Context-Free Grammars

Context-Free Grammars

Basic terminologies • Sentences – syntactically correct string • Terminal symbols – The elements of the alphabet (Jill, drives, frequently,......) • Variables or non-terminals • Rules • <sentences> -> <noun-phrase><verb-phrase> • <sentences> -> <noun-phrase><verb> <direct- object-phrase>

Rules, variables & terminals • <noun-phrase> -> <proper-noun> • -> <determiner><common-noun> • <proper-noun> -> John • -> Jill • <comon-noun> -> car • -> burger • <determiner> -> a • -> the • <verb-phrase> -> <verb><adverb> • -> <verb> • <verb> -> drives • -> eats • <adverb> -> slowly • -> frequently

derivations

More rules • <adjective-list> -> <adjective><adjective-list> • -> λ • <adjective> -> big • -> fresh • -> brown • <direct-object-phrase> -> <adjective-list><proper-noun> • -> <determiner><adjective-list><common-noun>

More derivations Example: John eats a big fresh burger

definition • Context-Free Grammar A context-free grammar is a quadruple {V, Σ, P, S} where V is a finite set of variables Σ {the alphabet} is a finite set of terminal symbols P is a finite set of rules And S is a distinguished element of V called start symbol. NB: The sets V and Σ are assumed to be disjoint

Conventions • Terminals are represented by lowercase occurring at the beginning of the alphabet, that is a, b, c, ……. • Variables will be denoted by capital letters. • Variables are referred as non-terminals • A rules is often called a production.

Terminologies • Context-free Languages • Directly recursive • A=> uAv • Indirectly recursive • A=> uB • B=>Av • Leftmost derivation • Rightmost derivation

A CFG Sample Derivation (leftmost and rightmost): a * (b00) • leftmost • E -> E * E • E -> I * E • E -> a * E • E -> a * (E) • E -> a * (E + E) • E -> a * (I+ E) • E -> a * (a + E) • E -> a * (a+ I) • E -> a * (a+ I0) • E -> a * (a+ I00) • E -> a * (a+ b00) • Reftmost ??? A CFG for simple expression: • E -> I • E -> E + E • E -> E * E • E -> ( E ) • I -> a • I -> b • I -> Ia • I -> Ib • I -> I0 • I -> I1

Sample Derivations • A CFG • G = {V, Σ, P, S} • V = {S, A} • Σ = { a, b} • P: S-> AA A -> AAA| bA| Ab |a

Sample Derivations Sentences/String = ababaa (Draw the Parse Tree)

Parse Tree • Draw the parse/derivation tree for : a * (b00)

Examples of Grammars and languages • S->aSa|aBa • B-> bB|b L(G) = {anbman| n>o, m>o} • Equal number a’s leading and trailing • Any number of b’s • At least one b.

Examples of Grammars and languages • S-> aSdd|A • A->bAc|bc L(G)={anbmcmd2n|n>=0, m>0} • No. of d’s are twice as many as a’s • Equal no. of b’s and c’s

Examples of Grammars and languages • Palindrome i.e. w=wR • Σ = {a,b} S->a|b|λ S->aSa|bSb

Examples of Grammars and languages • At least one b for every a and at most two. S-> aSb|aSbb| λ • Equal number of ab’s (leading) and c’s (trailing). • Any number of b+ outside S-> abScB| λ B-> bB|b

Examples of Grammars and languages • Grammar for a+b* • S->AB • A->aA|a • B->bB| λ • Grammar for a*ba*ba* • S->AbAbA • A->aA| λ

Examples of Grammars and languages • Even length string over {a,b} • S->aO|bO| λ • O->aS|bS • Even length string over {a} • S->aO| λ • O->aS

Examples of Grammars and languages • Odd length string over {a} • S->aSa| O • O->a

Context-Free Grammars