Extensional properties of sets of time bounded complexity (extended abstract)

Abstract

We analyze the fine structure of time complexity classes for RAM's, in particular the equivalence relation A= _C B (“A and B have the same time complexity”) ⇔ (for all time constructible f : A∈DTIME _RAM (f) ⇔ B∈DTIME _RAM (f)). The =_C-equivalence class of A is called its complexity type. We prove that every set X can be partitioned into two sets A and B such that X=t _C A= _C B, and that the partial order of sets in an arbitrary complexity type under $\subseteq *$ (inclusion modulo finite sets) is dense. The proofs employ a new strategy for finite injury priority arguments.

We consider the following set of time bounds:

T:={f : N→N|f(n)≥n and f is time constructible on a RAM}, where f is called time constructible on a RAM if some RAM can compute the function 1ⁿ↦1^f(n) in O(f(n)) steps. We do not allow arbitrary recursive functions as time bounds in our approach in order to avoid pathological phenomena (e.g. gap theorems [HU], [HH]). In this way we can focus on those aspects of complexity classes that are relevant for concrete complexity (note that all functions that are actually used as time bounds in the analysis of algorithms are time constructible). We use the random access machine (RAM) with uniform cost criterion as machine model (e.g. as defined in [CR]; see also [AHU], [MY]) because this is the most frequently considered model in algorithm design, and because a RAM allows more sophisticated diagonalization — constructions than a Turing machine. One defines DTIME _RAM (f):={A$\subseteq$ {0, 1}*| there is a RAM of time complexity O(f) that computes A∼. We write DTIME(f) for DTIME _RAM (f) in the following.

For sets A, B $\subseteq${0, 1}* we define

A= _C B(“A has the same det. time complexity as B”)

$$T: = \left\{ {f:N \to N|f(n) \geqslant n{\text{ }}and{\text{ }}f{\text{ }}is time constructible on a RAM} \right\},$$

A complexity type is an equivalence class of this equivalence relation = _C .

We write 0 for DTIME(n), which is the “minimal” complexity type. Note that for every complexity type C and every f∈T one has either C $\subseteq$ DTIME(f) or C∩DTIME(f)=0.

In this paper we investigate some basic properties of the partial order

$$PO(C): = \left\langle {\left\{ {X\left| {X \in C} \right.} \right\}, \subseteq ^* } \right\rangle ,$$

where C is an arbitrary complexity type and $\subseteq *$ denotes inclusion modulo finite sets (i.e. X$\subseteq *$Y : ↦ X−Y is finite).

This work is part of the long range project to study the relationship between extensional properties of a set and its computational complexity. Among other work in this direction we would like to mention in particular the study of the complexity of sparse sets (see e.g. [Ma]), and the investigation of the relationship between properties of recursively enumerable sets under $\subseteq *$ and their degree of computability (see e.g. Chapter XI in [So]). Our approach differs from this preceding work insofar as it also applies to “actually computable” sets (i.e. sets in P). Therefore it provides an opportunity to develop finer construction tools that can be used to examine also the structure of sets of small complexity. In this paper we introduce a new strategy for a finite injury priority argument that allows us to prove splitting and density theorems for sets of arbitrarily given complexity type C (for example sets of time complexity Θ(n ²)). Further results about the structure of complexity types can be found in [MS].

Written under partial support by NSF-Grant CCR 8703889.

Written under partial support by Presidential Young Investigator Award DMS-8451748 and NSF-Grant DMS-8601856.