CS 331 Spring 2013: Lecture Notes for Wednesday, January 23, 2013

Languages & Grammars (cont’d)

The Chomsky Hierarchy

In the late 1950s, linguist Noam Chomsky published a hierarchy of types of languages, defined in terms of the kinds of grammars that could describe them. Chomsky was trying to develop a framework for studying natural languages (e.g., English); however, his hierarchy has proved to be useful in the theoretical study of computing.

Below is the hierarchy. In the Grammar column, upper-case letters represent nonterminal symbols, while lower-case letters represent terminal symbols.

Language Category	Generator	Recognizer	Why We Care
Chomsky’s Number	Name
Type 3	Regular	Grammar in which each production is in one of the following forms: \(A \rightarrow \varepsilon\) \(A \rightarrow b\) \(A \rightarrow bC\) Another kind of generator: regular expressions.	Finite Automaton Think: Program that uses a small, fixed amount of memory	In this category lie things like the set of all legal identifiers in some programming language. Thus, lexical analysis: breaking a program into words. These are also the languages that can be described by regular expressions (used, for example, in text search & replace).
Type 2	Context-Free	Grammar in which the left-hand side of each production consists of a single nonterminal. \(A \rightarrow \textrm{[any]}\)	Nondeterministic Push-Down Automaton Think: Finite Automaton + Stack (roughly)	For most programming languages, the set of all syntactically correct programs, lies in this category. Thus, parsing: determining whether a program is syntactically correct and, if so, how it is structured.
Type 1	Context-Sensitive	Grammar in which each production is in the following form: \(\textrm{[str1]}A\textrm{[str2]} \rightarrow \textrm{[str1]}\textrm{[any]}\textrm{[str2]}\) where \(\textrm{[str1]}\), \(\textrm{[str2]}\) are fixed strings.	Don’t worry about it (“Linear Bounded Automaton”, if you must know)	Actually, we don’t care much.
Type 0	Recursively Enumerable	Arbitrary grammar. \(\textrm{[any]} \rightarrow \textrm{[any]}\)	Turing Machine Think: Computer program	For a language in this category, the task of analyzing a given string and saying “yes” precisely when it lies in the language, neatly encompasses those things that can be done using a computer program.

Language Category

Generator

Recognizer

Why We Care

Chomsky’s Number

Name

Type 3

Regular

Grammar in which each production is in one of the following forms:

\(A \rightarrow \varepsilon\)
\(A \rightarrow b\)
\(A \rightarrow bC\)

Another kind of generator: regular expressions.

Finite Automaton

Think: Program that uses a small, fixed amount of memory

In this category lie things like the set of all legal identifiers in some programming language. Thus, lexical analysis: breaking a program into words.

These are also the languages that can be described by regular expressions (used, for example, in text search & replace).

Type 2

Context-Free

Grammar in which the left-hand side of each production consists of a single nonterminal.

\(A \rightarrow \textrm{[any]}\)

Nondeterministic Push-Down Automaton

Think: Finite Automaton + Stack (roughly)

For most programming languages, the set of all syntactically correct programs, lies in this category. Thus, parsing: determining whether a program is syntactically correct and, if so, how it is structured.

Type 1

Context-Sensitive

Grammar in which each production is in the following form:

\(\textrm{[str1]}A\textrm{[str2]} \rightarrow \textrm{[str1]}\textrm{[any]}\textrm{[str2]}\)

where \(\textrm{[str1]}\), \(\textrm{[str2]}\) are fixed strings.

Don’t worry about it

(“Linear Bounded Automaton”, if you must know)

Actually, we don’t care much.

Type 0

Recursively Enumerable

Arbitrary grammar.

\(\textrm{[any]} \rightarrow \textrm{[any]}\)

Turing Machine

Think: Computer program

For a language in this category, the task of analyzing a given string and saying “yes” precisely when it lies in the language, neatly encompasses those things that can be done using a computer program.

Notes

Each of the above categories is contained in the next. So every regular language is context-free, etc.

In this class we are primarily interested in regular languages and context-free languages. See the “Why We Care” column for reasons.

There are languages that are not recursively enumerable, that is, that cannot be described by a grammar, and that we cannot write a computer program to recognize. One example of such a language is the set of all programs (in your favorite programming language) whose execution never stops.

For more on the Chomsky Hierarchy, see CS 451.

CS 331 Spring 2013: Lecture Notes for Wednesday, January 23, 2013 / Updated: 23 Jan 2013 / Glenn G. Chappell / ggchappell@alaska.edu

CS 331 Spring 2013 Lecture Notes for Wednesday, January 23, 2013

Languages & Grammars (cont’d)

The Chomsky Hierarchy

CS 331 Spring 2013
Lecture Notes for Wednesday, January 23, 2013