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
7. Sub-Areas of TOC
• automata theory and languages
• computability theory- undecidability
• computational complexity theory
Mrs.D.Jena
Catherine
Bel,
AP/CSE,
VEC
7
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
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
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
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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
