The characterization of nonexpansive grammars by rational power series

Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata

“…Using elimination theory and applying methods and results from algebra and algebraic geometry, Kuich, Salomaa [92J gave a procedure for the construction of the polynomial p. Results analogous to Theorem 10.5 and Corollary 10.6 are also valid for grammars of various kinds, e. g., tuple grammars (Kuich, Maurer [89,90]), phrase structure grammars and state grammars with context-free control languages (Kuich, Shyamasundar [93]), and context-free grammars with regular parallel control languages (Kuich, Prodinger, Urbanek [91]). Corollary 10.6 was used by Baron, Kuich [2], and, in a very refined manner, by Flajolet [31, 32J to The next result is an application of Corollary 10.6 or Corollary 1O.S and concerns infinite words in EW. Given w E EW, one defines its pf'efix language and its cop refix language by Pref( w) = {v I v is a finite prefix of w} and Copref(w) = E* -Pref(w), respectively.…”

“…For MI E Alt xI 2 and M2 E A!,x!o we define the product MlM2 E Altx!o by Furthermore, we introduce the matrix of unity E E AIXI. The diagonal entries Ei,i of E are equal to 1, the off-diagonal entries Ei1 ,i 2 , i l ¥-i2, of E are equal to 0, i, iI, i2 E I.…”

On the Commutative Equivalence of Algebraic Formal Series and Languages

“…Rational series are well-known and their structure has been thoroughly investigated. A result of Baron and Kuich [2] provides the characterization of the context-free grammars G such that, for every non-terminal X, the series G X is rational. This characterization is based upon the notion of non-expansive grammar.…”

“…Remark 2 Theorem 1 does not imply that an unambiguous context-free language whose characteristic series in commutative variables is rational, is generated by a non-expansive grammar. In fact, in [2] the following two conjectures were formulated:…”

“…A remarkable result by Baron and Kuich [2] provides a characterization of non-expansive grammars. In particular, an unambiguous grammar is nonexpansive if and only if all non-terminals generate languages whose characteristic series are rational.…”