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Theory of Automata

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Theory of Automata
(CS402)
Theory of Automata
Lecture N0. 11
Reading Material
Chapter 7
Introduction to Computer Theory
Summary
proof of Kleene's theorem part II (method with different steps), particular examples of TGs to determine
corresponding REs.
Proof(Kleene's Theorem Part II)
To prove part II of the theorem, an algorithm consisting of different steps, is explained showing how a RE can
be obtained corresponding to the given TG. For this purpose the notion of TG is changed to that of GTG i.e. the
labels of transitions are corresponding REs.
Basically this algorithm converts the given TG to GTG with one initial state along with a single loop, or one
initial state connected with one final state by a single transition edge. The label of the loop or the transition edge
will be the required RE.
Step 1 If a TG has more than one start states, then introduce a new start state connecting the new state to the old
start states by the transitions labeled by Λ and make the old start states the non-start states. This step can be
shown by the following example
Example
The above TG can be converted to
Step 2:
If a TG has more than one final states, then introduce a new final state, connecting the old final states to the new
final state by the transitions labeled by Λ.
This step can be shown by the previous example of TG, where the step 1 has already been processed
Example
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Theory of Automata
(CS402)
The above TG can be converted to
Step 3:
If a state has two (more than one) incoming transition edges labeled by the corresponding REs, from the same
state (including the possibility of loops at a state), then replace all these transition edges with a single transition
edge labeled by the sum of corresponding REs.
This step can be shown by a part of TG in the following example
Example
The above TG can be reduced to
Note
The step 3 can be generalized to any finite number of transitions as shown below
The above TG can be reduced to
Step 4 (bypass and state elimination)
If three states in a TG, are connected in sequence then eliminate the middle state and connect the first state with
the third by a single transition (include the possibility of circuit as well) labeled by the RE which is the
concatenation of corresponding two REs in the existing sequence. This step can be shown by a part of TG in the
following example
Example
r3
To eliminate state 5 the above can be reduced to
Consider the following example containing a circuit
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Theory of Automata
(CS402)
Example
Consider the part of a TG, containing a circuit at a state, as shown below
To eliminate state 3 the above TG can be reduced to
Example
Consider a part of the following TG
To eliminate state 3 the above TG can be reduced to
To eliminate state 4 the above TG can be reduced to
Note
It is to be noted that to determine the RE corresponding to a certain TG, four steps have been discussed. This
process can be explained by the following particular examples of TGs
Example
Consider the following TG
To have single final state, the above TG can be reduced to the following
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Theory of Automata
(CS402)
To eliminate states 2 and 3, the above TG can be reduced to the following
To eliminate state 1 the above TG can be reduced to the following
Hence the required RE is (ab+ba)(aa+b)*(aaa+bba)
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