevant semirings?

Key findings

• The paper successfully generalizes Conway semirings to partial Conway semirings by restricting the star operation to an ideal, enabling application to semirings like N and rational power series.
• The sum star and product star identities are preserved for elements in the ideal, ensuring consistency with the original Conway framework.
• A Kleene-type theorem is established for partial Conway semirings, proving that every recognizable series is expressible as a rational expression.
• The framework provides a mathematically sound foundation for rational expressions in semirings where a total star operation cannot be defined.
• The ideal-based star operation allows modeling of rational power series over N, resolving a key limitation of classical Conway semirings.
• The theory is robust enough to support closure and structural properties necessary for formal language and automata theory applications.