Some Quantitative Aspects of Fractional Computability

TL;DR

This paper explores the concept of fractional computability using effective measure and category theory, revealing that relaxing time constraints to a fraction does not diminish nondeterminism's power in random oracle settings.

Contribution

It introduces the notion of fractional computability classes based on effective density and analyzes their measure and category properties, extending complexity theory concepts.

Findings

The class of fractionally computable functions at density δ is effectively of the second category.

Fractional complexity classes are effectively meager under certain conditions.

Relaxing polynomial time to fractional time does not reduce nondeterminism's power in random oracle models.

Abstract

Motivated by results on generic-case complexity in group theory, we apply the ideas of effective Baire category and effective measure theory to study complexity classes of functions which are "fractionally computable" by a partial algorithm. For this purpose it is crucial to specify an allowable effective density, $\delta$, of convergence for a partial algorithm. The set $\mathcal{FC}(\delta)$ consists of all total functions $ f: \Sigma^\ast \to \{0,1 \}$ where $\Sigma$ is a finite alphabet with $|\Sigma| \ge 2$ which are "fractionally computable at density $\delta$". The space $\mathcal{FC}(\delta) $ is effectively of the second category while any fractional complexity class, defined using $\delta$ and any computable bound $\beta$ with respect to an abstract Blum complexity measure, is effectively meager. A remarkable result of Kautz and Miltersen shows that relative to an algorithmically random oracle $A$, the relativized class $\mathcal{NP}^A$ does not have effective polynomial measure zero in $\mathcal{E}^A$, the relativization of strict exponential time. We define the class $\mathcal{UFP}^A$ of all languages which are fractionally decidable in polynomial time at ``a uniform rate'' by algorithms with an oracle for $A$. We show that this class does have effective polynomial measure zero in $\mathcal{E}^A$ for every oracle $A$. Thus relaxing the requirement of polynomial time decidability to hold only for a fraction of possible inputs does not compensate for the power of nondeterminism in the case of random oracles.