The.Algorithm.Design.Manual.Springer-Verlag.1998
The.Algorithm.Design.Manual.Springer-Verlag.1998 The.Algorithm.Design.Manual.Springer-Verlag.1998
Lecture 23 - approximation algorithms and Cook's theorem A subset of the set of strings over some alphabet is a formal language. Formal language theory concerns the study of how powerful a machine you need to recognize whether a string is from a particular language. Example: Is the string a binary representation of a even number? A simple finite machine can check if the last symbol is zero: No memory is required, except for the current state. Observe that solving decision problems can be thought of as formal language recognition. The problem instances are encoded as strings and strings in the language if and only if the answer to the decision problem is YES! What kind of machine is necessary to recognize this language? A Turing Machine! A Turing machine has a finite-state-control (its program), a two way infinite tape (its memory) and a readwrite head (its program counter) So, where are we? Each instance of an optimization or decision problem can be encoded as string on some alphabet. The set of all instances which return True for some problem define a language. Hence, any problem which solves this problem is equivalent to a machine which recognizes whether an instance is in the language! The goal of all this is going to be a formal way to talk about the set of problems which can be solved in file:///E|/LEC/LECTUR16/NODE23.HTM (5 of 11) [19/1/2003 1:35:35]
Lecture 23 - approximation algorithms and Cook's theorem polynomial time, and the set that cannot be. Non-deterministic Turing Machines Suppose we buy a guessing module peripherial for our Turing machine, which looks at a Turing machine program and problem instance and in polynomial time writes something it says is an answer. To convince ourselves it really is an answer, we can run another program to check it. Ex: The Traveling Salesman Problem The guessing module can easily write a permutation of the vertices in polynomial time. We can check if it is correct by summing up the weights of the special edges in the permutation and see that it is less than k. The class of languages which we can recognize in time polynomial in the size of the string or a deterministic Turing Machine (without guessing module) is called P. The class of languages we can recognize in time polynomial in the length of the string or a nondeterministic Turing Machine is called NP. Clearly, , since for any DTM program we can run it on a non-deterministic machine, ignore what the guessing module is doing, and it will just as fast. P ?= NP Observe that any NDTM program which takes time P(n) can simulated in time on a deterministic machine, by running the checking program times, once on each possible guessed string. The $10,000 question is whether a polynomial time simulation exists, or in other words whether P=NP?. Do there exist languages which can be verified in polynomial time and still take exponential time on deterministic machines? This is the most important question in computer science. Since proving an exponential time lower bound for a problem in NP would make us famous, we assume that we cannot do it. file:///E|/LEC/LECTUR16/NODE23.HTM (6 of 11) [19/1/2003 1:35:35]
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Lecture 23 - approximation algorithms and Cook's theorem<br />
A subset of the set of strings over some alphabet is a formal language.<br />
Formal language theory concerns the study of how powerful a machine you need to recognize whether a<br />
string is from a particular language.<br />
Example: Is the string a binary representation of a even number? A simple finite machine can check if the<br />
last symbol is zero:<br />
No memory is required, except for the current state.<br />
Observe that solving decision problems can be thought of as formal language recognition. <strong>The</strong> problem<br />
instances are encoded as strings and strings in the language if and only if the answer to the decision<br />
problem is YES!<br />
What kind of machine is necessary to recognize this language? A Turing Machine!<br />
A Turing machine has a finite-state-control (its program), a two way infinite tape (its memory) and a readwrite<br />
head (its program counter)<br />
So, where are we?<br />
Each instance of an optimization or decision problem can be encoded as string on some alphabet. <strong>The</strong> set<br />
of all instances which return True for some problem define a language.<br />
Hence, any problem which solves this problem is equivalent to a machine which recognizes whether an<br />
instance is in the language!<br />
<strong>The</strong> goal of all this is going to be a formal way to talk about the set of problems which can be solved in<br />
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