1. TURING MACHINES AND COMPLEXITY By Mr.Neelamani Samal

2. INTRODUCING TURING MACHINES

Introduced by Alan Turing in 1936.

A simple mathematical model of a computer.

Models the computing capability of a computer.

3. THE TURING MACHINE MODEL a 1 a 2 a i a n B B Finite control 4. T uring Machine is represented by- M=(Q, ,,,q0,B,F), Where Q is the finite state of states a set of not including B, is the set of input symbols, is the finite state of allowable tape symbols, is the next move function, a mapping fromQ to Q {L,R} Q 0 in Q is the start state, Ba symbol of is the blank, F is the set of final states.Representation of Turing Machine 5. TYPES OF TURING MACHINES

• Two way infinite tape

• Multiple Turing Machines

• Nondeterministic Turing machines

• Multidimensional Turing machines

• Multihead Turing Machines

• Off-line Turing machines

6. SIMULATION Theorem- If L is accepted by a two dimensional TM M 2L is accepted by a one dimensional TM M 1 *BBB a 1 BBB*BB a 2a 3a 4a 5 B* a 6a 7 a 8 a 9 B a 10 B* a 11a 12a 13 B a 14a 15 *BB a 16a 17BBB** simulation of two dimensions by a)Two-dimensional tape b)One dimensional simulation B B B a 1 B B B B B a 2 a 3 a 4 a 5 B a 6 a 7 a 8 a 9 B a 10 B B a 11 a 12 a 13 B a 14 a 15 B B a 17 a 16 B B B 7. CHURCHS HYPOTHESIS The assumption that the intuitive notion ofcomputable function can be identified with the class of partial recursive function is known aschurchs hypothesisor thechurch Turing thesis Example-Random Access Memory.. 8. SIMULATION OF RAM BY TURING MACHINE Contd Theorem- A Turing machine can simulate a RAM provided that the elementary RAM instructions can themselves be simulated by a TM. The tape looks like- #0*v 0 #1*v 1 #10*v 2 #I*v i # Where v iis the contents in binary, of the ith word. 9. Contd COMPUTATIONAL COMPLEXITY

Some useful worst-case complexity classes:

DTIME(t(n)):languages accepted by a deterministic Turing Machine in time O(t(n)).

NTIME(t(n)):languages accepted by a nondeterministic Turing Machine in time O(t(n)).

DSPACE(s(n)):languages accepted by a deterministic Turing Machine in space O(s(n)).

NSPACE(s(n)):languages accepted by a nondeterministic Turing Machine in space O(s(n)).

10.

P:union of DTIME(n c ) for all c. Consists of those problems whose worst-case solution time is bounded by some polynomial.

NP:union of NTIME(n c ) for all c. Consists of problems for which a correct solution can be verified in worst-case time bounded by a polynomial.

PSPACE:union of DSPACE(n c ) for all c. Consists of those problems whose worst-case solution space is bounded by some polynomial.

COMPUTATIONAL COMPLEXITY contd. 11. RELATIONS AMONG COMPLEXITY MEASURES

If L is in DTIME(f(n)), then L is in DSPACE(f(n))

If L is in DSPACE(f(n)) and f(n)>log 2 n, then there is some constant c depending on L such that L is in DTIME(c f(n) ).

If L is in NTIME(f(n)), then there is some constant c depending on L, such that L is in DTIME(c f(n) ).

12. TAPE COMPRESSION Theorem- If L is accepted by S(n) space-bounded Turing machine with k storage tapes, then for any c>0, L is accepted by a cS(n) space bounded TM. Corollary- If L is in NSPACE(S(n)), then L is in NSPACE(cS(n)), where c is any constant greater than zero. 13. LINEAR SPEED UP Theorem- If L is accepted by a k-tape T(n) time bounded Turing machine M 1 , then L is accepted by a k-tape cT(n) time-bounded TM M 2for any c>0, provided that k>1 and inf n-> T(n)/ n->. Corollary-If inf n-> T(n)/ n=. And c>0, then DTIME(T(n))=DTIME(cT(n)). 14. THE UNION THEOREM Theorem- Let {f i (n)|i=1,2,.................} be a recursively enumerable collection of recursive functions. That is there is a TM that enumerates a list of TMs, the first computing f 1, the second computing f 2and so on. Also assume that for each i and n, f i (n)1DSPACE(f i (n)). 15. CONCLUSION

The Turing Machine is the most comprehensive, deep, and accessible model of computation extant, and its associated theories allow many ideas involving "complexity" to be profitably discussed.

16. THANK YOU !!