Phrase Structure Grammars and the Chomsky Hierarchy

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Phrase Structure Grammarsand theChomsky
Hierarchy

• Dr. David A. Workman
• School of EE and CS
• May 22, 2006

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Phrase Structure Grammar

• Definition. A phrase structure grammar (psg) is
  a 4-tuple, G (N,?, P, S),whereN nonterminal
  alphabet? terminal alphabet
  // N ? ? ?VG N ? ? (vocabulary of G)
  // use "V" when G is understoodS ? N, unique
  start symbolP finite set of productions
  (possibly empty)P u ? v u ? VNV and v
  ? V

leftpart
rightpart

• A psg really is a "program" describing an
  algorithm for generating a language over
  theterminal alphabet, ? that is, L(G) ? ?.
  To formalize this statement, we define a
  relation,G?, from V to V defined by G.
  Specifically,
• ? G? ? if and only if ? ?1 u ?2 , ? ?1 v
  ?2 , where u ? v ? G we say that " alpha
  directly rewrites (produces) beta". The strings
  ? and ? are called sentential forms of G.
• G defines a computation (is a "program") in the
  following way(1) initial state S (start
  symbol)(2) if ? is the "current state" of the
  computation, then any ? is a valid "next state of
  the computation" whenever ? G? ?(3) A
  computation ends whenever there does not exist a
  valid next state for a given current state.

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Language defined by G
Definition. Let G (N,?, P, S) be a psg. Then
the language generated by Gis the set L(G)
x ? ? S G? x that is, x ? L(G) if and
only if thereis a computation in G satisfying
S ?0 G? ?1 G? ?n x, for some n?1.
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The Chomsky Hierarchy of Languages
Technical NoteType-1 languages
explicitlyexclude the null string.All other
languagefamilies permit the null string as a
member. (The Venn diagram is in slight error
because of this technicality.)