Languages Grammars

8/11/2019 Languages Grammars

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Languages and Grammars

MSU CSE 260


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Outline

Introduction: Example

Phrase-Structure Grammars: Terminology, Definition,

Derivation, Language of a Grammar, Examples Exercise 10.1 (1)

Types of Phrase-Structure Grammars

Derivation Trees: Example,Parsing

Exercise 10.1 (2, 3)

Backus-Naur Form


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Introduction

In the English language, the grammar determineswhether a combination of words is a validsentence.

Are the following valid sentences?

The largerabbit hopsquickly. Yes

The frog writes neatly. Yes

Swims quickly mathematician. No Grammars are concerned with the syntax (form) of

a sentence, and NOT its semantics (or meaning.)


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English Grammar

Sentence: noun phrasefollowed by verb phrase;

Noun phrase: articleadjectivenoun,or article

noun; Verb phrase: verbadverb, or verb;

Article: a, or the;

Adjective: large, or hungry;

Noun: rabbit, or mathematician, orfrog;

Verb: eats, or hops, orwrites, orswims;

Adverb: quickly, or wildly, orneatly;


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Example

Sentence

Noun phrase verb phrase

Article adjective noun verb phrase Article adjective noun verb adverb

the adjective noun verb adverb

the largenoun verb adverb

the largerabbit verb adverb

the largerabbit hopsadverb

the largerabbit hopsquickly


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Grammars and Computation

Grammars are used as a model of

computation.

Grammars are used to:

generate the words of a language, and

determine whether a word is in a language.


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Phrase-Structure Grammars

Terminology Definitions.A vocabulary(or alphabet) Vis a

finite, nonempty set of elements calledsymbols.

A word(orsentence) over Vis a string of finitelength of elements of V.

The empty string(or null string,) denoted by , is

the string containing no symbols.

The set of all words over Vis denoted by V*.

A language over Vis a subset of V*.


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A language can be specified by:

listing all the words in the language, or

giving a set of criteria satisfied by its words, or using a grammar.

A grammar provides:

a set of symbols, and

a set of rules, called productions, for producing words

by replacing strings by other strings: w0w1.


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Phrase-Structure Grammar

DefinitionAphrase-structure grammarG = (V, T, S,P)consists of:

a vocabularyV,a subset Tof Vconsisting of terminalelements,

a start symbolSfrom V, and

a setPofproductions.

The setN = V-Tconsists of nonterminal symbols.

Every production inPmust contain at least onenonterminal on its left side.


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Phrase-structure Grammar

Example G= {V, T, S,P}, where

V = {a, b,A,B, S},

T = {a, b},Sis the start symbol, and

P= { SAba,

ABB,

Bab,

ABb}.


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Derivation Definition.

Let G = (V, T, S,P) be a phrase-structure grammar.

Let w0 = lz0rand w1 = lz1rbe strings over V. Ifz0z1is a production of G, we say that:

w1is directly derivablefrom w0(denoted: w0w1.)

If w0, w1, , wnare strings over Vsuch that:

w0w1, w1w2, ,wn-1wn, we say that:wnis derivablefrom w0(denoted: w0* wn.)

Note. * should be on top of.

The sequence of all steps used to obtain wnfrom w0iscalled a derivation.


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Example

In the previous example grammar, theproduction:Babmakes the stringAaba

directly derivable from stringABa.ABa Aaba

AlsoAaba BBaba Bababa abababa

using: ABB, Bab, andBab.

So:ABa* abababa

abababa is derivable fromABa.


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Language of a Grammar

Definition.

Let G = (V, T, S,P) be a phrase-structure

grammar.The language generatedby G(or thelanguageof G), denoted byL(G), is the set

of all strings of terminals that are derivablefrom the start symbol S.

L(G) = {wT* | S* w}.


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Example

Let G= {V, T, S,P} be the grammar where:

V = {S, 0, 1},

T = {0, 1},

P= { S11S,

S0}.

What isL(G)?

At any stage of the derivation we can either: add two 1s at the end of the string, or

terminate the derivation by adding a 0 at the end of the string.

L(G)={0, 110, 11110, 1111110, } = Set of all stringsthat begin with an even number of 1s and end with 0.


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Exercise 10.1 (1)


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Types of Grammars

A type 0(phrase-structure) grammar has norestrictions on its productions.

A type 1(or context-sensitive) grammar hasproductions only of forms:

w1w2with length of w2length of w1, or

w1.

A type 2(or context-free) grammar hasproductions only of the formAw2, whereAis asingle nonterminal symbol.


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Types of Grammarscont.

A type 3(or regular) grammar has productionsonly of the form:

AaB,orAa, where

AandBare nonterminal symbols, and

ais a terminal symbol, or

S.

Note.

Every type 3 grammar is a type 2 grammar




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Types of Grammars - Summary

Type Restrictions on productions w1w2

0 No restrictions

1 l(w1) l(w2), or w2=

2 w1=AwhereAN

3 w1=A, and w2=aBor w2=a, whereAN,BN, aT, or

w1=Sand w2=


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Derivation Trees

For type 2 (context-free) grammars:

A derivation(orparse)tree, is an ordered rooted

tree that represents a derivation in the languagegenerated by a context-free grammar, where:

the root represents the starting symbol;

the internal vertices represent nonterminal symbols;

the leaves represent the terminal symbols;

for a productionAw, the vertex representingAwill

have children vertices that represent each symbol in w.


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Example

Derivation tree for:

the hungry rabbit eats quickly

sentence

noun phrase verb phrase

article adjective noun verb adverb

the hungry rabbit eats quickly


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Exercise 10.1 (2, 3)


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Parsing

To determine whether a string is in the

language generated by a grammar, use:

Top-down parsing: Begin with S and attempt to derive the word by

successively applying productions, or

Bottom-up parsing:

Work backward: Begin by inspecting the word and

apply productions backward.


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Example

Let G= {V, T, S,P} be the grammar where: V = {a, b, c,A,B, C, S}, T = {a, b, c},

Productions: Determine whether cbabis inL(G)?

SAB Top-down parsing:ACa SAB

B Ba S AB CaB

B Cb SAB CaB cbaB

B b S AB CaB cbaBcbab

C cb Bottom-up parsing:C b Cab cbab

Ab Cabcbab

AB AbCab cbab

S AB AbCabcbab


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Backus-Naur Form

Used with type 2 (context-free) grammars; like forspecification of programming languages:

Use ::= instead of

Enclose nonterminal symbols within < >

Group productions with same left side with symbol |

Example.

::= ::=+ |-

::= |

::=0 |1 |2 |3 |4 |5 |6 |7 |8 |9

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