The Structure of In-Place Space-Bounded Computation

2025-10-13

AI Summary

This paper investigates the structural complexity of multi-output functions computable in logarithmic space by single-pass, read-write, in-place Turing machines. It formally defines the new complexity class *inplaceFL* (and its extension *inplaceFCL*) and systematically characterizes their computational power. Using relativization, circuit complexity analysis, and catalytic computation models, the paper establishes several unconditional and conditional separations under standard complexity assumptions: (1) *FL* ⊈ *inplaceFL*; (2) integer multiplication and NC⁰₄ circuit evaluation are not in *inplaceFL*, assuming cryptographic hardness; (3) NC⁰₂ circuit evaluation admits an *inplaceFL* algorithm; (4) matrix multiplication and inversion over finite fields are in *inplaceFCL*. This work provides the first structural complexity-theoretic characterization of inherent limitations of in-place computation, reveals a novel barrier to proving *CL* ⊆ *P*, and delivers foundational insights into computability under strict space constraints.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

In the standard model of computing multi-output functions in logspace ($mathsf{FL}$), we are given a read-only tape holding $x$ and a logarithmic length worktape, and must print $f(x)$ to a dedicated write-only tape. However, there has been extensive work (both in theory and in practice) on algorithms that transform $x$ into $f(x)$ in-place on a single read-write tape with limited (in our case $O(log n)$) additional workspace. We say $fin mathsf{inplaceFL}$ if $f$ can be computed in this model. We initiate the study of in-place computation from a structural complexity perspective, proving upper and lower bounds on the power of $mathsf{inplaceFL}$. We show the following: i) Unconditionally, $mathsf{FL} otsubseteq mathsf{inplaceFL}$. ii) The problems of integer multiplication and evaluating $mathsf{NC}^0_4$ circuits lie outside $mathsf{inplaceFL}$ under cryptographic assumptions. However, evaluating $mathsf{NC}^0_2$ circuits can be done in $mathsf{inplaceFL}$. iii) We have $mathsf{FL} subseteq mathsf{inplaceFL}^{mathsf{STP}}.$ Consequently, proving $mathsf{inplaceFL} otsubseteq mathsf{FL}$ would imply $mathsf{SAT} otin mathsf{L}$. We also consider the analogous catalytic class ($mathsf{inplaceFCL}$), where the in-place algorithm has a large additional worktape tape that it must reset at the end of the computation. We give $mathsf{inplaceFCL}$ algorithms for matrix multiplication and inversion over polynomial-sized finite fields. We furthermore use our results and techniques to show two novel barriers to proving $mathsf{CL} subseteq mathsf{P}$. First, we show that any proof of $mathsf{CL}subseteq mathsf{P}$ must be non-relativizing, by giving an oracle relative to which $mathsf{CL}^O=mathsf{EXP}^O$. Second, we identify a search problem in $mathsf{searchCL}$ but not known to be in $mathsf{P}$.

Problem

• Studies structural complexity of in-place logspace computation models
• Proves limitations on computing functions with limited workspace
• Analyzes relationships between in-place classes and standard complexity classes

Innovation

• In-place computation with logarithmic workspace
• Single read-write tape transformation model
• Structural complexity analysis of space-bounded computation