Greibach Normal Form in Algebraically Complete Semirings

Infinitary Axiomatization of the Equational Theory of Context-Free Languages

“…These properties also show that µ-continuous Chomsky algebras are algebraically complete semirings, and therefore also algebraic Conway semirings in the sense of [6,7]. The right-hand inequality can be proved by a symmetric argument.…”

“…In this section we show that the axiomatizations considered in [3,6,7] are valid in all µ-continuous Chomsky algebras.…”

“…These axioms can be viewed as imposing continuity properties of the semiring operators with respect to µ. All standard interpretations, including the context-free languages over an alphabet X, are continuous and satisfy the KAG axioms.Ésik and Leiß [6,7] show that conversion to Greibach normal form can be performed purely algebraically under these assumptions.…”

“…Various other axioms of [3,6,7] follow from the Park axioms. The µ-continuity condition (8) implies the Park axioms (5), but we must defer the proof of this fact until §3.…”

“…The characterization of the context-free languages as least solutions of algebraic inequalities involving µ goes back to a 1971 paper of Gruska [4]. More recently, several researchers have given equational axioms for semirings with µ and have developed fragments of the equational theory of context-free languages [3,5,6,7,8,9].…”