The Chomsky Hierarchy: Four Computers

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The Chomsky Hierarchy Four Computers

• Aimee Blackburn
• Chapter 7

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Outline of my Glorious Presentation

• (Brief!) History of Chomskys linguistic theories
• Overview of the Hierarchy itself
• How does this information pertain to the realm of
  computer science?

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Who is Noam Chomsky Anyway?

• Philosopher of Languages
• Professor of Linguistics at MIT
• Constructed the idea that language was not a
  learned behavior, but that it was cognitive and
  innate versus stimulus-response driven
• In an effort to explain these theories, he
  developed the Chomsky Hierarchy

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Chomsky Hierarchy

• Comprises four types of languages and their
  associated grammars and machines.
• Type 3 Regular Languages
• Type 2 Context-Free Languages
• Type 1 Context-Sensitive Languages
• Type 0 Recursively Enumerable Languages
• These languages form a strict hierarchy

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Chomsky Hierarchy
Language Grammar Machine Example
Regular Language Regular Grammar Right-linear grammar Left-linear grammar Deterministic or Nondeterministic Finite-state acceptor a
Context-free Language Context-free grammar Nondeterministic Pushdown automaton anbn
Context-sensitive Context-sensitive grammar Linear-bounded automaton anbncn
Recursively enumerable Unrestricted grammar Turing machine Any computable function
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Regular Languages

• A language is regular if and only if it is the
  accepted language of some DFA / NFA
• Construct an DFA / NFA for the language described
  by the regular expression (aba)
  

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Context-Free Languages

• Contains a finite alphabet, S
• Contains a finite set of non-terminals, N
• S is an element of N and is the Start symbol
• R Rules Grammar
  X? sY S
  a,b X? s
  N S
  X? l R S ? aSb,
  S ? l
• Uses a stack to hold infinite memory

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Context-Sensitive Languages

• The number of symbols on the LHS must not exceed
  the number of symbols on the RHS
• A ? l is not allowed unless A is the start symbol
  and does not occur on the RHS of any rule
• Since we allow more than one symbol on the LHS,
  we refer to those symbols other than the one we
  are replacing as the context of the replacement.
• Linear-bounded automaton a Turing Machine with a
  finite amount of tape
• The syntax of some natural languages (including
  Dutch, and Swiss German) is held to have
  structures of this type

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Recursively Enumerable

• Have no restrictions on their grammar rules
  (except, of course, that there must be at least
  one non-terminal on the LHS).
• Turing Machine is a finite-state machine in which
  a transition prints a symbol on a tape. The tape
  head may move in either direction, allowing the
  machine to read and manipulate the input as many
  times as desired.
• Predated and provided a model for the design and
  development of the stored-program computer.

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So, why should I care?

• Colorless green ideas sleep furiously
• Concepts of syntax and semantics used widely in
  computer science
• Basic compiler functions
• Development of computer languages
• Exploring the capabilities and limitations of
  algorithmic problem solving

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The Chomskybot

• Look On My Words Ye Mighty, And Despair!
• I suggested that these results would follow from
  the assumption that a descriptively adequate
  grammar does not affect the structure of the
  strong generative capacity of the theory. If the
  position of the trace in (99c) were only
  relatively inaccessible to movement, the natural
  general principle that will subsume this case
  cannot be arbitrary in the system of base rules
  exclusive of the lexicon. To provide a
  constituent structure for T(Z,K), the
  speaker-hearers linguistic intuition may remedy
  and, at the same time, eliminate an important
  distinction in language use. Notice,
  incidentally, that a case of semigrammaticalness
  of a different sort suffices to account for
  problems of phonemic and morphological analysis.
  It must be emphasized, once again, that this
  analysis of a formative as a pair of sets of
  features is not to be considered in determining
  the requirement that branching is not tolerated
  within the dominance scope of a complex symbol.

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Thank You!

• So, you want homework questions, eh?
• (1.) Construct a parse tree for the grammar S
  ? aA
• A ? bC
• C ? aS a
• (2.) Construct a NFA for (ab)b