ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 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.

Composing two logspace reductions through a virtual tape

Example

Work over $Σ = {0, 1, #}$ . Let $f (x) = 0 x # x^{r e v}$ . Let $g (y)$ output the first and last symbols of $y$ when $y \neq ϵ$ , and set $g (ϵ) = 00$ . Both total functions are deterministic logspace transductions. To compute $g (f (x))$ , a composed machine asks for virtual cells $1$ and $∣ f (x) ∣$ rather than storing the string $f (x)$ . Let $A$ be the words starting with $1$ , let $B$ be the nonempty words starting with $0$ and ending with $1$ , and let $C = {01}$ . Then $f$ reduces $A$ to $B$ and $g$ reduces $B$ to $C$ .

Facts & Assumptions

Given: an input $x$ of length $n$ .

[L1]

A reduction output can be recomputed at a requested position in logarithmic work space (Logspace many-one reduction).

[L2]

If $A \leq_{\log} B$ and $B \leq_{\log} C$ , then $A \leq_{\log} C$ (Logspace reductions compose).

Verification

technique · direct computation

1.1

The first requested virtual symbol is $0$ . Maintain a counter to find $∣ f (x) ∣ = 2 n + 2$ and recompute the final virtual symbol, which is $x_{1}$ when $n > 0$ and $#$ when $n = 0$ . Forward and backward scans emit $f$ , while two endpoint scans emit $g$ ; both use at most logarithmic work space and halt on every input.

L1construct

2.1

Thus the composed output is $0 x_{1}$ for $n > 0$ and $0 #$ for $n = 0$ , exactly $g (f (x))$ ; it used only counters and a reread of $x$ .

L1step 1.1cases

3.1

The word $f (x)$ lies in $B$ exactly when $x$ starts with $1$ , and $g (y) = 01$ exactly when $y$ starts with $0$ and ends with $1$ ; the empty input maps to $00$ . Thus these are total logspace reductions $A \leq_{\log} B \leq_{\log} C$ , and the explicit virtual computation illustrates their composition in [L2].

L2step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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, Figure 3.3 (standard reference, not scraped)