On the Structure of Polynomial Time Reducibility

@article{Ladner1975OnTS,
  title={On the Structure of Polynomial Time Reducibility},
  author={Richard E. Ladner},
  journal={J. ACM},
  year={1975},
  volume={22},
  pages={155-171}
}
  • R. Ladner
  • Published 1975
  • Mathematics, Computer Science
  • J. ACM
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… Expand
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References

SHOWING 1-10 OF 40 REFERENCES
Comparison of polynomial-time reducibilities
TLDR
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. Expand
The complexity of theorem-proving procedures
  • S. Cook
  • Computer Science, Mathematics
  • STOC
  • 1971
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… Expand
Every Prime has a Succinct Certificate
  • V. Pratt
  • Mathematics, Computer Science
  • SIAM J. Comput.
  • 1975
TLDR
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 $. Expand
Augmented Loop Languages and Classes of Computables Functions
TLDR
A classification of all the computable functions is given in terms of subrecursive programming languages and it is shown that any countable partial ordering can be embedded in the dishonest classes, and that the dishonestclasses are dense in the honest classes. Expand
The Upper Semi-Lattice of Degrees of Recursive Unsolvability
The concept 'degree of recursive unsolvability' was introduced briefly in Post [16]. In his abstract [17] the concept was formulated precisely via an extension of [15], and a resulting partial scale… Expand
Word problems requiring exponential time(Preliminary Report)
TLDR
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. Expand
Recursively enumerable sets of positive integers and their decision problems
Introduction. Recent developments of symbolic logic have considerable importance for mathematics both with respect to its philosophy and practice. That mathematicians generally are oblivious to the… Expand
On a subrecursive hierarchy and primitive recursive degrees
1. Prior to the Herbrand-Godel-Kleene definition of general recursive function, certain classes of functions defined more restrictively by particular "recursions" had been studied. Subsequently… Expand
CLASSES OF PREDICTABLY COMPUTABLE FUNCTIONS
0. Introduction. In this paper we study a sequence of classes of computable functions for which a prediction of the complexity of the calculation may be made in a comparatively simple fashion. The… Expand
The Honest Subrecursive Classes Are a Lattice
TLDR
It is shown that the honest subrecursive classes are a distributive lattice under the partial ordering of set inclusion, and every honest sub Recursion class is the greatest lower bound of two setwise incomparable honest sub recursion classes. Expand
...
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