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P=NP10290512pt - 1 The manuscript below corresponds to my...

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1 The manuscript below corresponds to my talk presented to the mathematics Colloquium, but with references added later. The references are by no means complete, particularly with regard to books that I could not conveniently locate. I want to thank Dan Burghelea for inviting me to give a talk on P = NP (as part of a series of lectures on the Clay Millenium Problems at Ohio State University) as this is not something that I would normally do. CLAY MILLENIUM PROBLEM: P = NP by Harvey M. Friedman Mathematics Colloquium Ohio State University October 20, 2005 The equation P = NP concerns algorithms for deciding membership in sets. The consensus is that P NP, although some prominent experts guess otherwise. P = NP is one of many questions of the form: If there is an algorithm for determining membership in a given set using specified limited resources, is there automatically such an algorithm using yet more limited specified resources? P = NP sits in discrete complexity theory - based on sets of finite strings in a finite alphabet of letters. Sets of integers, sets of rationals, and sets of finite sequences of such, are all treated as special cases of sets of finite strings. Discrete complexity theory is dependent on a formal treatment of algorithms - initiated in the 1930’s by Alan Turing. The official problem description [Co] contains much detailed information that will not be mentioned here. 1. TURING MACHINES. 2. SOME R.E. COMPLETENESS. 3. TIME, SPACE, NONDETERMINISM. 4. SOME NP COMPLETENESS. 5. OPINIONS.
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2 1. TURING MACHINES. There are many sources that discuss Turing machines. E.g., [Tu36], [Tu37], [Ro67], [Co71], [Da73], [AHU74], [HU79], [GJ79], [Od89], [Pa94], [Da00], [CLR01], [Si05], [Co], and many dozens of others. Turing machines form the most common primitive model of computation used in theoretical studies. Micro Center does not have Turing machines for sale. TM’s process symbols from a finite alphabet. When given a finite string from a finite alphabet as input, it grinds away, perhaps halting after finitely many steps, or perhaps going on forever. TM’s have a two way infinite tape divided into squares indexed by the integers, and a reading head that always sits on exactly one square of tape. Every TM comes with a finite set of distinct symbols a 0 ,...,a k ,b 0 ,...,b n , k,n 0. {a 0 ,...,a k } is the input alphabet. b 0 ,...,b n are the auxiliary symbols. b 0 is the “blank”. Every TM also comes with a finite set of distinct states q 0 ,...,q m , m 2. q 0 is the starting state, q 1 is “accept”, and q 2 is “reject”. Every TM comes already programmed. The program consists of a finite list of instructions that tell the reading head what to do. Each instruction takes the following form: If the reading head sees symbol x and TM is in state q, then replace x with symbol x’, put TM into state q’, and move the reading head one square to the left (right).
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