The Chomsky Hierarchy

1
The Chomsky Hierarchy

• Richard Kelley

2
Overview

• Mini Biography Chomsky
• Formal Grammars
• Formal Languages
• The Chomsky Hierarchy
• How does it relate to CS?

3
Biography Noam Chomsky

• Avram Noam Chomsky was born 17 December 1928
• The son of a Hebrew scholar he was exposed to
  many different languages at an early age
• Yiddish
• English
• Made to study Hebrew at least once a week
• Although claims that The only language I speak
  and write proficiently English

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Biography (cont)

• Not a computer scientist at all
• Hes a linguist
• Institute Professor of linguistics at MIT
• Main work is in generative linguistics
• Defn. The study of linguistic syntax using formal
  languages that can generate the well-formed
  expressions of a natural language

5
Biography (cont)

• 1945 he began studying philosophy and linguistics
  at the University of Pennsylvania
• conducted much of his doctoral research at
  Harvard, but received his Ph.D from U of P
• At this time he started to develop is linguistic
  ideas

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Biography (cont)

• Book Syntactic Structures
• The most well known work in the field of
  linguistics
• "Colorless green ideas sleep furiously
• grammar is correct, but the sentence is complete
  nonsense
• idea that syntax is separate from grammar

7
Biography (cont)

• Hes not a computer scientist, so why do we care?
• ideas became the basis for machine translation
• Came up with the idea of generative grammars
• Study of automata

8
Formal Grammars

• Formal Grammar is an abstract structure that
  describes a formal language
• in other words it's a set of rules that defines a
  set of strings over an alphabet
• In CS
• Grammars generate
• Languages describe

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Formal Grammars (cont)

• Generative grammars are the ones were familiar
  with
• How about an example?

10
Formal Grammars (cont)
1. S - aSb
2. S - ba
So we can generate strings like this
(1) aSb
(1)- aaSbb
(1)- aaaSbbb
(2)- aaababbb
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Formal Grammars (cont)

• Formally, Chomsky defined generative grammars as
• A finite set N of nonterminal symbols
• A finite set S of terminal symbols
• A finite set of production rules like the ones on
  the previous slide, P
• A start symbol, S

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Formal Grammars (cont)

• G N, S, P, S
• Where, P

• S - aBSc
• S - abc
• Ba - aB
• Bb - bb

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Formal Grammars (cont)

• S - aBSc
• S - abc
• Ba - aB
• Bb - bb

S - aBSc
- aBabcc
- aaBbcc
- aabbcc
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Formal Languages

• Formal languages are used to describe the strings
  that the grammars generate
• Regular Languages
• A regular language is an language that can be
  expressed using a regular expressions
• In other words, a finite state machine accepts it
• EX. L a is regular

15
Formal Languages (cont)

• Context Free Languages
• context free languages are generated by context
  free grammars
• In context free grammars every production (P)
  rule is of the form
• V - w
• V nontermimal W string
• The term context free comes from the fact that V
  can always be replaced by w regardless context
• Contrast Context-sensitive languages

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Formal Languages (cont)
Context free Grammar S - aSb e generates
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Formal Languages

• Context-Sensitive Languages
• In these languages all the rules in P are in the
  form
• aAb - ayb
• A can only be replaced based on what context
  their in because of a b
• Where a, b, and y are strings of terminals and
  non-terminals

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Formal Languages (cont)
Context-Sensitive Grammar S - abc aSBc cB -
Bc bB - bb generates
19
Turing Machines
Basically just Finite State Machines
TM
?
?
x y
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Formal Languages

• Turing-recognizable
• or recursively enumerable
• A language is TR if some TM recognizes it
• Meaning the problem can be represented on a TM
• Turing-Decidable (decidable)
• or recursive
• A language is decidable if some TM halts on every
  string in the language

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Chomsky Hierarchy

• Type 0 Recursively-enumerable
• No restrictions
• Type 1 Context-Sensitive
• aCb - acB
• Type 2 Context Free
• A -b
• Type 3 Regular

22
Chomsky Hierarchy

• Every regular language is context-free
• Every context free language is context sensitive
• Every context-sensitive language is recursively
  enumerable

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Chomsky Hierarchy
?(?)
Turing-Recognizable
Type 0
Decidable
Type 1
Context-Free
Regular
Type 3
Type 2
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Direct Application to CS

• Ability to reduce problems
• A way of converting one problem into another
  problem in such a way that the second problem is
  used to solve the first
• NP-C problems can be reduced to the halting
  problem
• M is a TM and M accepts w
• Keeps you from wasting your time!

25
Summary

• Mini Biography Chomsky
• Formal Grammars
• Formal Languages
• The Chomsky Hierarchy
• How does it relate to CS?

26
References

• Sipser, Micheal. Introduction to the theory of
  computation. Boston PWS Publishing Company, 1997
• Context-Sensitive Grammars http//en.wikipedia.o
  rg/wiki/Context-sensitive_grammar March 2005
• Altenkirch, Thorsten Turing machines and the
  rest http//www.cs.nott.ac.uk/txa/g51mal.2001/no
  tes/node41.html March 2005

27
References

• Context-Free Grammars http//en.wikipedia.org/wi
  ki/Context-free_grammar March 2005
• Noam Chomsky http//en.wikipedia.org/wiki/Chomsk
  y March 2005