Difference between revisions of "Category:Automata Theory Notes"

Revision as of 09:37, 14 July 2014

Video Lectures by Shai Simonson

NP Complete

Recursively Enumerable Sets

Recursively Enumerable Sets

Notes from CMU

Definitions

Push Down Automata

Linear Bounded Automata

Turing Machine Problems

Turing Machine Handout

Rice's theorem, Undecidable and unrecognizable problems on TM

Undecidable problems about L(CFG)

Trivial and Non-trivial properties of L(M)

Undecidable Problems

Closure property of CFL

NP Complete

A good lecture going deep into Abstract Machines

Identify the class of the language

It might be because of the name but many graduate students find it difficult to understand <math>NP</math> problems. So, I thought of explaining them in an easy way. (When explanation becomes simple, some points may be lost. So, please do refer standard text books for more information)

<math>P</math> Problems

As the name says these problems can be solved in polynomial time, i.e.; <math>O(n)</math>, <math>O(n^2)</math> or <math>O(n^k)</math>, where <math>k</math> is a constant.

<math>NP</math> Problems

Some think <math>NP</math> as Non-Polynomial. But actually it is Non-deterministic Polynomial time. i.e.; "yes" instances of these problems can be solved in polynomial time by a non-deterministic Turing machine and hence can take up to exponential time (some problems can be solved in sub-exponential but super polynomial time) by a deterministic Turing machine. In other words these problems can be verified (if a solution is given, say if it is correct or wrong) in polynomial time. Examples include all P problems. One example of a problem not in <math>P</math> but in <math>NP</math> is Integer Factorization.

<math>NP</math> Complete Problems<math>(NPC)</math>

Over the years many problems in <math>NP</math> have been proved to be in <math>P</math> (like Primality Testing). Still, there are many problems in <math>NP</math> not proved to be in <math>P</math>. i.e.; the question still remains whether <math>P = NP</math> (i.e.; whether all <math>NP</math> problems are actually <math>P</math> problems).

<math>NP</math> Complete Problems helps in solving the above question. They are a subset of <math>NP</math> problems with the property that all other <math>NP</math> problems can be reduced to any of them in polynomial time. So, they are the hardest problems in <math>NP</math>, in terms of running time. If it can be showed that any <math>NPC</math> Problem is in <math>P</math>, then all problems in <math>NP</math> will be in <math>P</math> (because of <math>NPC</math> definition), and hence <math>P = NP = NPC</math>.

All <math>NPC</math> problems are in <math>NP</math> (again, due to <math>NPC</math> definition). Examples of <math>NPC</math> problems

<math>NP</math> Hard Problems <math>(NPH)</math>

These problems need not have any bound on their running time. If any <math>NPC</math> Problem is polynomial time reducible to a problem <math>X</math>, that problem <math>X</math> belongs to <math>NP</math> Hard class. Hence, all <math>NP</math> Complete problems are also <math>NPH</math>. In other words if a <math>NPH</math> problem is non-deterministic polynomial time solvable, its a <math>NPC</math> problem. Example of a <math>NP</math> problem that's not <math>NPC</math> is Halting Problem.

From the diagram, its clear that <math>NPC</math> problems are the hardest problems in <math>NP</math> while being the simplest ones in <math>NPH</math>. i.e.; $NP ∩ NPH = NPC$

Note

Given a general problem, we can say its in <math>NPC</math>, if and only if we can reduce it to some <math>NP</math> problem (which shows its in NP) and also some <math>NPC</math> problem can be reduced to it (which shows all NP problems can be reduced to this problem).

Also, if a <math>NPH</math> problem is in <math>NP</math>, then it's <math>NPC</math>

Some Reduction Inferences

--Arjun (talk) 22:48, 16 November 2013 (UTC)