The Complexity of Computing the Size of an Interval
We study the complexity of counting the number of elements in intervals of feasible partial orders. Depending on the properties that partial orders may have, such counting functions have different complexities. If we consider total, polynomial-time decidable orders then we obtain exactly the #P functions. We show that the interval size functions for polynomial-time adjacency checkable orders are tightly related to the class FPSPACE(poly): Every FPSPACE(poly) function equals a polynomial-time function subtracted from such an interval size function. We study the function #DIV that counts the nontrivial divisors of natural numbers, and we show that #DIV is the interval size function of a polynomial-time decidable partial order with polynomial-time adjacency checks if and only if primality is in polynomial time.
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- 1.D. P. Bovet and P. Crescenzi. Introduction to the Theory of Complexity. International Series in Computer Science. Prentice Hall, New York, 1994.Google Scholar
- 3.S. A. Cook. The complexity of theorem-proving procedures. In Proceedings 3rd ACM Symposium on Theory of Computing, pages 151–158, 1971.Google Scholar
- 14.L. Levin. Universal sorting problems. Problems of Information Transmission, 9:265–266, 1973.Google Scholar
- 15.A. R. Meyer and M. Paterson. With what frequency are apparently intractable problems difficult? Technical Report MIT/LCS/TM-126, Laboratory for Computer Science, MIT, Cambridge, MA, 1979.Google Scholar
- 19.C. H. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, 1994.Google Scholar
- 20.J. Simon. On Some Central Problems in Computational Complexity. PhD thesis, Cornell University, Ithaca, 1975.Google Scholar