Catalytic Space: Non-determinism and Hierarchy

Harry Buhrman,Michal Koucký,Bruno Loff,Florian Speelman

doi:10.1007/s00224-017-9784-7

Abstract

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation.

Highlights

Buhrman et al [3] define the notion of catalytic computation, a space-bounded model of computation in which the usual Turing machine has, in addition to its work tape, access to a large auxiliary memory which is full
In this paper we show that non-deterministic catalytic space is closed under complement under a widely accepted derandomization assumption
The model for catalytic computation is defined in terms of deterministic Turing machines

Summary

Introduction

Buhrman et al [3] define the notion of catalytic computation, a space-bounded model of computation in which the usual Turing machine has, in addition to its work tape, access to a large auxiliary memory which is full. The catch with the auxiliary memory is that it may contain arbitrary content, possibly incompressible, which has to be preserved in some way during the computation. We establish hierarchy theorems for catalytic computation in the deterministic and non-deterministic settings. The full description of a configuration is exponentially bigger than our work space, so we cannot possibly store it in full This is one of the hurdles that prevents us from carrying out Savitch’s algorithm for catalytic computation. For semantic models of computation, like bounded-error randomized computation, the only hierarchy theorems that we know of are in the setting with advice.

Preliminaries

Non-deterministic Catalytic Computation

An Analog of the Immerman–Szelepcsényi Theorem

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Hierarchies for Catalytic Computation