2.4 derivations and languages

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

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
2
Derivation using Grammar
Recursive Inference Derivation
Left Most
Derivation
Right Most
Derivation

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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நன்றி
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