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The Chomsky Hierarchy

- Richard Kelley

Overview

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

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

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

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

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

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

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

Formal Grammars (cont)

- Generative grammars are the ones were familiar

with - How about an example?

Formal Grammars (cont)

1. S - aSb

2. S - ba

So we can generate strings like this

(1) aSb

(1)- aaSbb

(1)- aaaSbbb

(2)- aaababbb

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

Formal Grammars (cont)

- G N, S, P, S
- Where, P

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

Formal Grammars (cont)

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

S - aBSc

- aBabcc

- aaBbcc

- aabbcc

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

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

Formal Languages (cont)

Context free Grammar S - aSb e generates

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

Formal Languages (cont)

Context-Sensitive Grammar S - abc aSBc cB -

Bc bB - bb generates

Turing Machines

Basically just Finite State Machines

TM

?

?

x y

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

Chomsky Hierarchy

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

Chomsky Hierarchy

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

enumerable

Chomsky Hierarchy

?(?)

Turing-Recognizable

Type 0

Decidable

Type 1

Context-Free

Regular

Type 3

Type 2

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!

Summary

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

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

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

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