LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Logspace machines have polynomially many configurations

Statement

For a fixed read-only-input logspace machine $M$ , there is a polynomial $p_{M}$ such that on every input $x$ of length $n$ , $M$ has at most $p_{M} (n)$ reachable configurations. Consequently, any accepting branch has an accepting branch with no repeated configuration and hence with fewer than $p_{M} (n)$ transitions.

Facts & Assumptions

Given: a fixed read-only-input logspace machine $M$ and an input $x$ of length $n$ .

[L1]

A logspace machine charges only $O (\log_{2} (n + 2))$ visited work cells and has a read-only input tape (A read-only-input logarithmic-space machine).

Proof

technique · direct counting

1.1

By [L1], a configuration is specified by a state from a fixed finite set, one of $n + 2$ input-head positions, the contents of $O (\log (n + 2))$ work cells over fixed finite alphabets, and finitely many work-head positions among those cells.

L1construct

2.1

The product of these choices is $(n + 2)^{O (1)}$ ; enlarge its fixed constant to obtain a polynomial $p_{M} (n)$ bounding the reachable configurations.

step 1.1algebra

3.1

If an accepting branch repeats a configuration, delete the segment between its two occurrences; the identical later configuration has the same possible continuation. Repeating this deletion yields a simple accepting path with fewer than $p_{M} (n)$ transitions.

step 2.1cases∎

Depends on

A read-only-input logarithmic-space machine

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Arora and Barak, Computational Complexity, §3.1 (standard reference, not scraped)