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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”

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2 DFA versus Normal DFA

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  1. 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.

  2. 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.

  3. Understanding the 2DFA Behavior • Consider a String w: which is broken down into two parts say x and 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(.). P x z Q

  4. 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.

  5. Using MyhillNerode 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.

  6. 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.

  7. 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≡

  8. 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 ЄL(M) }

  9. Advantages Of 2 DFA • 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.

  10. 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.

  11. Thank Q

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