Theory of Computation Lecture 6

Questions and Answers

What regular expression represents the language over ∑ = {0} having even length of the string?

(00)* (correct)

(000)*

(0)*

(0+)*

What is the description of the language denoted by the regular expression R = (b* (aaa)* b)?

Strings with exactly one 'aaa' surrounded by b's

Strings with zero or more 'aaa' and any number of b's (correct)

Only the string 'aaa' with no b's

Strings containing exactly three 'aaa's and no b's

Which regular expression restricts the language to strings that do not contain the substring '01'?

(00)*

(10)*

(1* 0*) (correct)

(0+ 1+)*

What regular expression represents a language where every 0 is followed immediately by two 1's?

(011 + 1)* Signup and view all the answers

What is the regular expression for strings having at least two occurrences of '1's between any two occurrences of '0's?

(01110) Signup and view all the answers

What does the grammar G = (V, T, S, P) for Example 5 generate?

Strings with an equal number of 0s and 1s Signup and view all the answers

In Example 6, what do the variables V in grammar G1 consist of?

{S, 0, 1} Signup and view all the answers

Which of the following productions is NOT part of grammar G2 in Example 6?

S → 0S Signup and view all the answers

What does the production S → λ in both grammars indicate?

It signifies the end of a string Signup and view all the answers

How does grammar G1 ensure that the generated strings consist of both 0s and 1s?

By using one production to add 0s and another to add 1s Signup and view all the answers

Study Notes

Regular Expressions

Regular expressions describe languages accepted by finite automata, defining patterns for matching strings.
Languages corresponding to regular expressions are termed Regular Languages.
Regular expressions include symbols representing different occurrences:
- x* indicates zero or more occurrences of x
- x+ indicates one or more occurrences of x

Operations on Regular Languages

Union (L U M): Combines two regular languages, allowing strings to be included if present in either language.
Intersection (L ∩ M): Identifies common strings present in both regular languages.
Kleene Closure (L)*: Represents zero or more occurrences of a language, creating new regular languages from existing ones.

Examples of Regular Expressions

To represent all combinations of a's: R = a*
To represent combinations of a's excluding the null string: L = {a, aa, aaa, ...}
To describe strings with any number of a's and b's: R = (a + b)*

Specific Regular Expression Examples

For strings starting with 1 and ending with 0: R = 1(0 + 1)*0
For strings starting and ending with a, with any combination of b's in between: R = ab* a
For strings starting with a and avoiding consecutive b's: R = (a + ab)*
For strings with a's followed by b's and then c's: R = a* b* c*
For strings over {0} with even lengths: R = (00)*
For the language denoting R = (b* (aaa)* b*)*: strings where a's appear in triples with no restriction on b's.
For strings avoiding the substring 01: R = (1* 0*)
For strings with at least two 1's between 0's: R = (1 + (0111*0))*
For strings where every 0 is followed by 11: R = (011 + 1)*

Converting Regular Expressions to NFA

Conversion of a regular expression, e.g., (ab U a)*, into a Non-deterministic Finite Automaton (NFA) can be performed through a series of stages.

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.

Description

This quiz focuses on Regular Expressions, a crucial concept in the Theory of Computation. It explores how finite automata accept languages described by these expressions. Dive into the world of computation and enhance your understanding of automata theory.