Complexity of grammars by group theoretic methods

John Rhodes*, Eliahu Shamir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Let G be a context free (phrase) structure grammar generating the context free language L. The set P = P G of all "generation histories" of words in L can be coded as words in some augmented alphabet. It is proved here that P = R∩G where R is a regular (finite automaton definable) set and G is a "free group kernel" or Dyck set, a result first proved by Chomsky and Schützenberger [3]. We can construct the Lower central series of the free group kernel G1∼G2∼ ... ∼Gn∼ ..., so ∩Gn= G. Let Pn= R∩Gn, so ∩Pn=P.Pn is the n-th order approximation of P.Pn need not be a context free language but it can be computed by n cascade or sequential banks of counters (integers). We give two equivalent characterizations of Pn, one "grammatical" and one "statistical", which follow from the theorems of Magnus, Witt, M. Hall, etc. for free groups. The main new theoretical tool used here for the study of grammars is the Magnus transform on the free group, a→1+a, a-1→1-a+a2- a3+a4..., which acts like a non-commutative Fourier transform.

Original languageEnglish
Pages (from-to)222-239
Number of pages18
JournalJournal of Combinatorial Theory
Volume4
Issue number3
DOIs
StatePublished - Apr 1968
Externally publishedYes

Fingerprint

Dive into the research topics of 'Complexity of grammars by group theoretic methods'. Together they form a unique fingerprint.

Cite this