Theory of Computation Lecture 7: Grammars

Questions and Answers

What does the production S → 0S1 represent in the context of the first grammar?

It generates a string with an equal number of 0s and 1s. (correct)

It generates a string with no 0s.

It generates any string of 1s.

It generates a string with more 0s than 1s.

Which of the following sets of strings does grammar G1 generate?

{0^m 1^n | m = 0, n = 0}

{0^m 1^n | m, n are any integers}

{0^m 1^n | m and n are nonnegative integers} (correct)

{0^m 1^n | m = n}

What is the purpose of the production S → λ in the grammar?

To remove all 1s from the string.

To ensure that there are at least one 0 and one 1.

To generate strings only containing 1s.

To allow for the creation of the empty string. (correct)

Which grammar generates strings consisting of any number of 0s followed by 1s that may be different in quantity?

Both Grammar G1 and G2 Signup and view all the answers

What is the effect of using the first production in grammar G2, S → 0S?

It adds 0s to the beginning of the string. Signup and view all the answers

The grammar G is defined as (V, T, S, P) where V = {0, 1, ______}

S Signup and view all the answers

In grammar G1, the production S → ______S adds 1s to the end of the string.

S Signup and view all the answers

The productions in grammar G2 include S → 1A, S → 1, and S → ______.

λ Signup and view all the answers

Example 5 describes a set of strings made up of 0s followed by ______, where the number of 0s and 1s are the same.

1s Signup and view all the answers

Grammar G1 generates strings where m and n are ______ integers.

nonnegative Signup and view all the answers

Study Notes

Grammars

Grammars generate words of a language and determine word validity within that language.
Formal languages, defined by grammars, model both natural languages (like English) and programming languages (like C, Java).
Key role in the construction and theory of compilers.
Noam Chomsky introduced grammars in the 1950s.

Syntax and Semantics

Syntax refers to the structure of a sentence, regardless of meaning (semantics).
Example of valid syntax: "the frog writes neatly."
Invalid syntax: "swims quickly mathematics."

Formal vs. Natural Languages

Natural language syntax is complicated and not fully specified.
Formal languages have a well-defined syntax and rules.

Problems Addressed by Grammars

How to determine if a word is a valid sentence in a formal language.
How to generate valid sentences in a formal language.

Example Grammar for English Sentences

A sentence consists of a noun phrase followed by a verb phrase.
Noun phrase options:
- Article + Adjective + Noun
- Article + Noun
Verb phrase options include:
- Verb + Adverb
- Verb
Terminal symbols include articles (the, a), adjectives (large, hungry), nouns (rabbit, mathematician), verbs (eats, hops), and adverbs (quickly, wildly).

Phrase-Structure Grammars

Vocabulary (alphabet) is a finite, nonempty set of symbols.
Words are strings made from vocabulary elements.
The empty string (λ) contains no symbols, distinct from the empty set (∅).

Grammar Definition

A phrase-structure grammar G is expressed as (V, T, S, P):
- V: Vocabulary
- T: Subset of terminal symbols from V
- S: Start symbol from V
- P: Finite set of productions
Nonterminal symbols are elements of V not in T.

Derivations

A string is directly derivable if it can be produced from another string using a production rule.
Derivable strings can form a sequence of derivations called a derivation.

Language Generated by a Grammar

The language L(G) generated by grammar G is the set of all terminal strings derivable from the starting symbol S.

Examples

Example 1 demonstrates how a specific grammar can generate valid combinations based on defined productions.
Example 2 illustrates deriving strings by applying production rules in sequence.

Constructing a Grammar

Example 5 constructs a grammar for strings {0^n1^n | n = 0, 1, 2...}, using productions to ensure an equal number of 0s and 1s.
Example 6 involves grammars generating sets where the number of 0s and 1s can differ, showcasing multiple grammars can generate the same language.