2 DFA versus Normal DFA

```2 DFA VERSUS NORMAL DFA
Pavan Kumar Akulakrishna (M.Tech, SERC)
SR No: 06-02-01-10-51-12-1-09469
Under Guidance of : Prof. Deepak D’souza.
FUNDAMENTAL QUESTION
Is 2 way DFA more powerful than a one way
DFA?
“OR”
 Does the ability to do multiple scans on the tape
gives rise to an increase in ‘POWER’ of the
2DFA?

Not Intuitively clear!
 Need to develop a formal model.

UNDERSTANDING THE 2DFA BEHAVIOR

Consider a String w: which is broken down into two
parts say x and z.
x





z
Right to Left  Q
Left to Right  P
If next time Right to Left Q, then Left to Right will be
P. i.e., Tx (Q)=P
If never emerges? Let Tx (Q) = ┴
The first time it emerges from x without having been
to z. Call the state Tx(.).
CONSTRUCTING A TABLE
Tx (q) depends only on x and q.
 Write down Tx(q) for all possible states q
• T: (Q U {.})  (Q U {┴})
 Possible such tables taking |Q| = k are,
(k+1)^(k+1).
 Hence, a finite information can be passed.

Q1
Q1
Qk
.
…….
Qk
┴
√
Q2
……
Q2
√
√
……..
√
√
USING MYHILL NERODE THEOREM



•
If there are tables Tx == Ty and M accepts xz.
Then, yz is also said to be accepted.
Let L(M) denote the language accepted by the
machine.
Define a Equivalence relation on strings x and y:
If Tx = Ty, x ≡L(M) y.
PROPERTIES OF ≡L(M)
Right congruence,
• x ≡L(M) y then xa ≡L(M) ya
 Refines L(M)
•
If Tx = Ty , either both x and y accepted or
rejected.
 Finite Index
•
No. of Equivalence classes = No. of Unique
Tables [i.e., (k+1)^(k+1)].


Hence, by Myhill-Nerode theorem, L(M) is a
Regular Language.
INFERENCES DRAWN


A 2DFA is NO more powerful than a normal DFA
We can construct a one way DFA equivalent to a
2 DFA viz.
•
Indentify equivalence classes.
•
Use construction defined by ≡  M≡
CONSTRUCTING EQUIVALENT DFA

Formally, a DFA equivalent to 2 DFA looks as:
D=(Q’,S’,Σ U{├, ┤}, δ,F’)
 Q’ = { T: (Q U {.})  (Q U {┴}) }
 S’ = Tε
 δ (Tx, a) = Txa
 F’ = {Tx | x Є L(M) }


A 2DFA is more compact than a DFA
•
i.e., L(M)= {x Є {a, b}* | #a(x) is multiple of 7 and
#b(x) is multiple of 5}
Normal DFA would have 35 states.
2 DFA can be constructed using 14 states, including
the accept and reject states (t, r).
•

•
•
•
A 2DFA has single accept and reject states
Whereas, a DFA may have multiple final states.
E.g. L(M) = {x Є {a, b}* | #a(x) is multiple of 7 or #b(x)
is multiple of 5}.
Normal DFA has 12 final states.
CONCLUSIONS


2 way DFA is a convenient representation for
some regular languages, much like NFA
Converting it to a normal DFA can be tedious as
it involves constructing and identifying identical
tables.
```