Journal Article10.1145/321864.321877
On the Structure of Polynomial Time Reducibility
875
TL;DR: The method of showing density ymlds the result that if P ~ NP then there are members of NP -P that are not polynomml complete is shown, which means there is a strictly ascending sequence with a minimal pair of upper bounds to the sequence.
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Abstract: Two notions of polynomml time reduclbihty, denoted here by ~ T e and <.~P, were defined by Cook and Karp, respectively The abstract propertms of these two relatmns on the domain of computable sets are investigated. Both relations prove to be dense and to have minimal pairs. Further , there is a strictly ascending sequence with a minimal pair of upper bounds to the sequence. Our method of showing density ymlds the result that if P ~ NP then there are members of NP -P that are not polynomml complete
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References
Reducibility Among Combinatorial Problems.
Richard M. Karp
- 01 Jan 1972
TL;DR: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.
13.6K
The complexity of theorem-proving procedures
Stephen A. Cook
- 03 May 1971
TL;DR: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology.
7.4K
Word problems requiring exponential time(Preliminary Report)
L. J. Stockmeyer,A. R. Meyer +1 more
- 30 Apr 1973
TL;DR: A number of similar decidable word problems from automata theory and logic whose inherent computational complexity can be precisely characterized in terms of time or space requirements on deterministic or nondeterministic Turing machines are considered.
Every Prime Has a Succinct Certificate
TL;DR: It remains an open problem whether a prime n can be recognized in only $\log _2^\alpha n$ operations of a Turing machine for any fixed $\alpha $.
327
Comparison of polynomial-time reducibilities
Richard E. Ladner,Nancy Lynch,Alan L. Selman +2 more
- 30 Apr 1974
TL;DR: Comparison of the polynomial-time-bounded reducibilities introduced by Cook [1] and Karp] leads naturally to the definition of several intermediate truth-tableredcibilities, and it is noted that all redu cibilities of this type which do not have obvious implication relationships are in fact distinct in a strong sense.
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