Decision, Computation and Language
Pumping Lemma
Dr. Muhammad S Khan
([email protected])
Ashton Building, Room G22
http://www.csc.liv.ac.uk/~khan/comp218
Exercise
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COMP218 Decision, Computation and Language
2
Exercise
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COMP218 Decision, Computation and Language
3
Regular Languages
Regular languages has at least three different
descriptions:
Languages accepted by DFA
Languages accepted by NFA
Languages defined by regular expressions
NOT every language is a regular language
Powerful technique for showing certain languages
not to be regular: Pumping Lemma
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COMP218 Decision, Computation and Language
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Limitation of FSA
Cannot recognise
𝐿 = {0𝑛 1𝑛 : 𝑛 = 1,2,3, … }
Palindromes 𝑤 = 𝑤 𝑅
RADAR
KAYAK
NOON
EYE
DEED
ROTATOR
Only finite number of states, so it can’t count
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COMP218 Decision, Computation and Language
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Infinite Languages
First of all, we note one very important point:
Every finite language is regular
How do we form an infinite regular language?
The union or concatenation of two finite languages will still
be a finite language;
thus, any infinite regular language must be formed from a
regular expression which uses Kleene Closure or Loop.
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COMP218 Decision, Computation and Language
6
Algorithm for infinite languages
So, the input to the algorithm is a DFA, represented
(say) as a list of states and transitions. The algorithm
should look out for a cycle (or loop), a sequence of
states that are connected by transitions that form a
loop.
Furthermore, the cycle should be accessible
(reachable from the initial state), and there should
be a way to get from some state on the cycle, to an
accepting state.
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COMP218 Decision, Computation and Language
7
Pumping Lemma
Every infinite regular language must have as a subset,
some set of words of the form 𝑥𝑦 𝑖 𝑧, for all 𝑖 ≥ 0,
where 𝑥, 𝑦 and 𝑧 are strings which correspond to 𝑥, 𝑦
and 𝑧 respectively.
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COMP218 Decision, Computation and Language
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The Pumping Lemma
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COMP218 Decision, Computation and Language
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Proof of Pumping Lemma
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COMP218 Decision, Computation and Language
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Reminder: accepting/rejecting paths notation
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COMP218 Decision, Computation and Language
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Reminder: accepting/rejecting paths notation
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COMP218 Decision, Computation and Language
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Proof of Pumping Lemma
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COMP218 Decision, Computation and Language
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Proof contd.
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Proof
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Proof (cont.)
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COMP218 Decision, Computation and Language
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Proof (cont.)
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COMP218 Decision, Computation and Language
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Summary
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Example
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Another example
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COMP218 Decision, Computation and Language
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The Pumping Lemma as an Adversarial Game
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COMP218 Decision, Computation and Language
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