Formal Grammars Real

REYNALD JAY F. HIDALGO, MS Computer ScienceProfessor

A set of atomic/non-divissible symbols

Eg. A = {a,b,c,d,e,f,g,h,I,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}A = {a,b,c}A = {0, 1}A = {+,-,*,/}

A sequence of symbol/s from a given alphabet

Eg.A = {a,b,c}w = aw = bw = cw = aaw = abcw = ccc

Denotes the empty string

w=

Denotes the length of a given string (an integer value)

w = abcd/w/ = 4w = 0/w/ = 1w = 12/w/ = 2w = /w/ = 0

Fundamental operation on strings

Let be the first string and be the 2nd stringThe concatenation of and is denoted by = .

= .

Where is the prefix of and is a proper prefix if while is the suffix of and is a proper suffix if

Any string concatenated with will result to the same string

denotes k copies of w

Given w = abcw3 = 3 copies 0f w = abcabcabc

Is a subset of the powerset W

S L EL V :=E if B then A else EE AB A < AB A = AA T + AA TT VT 0T 1T (E)V xV yV w

1. Generate the Algol statement w:= if x < y then 0 else x + y + 1

Prove that the following Algol statement is valid

x:= ( 1 + ( 0 + x ) )

Generate the Algol Statement

y := if ( x + y ) = 1 then 0 else ( if x < 0 then x else 0 )

NTP

It is a four-tupleG = (N, T, P, )Where N is a finite set of nonterminal symbolsT is a finite set of terminal symbolsN and T are disjoint: N T = P is a finite set of productions is the sentence symbol; (NT)

Each production P is an ordered pair of strings (,) = = In which ,, are possible empty strings in (NT)* and A is or a nonterminal letter. We usually write production () as

Let G1 have N = {A,B,C}, T =(a,b) and the set of productions A A aABC A abC CB BC bB bb bC bIdentify the Language L generated by G1

Let G2 have N = {A,B,C}, T (a,b,c) and the set of productions AA aABCA abCCB BCbB bbbC bccC ccIdentify the Language L generated by G2

Let G3 have N = {S}, T= (a,b) and the set of productions SS aSbS abdentify the Language L generated by G3

The Language L(G) generated by a formal grammar G is the set of terminal strings derivable from

Given a G an L can be derived

1. Let G4 have N = {A,B}, T= (1,0) and the set of productions 1B 1 A 1B B 0A A 1 Identify the Language L generated by G4

2. Let G5 have N = {A,B}, T= (1,0) and the set of productions B1 1 A B1 B A0 A 1 Identify the Language L generated by G5

Occurs when 2 or more derivation trees can be drawn from a single w

StructuralLabeling

Let G6 have N = {A}, T (0,1) and the set of productions AA A0AA 1Draw a derivation tree for the string 10101

Let G7 have N = {A,B}, T (1,0) and the set of productions A A B0 A A0 B B0 A 1 B 1Draw a derivation tree for the string 100

S S ScS S c 2. A B A A a A b A c B + B * Prove that the following Grammars are ambiguous. How? ConceptsN = {A,B,S}

AN {}, , and (N T)*, B N, a T

TYPEFORMAT OF PRODUCTIONSREMARKS0AUnrestricted Substitution RulesContracting1A, Context SensitiveNon-Contracting2A , Context FreeNon-Contracting

3A aBA aA BaA aRegular, Right Linear

Regular, Left LinearNon-Contracting