This document discusses derivations and languages in context-free grammars. It covers recursive inference versus derivation to determine if a string belongs to a language. It also discusses leftmost and rightmost derivations, derivation trees, sentential forms, and left and right recursive grammars. Several example problems are provided to demonstrate derivations, parse trees, and generating strings from given grammars.

1. Derivations
and Languages
-Sampath Kumar S,
AP/CSE, SECE

2. Derivations and Languages
 While inferring whether the given input strings
belongs to the given CFG, we can have 2
approaches:
 Using the rules from body to head (Recursive
Inference)
 Using the rules from head to body (Derivation)
11/21/2017
Sampath Kumar S, AP/CSE, SECE
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Derivation using Grammar
Recursive Inference Derivation
Left Most
Derivation
Right Most
Derivation

3. Recursive Inference:
 Here we take string from each variables,
concatenate them in proper order and infer that the
resulting string is in the language of the variable in
the head
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Sampath Kumar S, AP/CSE, SECE

4. Derivation:
 Here we use the production from head to body i.e.,
from start symbol expanding till reaches the given
string. This use of grammar is called derivation.
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Sampath Kumar S, AP/CSE, SECE

5. Problems to discuss:
61. For the language L = {wcwR | w (0+1)*} check
whether the string 01c10 belongs to the language
L or not.
Solution:
Step 1: For the language L, the productions are
S → 0S0 | 1S1 | C
Step 2: Derivation
S → 0S0 [∵ S → 0S0]
→ 01E10 [∵ S → 1S1]
→ 01c10 [∵ S → c]
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Sampath Kumar S, AP/CSE, SECE
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6. Understanding the Language
defined by CFG:
 The only way to recognize the language, is to try
out various strings from the given production rules.
 Then by observing the derived strings, one can
find out the language generated from the given
CFG.
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Sampath Kumar S, AP/CSE, SECE
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7. Problems to discuss:
62. Derive ‘a4’ from the grammar G=({S}, {a},P,S)
where P: S → aS|ε.
63. Find L(G) and derive the string ‘abbab’ for the
grammar G=({S}, {a},P,S) where
P: S → aS|bS|a|b.
64. Find L(G) and derive the string ‘abbaaba’ for the
grammar G=({S,X}, {a},P,S) where
P: S → XaaX, X → aX|bX|ε.
65. Find L(G) for the grammar G=({S,C}, {a,b},P,S)
where P is given by S → aCa, C → aCa|b.
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Sampath Kumar S, AP/CSE, SECE
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8. Leftmost and Rightmost Derivation:
 If a word w is generated by a CFG by a certain
derivation and at each step in the derivation, a rule
of production is applied to the leftmost non
terminal in the working string, then this derivation
is called leftmost derivation (LMD).
 If a word w is generated by a CFG by a certain
derivation and at each step in the derivation, a rule
of production is applied to the rightmost non
terminal in the working string, then this derivation
is called rightmost derivation (RMD).
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Sampath Kumar S, AP/CSE, SECE
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9. Problems to discuss:
66. Consider the CFG ({S,X}, {a,b}, P, S} where
productions are S → baXaS|ab, X →Xab|aa. Find
the LMD and RMD for the string
w=baaaababaab.
67. Consider the CFG S → aB|bA
A → a | aS | bAA
B → b | bS | aBB
Find the LMD and RMD for the string
W=aabbabba.
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Sampath Kumar S, AP/CSE, SECE
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10. Derivation Tree:
 A derivation tree or parse tree is an ordered
rooted tree that graphically represents the
semantic information a string derived from a
Context - Free Grammar.
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11. Representation Technique
 Root vertex − Must be labeled by the start
symbol.
 Vertex − Labeled by a non-terminal symbol.
 Leaves − Labeled by a terminal symbol or ε.
If S → x1x2 …… xn is a production rule in a
CFG, then the parse tree / derivation tree will be
as follows −
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Sampath Kumar S, AP/CSE, SECE
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12. Approaches for constructing
Derivation Tree:
There are two different approaches to draw a
derivation tree −
 Top-down Approach −
 Starts with the starting symbol S
 Goes down to tree leaves using productions
 Bottom-up Approach −
 Starts from tree leaves
 Proceeds upward to the root which is the starting
symbol S
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Sampath Kumar S, AP/CSE, SECE
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13. Derivation or Yield of a Tree:
 The derivation or the yield of a parse tree is the
final string obtained by concatenating the labels of
the leaves of the tree from left to right, ignoring the
Nulls.
 However, if all the leaves are Null, derivation is
Null.
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Sampath Kumar S, AP/CSE, SECE
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14. Sentential Form and Partial
Derivation Tree:
 A partial derivation tree is a sub-tree of a
derivation tree/parse tree such that either all of its
children are in the sub-tree or none of them are in
the sub-tree.
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Sampath Kumar S, AP/CSE, SECE
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15. Leftmost and Rightmost
Derivation of a String
 Leftmost derivation − A leftmost derivation is
obtained by applying production to the leftmost
variable in each step.
 Rightmost derivation − A rightmost derivation is
obtained by applying production to the rightmost
variable in each step.
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Sampath Kumar S, AP/CSE, SECE
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16. Left and Right Recursive Grammars
 In a context-free grammar G, if there is a
production in the form X → Xa, where X is a non-
terminal and ‘a’ is a string of terminals, it is called
a left recursive production. The grammar having
a left recursive production is called a left
recursive grammar.
 If in a context-free grammar G, if there is a
production is in the form X → aX where X is a non-
terminal and ‘a’ is a string of terminals, it is called
a right recursive production. The grammar
having a right recursive production is called a right
recursive grammar.
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Sampath Kumar S, AP/CSE, SECE
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17. Problems to discuss:
68. Let a CFG G= { {S}, {a, b}, P, S} Where
P = S → SS | aSb | ε. Draw derivation tree for the
string “abaabb”
 One derivation form of the above CFG is
S → SS → aSbS → abS → abaSb → abaaSbb → abaabb
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Sampath Kumar S, AP/CSE, SECE
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18. Problems to discuss:
69. Let any set of production rules in a CFG be X →
X+X | X*X |X| a over an alphabet {a}. Find LMD and
RMD for string "a+a*a“.
70. Construct Parse tree for the string a*(a+b00) for the
following CFG using left and right most derivation.
E → A | E + E | E * E | (E)
A → a | b |Aa | Ab | A0 | A1
71. Construct Parse tree for the string a*(a+a) for the
following CFG using left and right most derivation.
E → E + E | E * E | (E) | a
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19. Problems to discuss:
72. Generate parse tree for the string 00101 using
left and right most derivation for following CFG
S → A1B, A → 0A|ε | B → 0B | 1B | ε
73. Generate parse tree for the string babbab using
left and right most derivation for following CFG
S → aSa | bSb | a |b | ε
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21. நன்றி
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