{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

DecisonProbs042399

DecisonProbs042399 - 1 Some Decision Problems of Enormous...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Some Decision Problems of Enormous Complexity Harvey M. Friedman Department of Mathematics The Ohio State University [email protected] Abstract We present some new decision and comparison problems of unusually high computational complexity. Most of the problems are strictly combinatorial in nature; others involve basic logical notions. Their complexities range from iterated exponential time completeness to 0 time completeness to q ( W w ,0) time completeness to q ( W w ,,0) time completeness. These three ordinals are well known ordinals from proof theory, and their associated complexity classes represent new levels of computational complexity for natural decision problems. Proofs will appear in an extended version of this manuscript to be published elsewhere. 1. Iterated exponential time - universal relational sentences Let F be a function from A* into B*, where A,B are finite alphabets. We say that F is iterated exponential time computable if and only if there is a multitape Turing machine TM (which processes inputs from A* and outputs from B*) and an integer constant c > 0 such that TM computes F(x) with run time at most 2 [c|x|] . Here 2 [k] is the exponential stack of 2's of height k and |x| is the length of the string x. More generally, 2 [k] (n) is the exponential stack of k 2's with n placed on top. Define 2 [0] = 1 and 2 [0] (n) = n. Hence 2 [k] = 2 [k] (1) and 2 [1] (n) = 2 n . The iterated exponential time computable sets strictly include those sets in the more familiar class of elementary time computable sets - where the stack of 2's is of fixed height and |x| appears at the top of the stack. We say that X is iterated exponential time complete if and only if X is in iterated exponential time and every Y in iterated exponential time is polynomial time reducible to X. It is well known that for every finite alphabet A there exists an iterated exponential time complete X A*. As is customary, these definitions extended to include sets of strings in a finite alphabet using characteristic functions. A decision problem is given by a set of strings in a finite alphabet, where the “decision” is to decide membership. Suppose we are given a map G:A* w , where w is the set of all nonnegative integers. We can consider the associated equality problem: decide, given two strings x,y A*, whether G(x) = G(y). We can also consider the comparison problem: given x,y A*, compare the numbers G(x) and G(y). We find this terminology convenient and suggestive. Strictly speaking, this comparison problem is not a decision problem. It amounts to considering the function G’(x,y) = 0 if F(x) = F(y); 1 if F(x) < F(y); 2 if F(x) > F(y). It is easy to see that it is computationally equivalent, in the strongest possible sense, to the related decision problem: decide, given x,y A*, whether F(x) < F(y).
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern