Results 1 
4 of
4
Instance Complexity
, 1994
"... We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest specialcase program for A that ..."
Abstract

Cited by 30 (1 self)
 Add to MetaCart
We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest specialcase program for A that runs in time t, decides x correctly, and makes no mistakes on other strings ("don't know" answers are permitted). We prove that a set A is in P if and only if there exist a polynomial t and a constant c such that ic t (x : A) c for all x
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Random Strings Make Hard Instances
, 1996
"... We establish the truth of the "instance complexity conjecture" in the case of DEXTcomplete sets w.r.t. polynomial time computations, and r.e. complete sets w.r.t. recursive computations. Specifically, we obtain for every DEXTcomplete set A an exponentially dense subset C such that for every nondec ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
We establish the truth of the "instance complexity conjecture" in the case of DEXTcomplete sets w.r.t. polynomial time computations, and r.e. complete sets w.r.t. recursive computations. Specifically, we obtain for every DEXTcomplete set A an exponentially dense subset C such that for every nondecreasing polynomial t(n) = !(n log n), ic t (x : A) K t (x) \Gamma c holds for some constant c and all x 2 C, where ic t and K t are the tbounded instance complexity and Kolmogorov complexity measures, respectively. For r.e. complete sets A we obtain an infinite set C ` A such that ic 1 (x : A) K 1 (x) \Gamma c holds for some constant c and all x 2 C. The proofs are based on the observation that Kolmogorov random strings are individually hard to recognize by bounded computations. 1 Introduction The notion of "instance complexity" was introduced in [5] to quantify the complexity of solving individual instances of decision problems. The basic idea here is to measure the complex...
DSPACE(n) ? = NSPACE(n): A degree theoretic characterization
 in Proc. 10th Structure in Complexity Theory Conference
, 1995
"... It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1 ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1