Understanding Right-Linear Grammars and Their Relation to Regular Languages, Slides of Theory of Automata

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Chapter Ten:

Grammars

Grammar is another of those common words for which

the study of formal language introduces a precise

technical definition. For us, a grammar is a certain

kind of collection of rules for building strings. Like

DFAs, NFAs, and regular expressions, grammars are

mechanisms for defining languages rigorously.

A simple restriction on the form of these grammars

yields the special class of right-linear grammars. The

languages that can be defined by right-linear grammars

are exactly the regular languages. There it is again!

A Little English

• An article can be the word a or the :

A → a A → the

• A noun can be the word dog , cat or rat : N → dog N → cat N → rat A noun phrase is an article followed by a noun: P → AN

A Little English

• An verb can be the word loves, hates or eats :

V → loves V → hates V → eats

A sentence can be a noun phrase, followed by a verb, followed by another noun phrase:

S → PVP

• Start from S and follow the productions of G 1
• This can derive a variety of (unpunctuated) English sentences:

S ⇒ PVP ⇒ ANVP ⇒ theNVP ⇒ thecatVP ⇒ thecateatsP ⇒ thecateatsAN ⇒ thecateatsaN ⇒ thecateatsarat S ⇒ PVP ⇒ ANVP ⇒ aNVP ⇒ adogVP ⇒ adoglovesP ⇒ adoglovesAN ⇒ adoglovestheN ⇒ adoglovesthecat S ⇒ PVP ⇒ ANVP ⇒ theNVP ⇒ thecatVP ⇒ thecathatesP ⇒ thecathatesAN ⇒ thecathatestheN ⇒ thecathatesthedog

S → PVP A → a P → AN A → the V → loves N → dog V → hates N → cat V → eats N → rat

• Often there is more than one place in a string where a production could be applied
• For example, PlovesP :
  • PlovesP ⇒ ANlovesP
  • PlovesP ⇒ PlovesAN
• The derivations on the previous slide chose the leftmost substitution at every step, but that is not a requirement
• The language defined by a grammar is the set of lowercase strings that have at least one derivation from the start symbol S

S → PVP A → a P → AN A → the V → loves N → dog V → hates N → cat V → eats N → rat

Informal Definition

• Productions define permissible string substitutions
• When a sequence of permissible substitutions starting from S ends in a string that is all lowercase, we say the grammar generates that string
• L ( G ) is the set of all strings generated by grammar G

A grammar is a set of productions of the form x → y. The strings x and y can contain both lowercase and uppercase letters; x cannot be empty, but y can be ε. One uppercase letter is designated as the start symbol (conventionally, it is the letter S ).

• That final production for X says that X may be replaced by the empty string, so that for example abbX ⇒ abb
• Written in the more compact way, this grammar is:

S → aS | X

X → bX | ε

S → aS S → X X → bX

X → ε

• For this grammar, all derivations of lowercase

strings follow this simple pattern:

• First use S → aS zero or more times
• Then use S → X once
• Then use X → bX zero or more times
• Then use X → ε once
• So the generated string always consists of

zero or more a s followed by zero or more b s

• L ( G ) = L ( ab )

S → aS | X

X → bX | ε

Untapped Power

• All our examples have used productions with a single uppercase letter on the left-hand side
• Grammars can have any non-empty string on the left-hand side
• The mechanism of substitution is the same
  • Sb → bS says that bS can be substituted for Sb
• Such productions can be very powerful, but we won't need that power yet
• We'll concentrate on grammars with one uppercase letter on the left-hand side of every production

Formalizing Grammars

• Our informal definition relied on the difference between lowercase and uppercase
• The formal definition will use two separate alphabets:
  • The terminal symbols ( typically lowercase)
  • The nonterminal symbols (typically uppercase)
• So a formal grammar has four parts…

4-Tuple Definition

• A grammar G is a 4-tuple G = ( V , Σ, S , P ), where:
  • V is an alphabet, the nonterminal alphabet
  • Σ is another alphabet, the terminal alphabet , disjoint from V
  • S ∈ V is the start symbol
  • P is a finite set of productions, each of the form x → y , where x and y are strings over Σ ∪ V and x ≠ ε

Outline

• 10.1 A Grammar Example for English
• 10.2 The 4-Tuple
• 10.3 The Language Generated by a

Grammar

• 10.4 Every Regular Language Has a

Grammar

• 10.5 Right-Linear Grammars
• 10.6 Every Right-linear Grammar Generates

a Regular Language

The Program

• For DFAs, we derived a zero-or-more-step δ*

function from the one-step δ

• For NFAs, we derived a one-step relation on

IDs, then extended it to a zero-or-more-step

relation

• We'll do the same kind of thing for

grammars…