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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Logspace reductions compose

Statement

If $A \leq_{\log} B$ and $B \leq_{\log} C$ , then $A \leq_{\log} C$ . In particular, L and NL are closed under inverse logspace many-one reductions: if $A \leq_{\log} B$ and $B$ is in the indicated class, then $A$ is too.

Facts & Assumptions

Given: logspace reductions $f : A \to B$ and $g : B \to C$ .

[L1]

A logspace reduction has polynomial output length and permits bitwise recomputation of its write-only output (Logspace many-one reduction).

Proof

technique · virtual-tape simulation

1.1

Simulate the transducer for $g$ on a virtual input tape containing $f (x)$ , retaining its work tape and its virtual input-head index $j$ .

givenconstruct

2.1

Whenever the simulation reads virtual cell $j$ , rerun the transducer for $f$ on $(x, j)$ and supply its $j$ th output bit; [L1] makes this a logarithmic-work-space subroutine.

L1step 1.1construct

3.1

Since $∣ f (x) ∣$ is polynomial in $∣ x ∣$ , both $j$ and the simulated work tape use $O (\log ∣ x ∣)$ space. The streamed output is $g (f (x))$ , so $x \in A$ iff it lies in $C$ .

L1step 2.1algebra

4.1

To decide $A$ from a deterministic or nondeterministic decider for $B$ , run that decider on the same virtual input $f (x)$ rather than the transducer for $g$ . The virtual-input access above preserves its deterministic or nondeterministic choice and its logarithmic work bound, proving the stated L or NL closure.

step 2.1step 3.1cases∎

Depends on

Used by

Composing two logspace reductions through a virtual tape Example

Dependency tree · two levels

4 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, Lemma 3.15 (standard reference, not scraped)