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Fully Characterizing Lossy Catalytic Computation

Marten Folkertsma, Ian Mertz, Florian Speelman, Quinten Tupker

TL;DR

This work completely characterize lossy catalytic space ($LCSPACE[s,c,e]$) in terms of ordinary catalytic space ($CSPACE[s,c,e) in terms of ordinary catalytic space, showing that for any e, $LCL[e] = CL$ implies $SPACE[e \log n] \subseteq ZPP$, thus giving a barrier to any improvement beyond $LCL[O(1)] = CL$.

Abstract

A catalytic machine is a model of computation where a traditional space-bounded machine is augmented with an additional, significantly larger, "catalytic" tape, which, while being available as a work tape, has the caveat of being initialized with an arbitrary string, which must be preserved at the end of the computation. Despite this restriction, catalytic machines have been shown to have surprising additional power; a logspace machine with a polynomial length catalytic tape, known as catalytic logspace ($CL$), can compute problems which are believed to be impossible for $L$. A fundamental question of the model is whether the catalytic condition, of leaving the catalytic tape in its exact original configuration, is robust to minor deviations. This study was initialized by Gupta et al. (2024), who defined lossy catalytic logspace ($LCL[e]$) as a variant of $CL$ where we allow up to $e$ errors when resetting the catalytic tape. They showed that $LCL[e] = CL$ for any $e = O(1)$, which remains the frontier of our understanding. In this work we completely characterize lossy catalytic space ($LCSPACE[s,c,e]$) in terms of ordinary catalytic space ($CSPACE[s,c]$). We show that $$LCSPACE[s,c,e] = CSPACE[Θ(s + e \log c), Θ(c)]$$ In other words, allowing $e$ errors on a catalytic tape of length $c$ is equivalent, up to a constant stretch, to an equivalent errorless catalytic machine with an additional $e \log c$ bits of ordinary working memory. As a consequence, we show that for any $e$, $LCL[e] = CL$ implies $SPACE[e \log n] \subseteq ZPP$, thus giving a barrier to any improvement beyond $LCL[O(1)] = CL$. We also show equivalent results for non-deterministic and randomized catalytic space.

Fully Characterizing Lossy Catalytic Computation

TL;DR

This work completely characterize lossy catalytic space () in terms of ordinary catalytic space (LCL[e] = CLSPACE[e \log n] \subseteq ZPPLCL[O(1)] = CL$.

Abstract

A catalytic machine is a model of computation where a traditional space-bounded machine is augmented with an additional, significantly larger, "catalytic" tape, which, while being available as a work tape, has the caveat of being initialized with an arbitrary string, which must be preserved at the end of the computation. Despite this restriction, catalytic machines have been shown to have surprising additional power; a logspace machine with a polynomial length catalytic tape, known as catalytic logspace (), can compute problems which are believed to be impossible for . A fundamental question of the model is whether the catalytic condition, of leaving the catalytic tape in its exact original configuration, is robust to minor deviations. This study was initialized by Gupta et al. (2024), who defined lossy catalytic logspace () as a variant of where we allow up to errors when resetting the catalytic tape. They showed that for any , which remains the frontier of our understanding. In this work we completely characterize lossy catalytic space () in terms of ordinary catalytic space (). We show that In other words, allowing errors on a catalytic tape of length is equivalent, up to a constant stretch, to an equivalent errorless catalytic machine with an additional bits of ordinary working memory. As a consequence, we show that for any , implies , thus giving a barrier to any improvement beyond . We also show equivalent results for non-deterministic and randomized catalytic space.
Paper Structure (20 sections, 21 theorems, 22 equations, 4 algorithms)

This paper contains 20 sections, 21 theorems, 22 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Catalytic computation
  3. Lossy catalytic computation
  4. Our results
  5. Open problems
  6. Errors in expectation.
  7. Randomized error-prone catalytic space.
  8. Lossy catalytic branching programs.
  9. Exact Simulation Space Requirements
  10. Preliminaries
  11. Main theorem
  12. Simulating errors with space
  13. Simulating space with errors
  14. Further consequences
  15. Lossy catalytic logspace with superconstant errors
  16. ...and 5 more sections

Key Result

Theorem 1

Let $s := s(n), c := c(n), e := e(n)$ be such that $e \leq c^{1-\Omega(1)}$. Then

Theorems & Definitions (50)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Definition 1: Catalytic space
  • Definition 2: Lossy catalytic space
  • Definition 3
  • Theorem 4
  • proof
  • Lemma 5
  • Theorem 6
  • ...and 40 more