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Theory of Computation Unit 5
- 1. UNIT - V Undecidability Mrs. D.Jena Catherine Bel, Assistant Professor, CSE, Velammal Engineering College
- 2. UNIT V UNDECIDABILITY Non Recursive Enumerable (RE) Language – Undecidable Problem with RE – Undecidable Problems about TM – Post‘s Correspondence Problem, The Class P and NP. Mrs.D.Jena Catherine Bel, AP/CSE, VEC 2
- 3. Undecidability-languages • Recursive • Recursively Enumerable • Non-RE Mrs.D.Jena Catherine Bel, AP/CSE, VEC 3
- 6. Non- Recursively Enumerable There are languages which does not have a Turing machine at all Mrs.D.Jena Catherine Bel, AP/CSE, VEC 6
- 7. Sub-Areas of TOC • automata theory and languages • computability theory- undecidability • computational complexity theory Mrs.D.Jena Catherine Bel, AP/CSE, VEC 7
- 8. 10/3 = 3.33333333 =3.3 22/7 = 3.142857142857143 = ? Mrs.D.Jena Catherine Bel, AP/CSE, VEC 8
- 9. Based on the extent to which a problem can be solved Decidable Problems - A decidable problem has an algorithm to determine the answer for a given input Example: Find whether P is prime or not Mrs.D.Jena Catherine Bel, AP/CSE, VEC 9
- 10. 2. Undecidable problem – problems that has no algorithm to determine the answer for a given input or which have an algorithm that answers for some input Example - no three positive integers a, b and c for any n>2 can ever satisfy the equation: a^n + b^n = c^n. Mrs.D.Jena Catherine Bel, AP/CSE, VEC 10
- 11. The mortal matrix problem: Determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. This is known to be undecidable for a set of six or more 3 × 3 matrices, or a set of two 15 × 15 matrices. Mrs.D.Jena Catherine Bel, AP/CSE, VEC 11
- 12. Languages 1. Recursive ------------------------- Decidable 2. Recursively Enumerable (RE) 3. Non-RE Undecidable Mrs.D.Jena Catherine Bel, AP/CSE, VEC 12
- 13. Recursive - Decidable • A language ‘L’ is said to be recursive if there exists a Turing machine which will • accept all the strings in ‘L’ and • reject all the strings not in ‘L’. • The Turing machine will halt every time and give an answer(accepted or rejected) for each and every string input. Mrs.D.Jena Catherine Bel, AP/CSE, VEC 13
- 14. Recursively Enumerable- Undecidable Recursively enumerable language(RE) – A language ‘L’ is said to be a recursively enumerable language if there exists a Turing machine which will • accept (and therefore halt) for all the input strings which are in ‘L’ • but may or may not halt for all input strings which are not in ‘L’. Mrs.D.Jena Catherine Bel, AP/CSE, VEC 14
- 15. Non-RE -Undecidable Non- Recursively enumerable language(RE) – A language ‘L’ is said to be a non recursively enumerable language if there doesn't exists a Turing machine Mrs.D.Jena Catherine Bel, AP/CSE, VEC 15
- 16. Recursive, RE, Non-RE • Recursive --------Regular, Context Free • RE -----Lu-----------Universal Language • Non-RE -----Ld ------- Diagonalization language Mrs.D.Jena Catherine Bel, AP/CSE, VEC 16
- 19. Machine as strings Turing Machine/program M can itself be encoded as a binary string. Moreover every binary string can be thought of as encoding a TM/program. (If not the correct format, considered to be the encoding of a default TM.) Mrs.D.Jena Catherine Bel, AP/CSE, VEC 19
- 20. TM-Encoding A Turing machine M- 7- Tuple (Q, F, q0, Σ, Γ, δ ,blank) – The encoding of a TM - is a binary string that has all the information of the 7- tuple describing TM. So the encoding of M, is just a string that describes how the TM works. Mrs.D.Jena Catherine Bel, AP/CSE, VEC 20
- 22. Every binary string is a TM Mrs.D.Jena Catherine Bel, AP/CSE, VEC 22
- 24. Ld - Non- RE Mrs.D.Jena Catherine Bel, AP/CSE, VEC 24
- 28. Lu in RE • RE-- Lu has a TM and Lu’ is Non RE Mrs.D.Jena Catherine Bel, AP/CSE, VEC 28
- 29. Properties of RE Languages Mrs.D.Jena Catherine Bel, AP/CSE, VEC 29
- 30. Reductions If we are able to do a mathematical operation for binary numbers than can do for decimal , hexadecimal number system-> as we have an alg to convert decimal to binary and viceversa Mrs.D.Jena Catherine Bel, AP/CSE, VEC 30
- 32. Reductions • Let L1 and L2 be two decision problems. Suppose algorithm A2 solves L2. - if y is an input for L2 then algorithm A2 will answer Yes or No depending upon whether y belongs to L2 or not. • The idea is to find a transformation from L1 to L2 so that the algorithm A2 can be part of an algorithm A1 to solve L1. Mrs.D.Jena Catherine Bel, AP/CSE, VEC 32