ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-02

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.

Lexicographic certificate search uses polynomial space and exponential time

Example

Suppose a language $L$ has a verifier whose certificates have length at most $n^{k}$ . A deterministic search through those certificates illustrates why the containment $N P \subseteq P S P A C E$ can hold even though the resulting algorithm may still take exponential time.

Facts & Assumptions

Given: A verifier with certificate length bound $n^{k}$ .

[L1]

The verifier search problem asks for a certificate witnessing acceptance, by The search problem attached to an NP verifier.

[L2]

Every language in $N P$ lies in $P S P A C E$ , and every language in $P S P A C E$ lies in $E X P$ , by $N P \subseteq P S P A C E \subseteq E X P$ .

Verification

technique · direct

1.1

Enumerate all binary certificates of lengths at most $n^{k}$ in lexicographic order and run the verifier on each until one succeeds. At every stage the algorithm stores only the input, the current certificate, and the verifier workspace, so the space usage is polynomial. This is the concrete mechanism behind the first inclusion in [L2].

L1L2givenconstruct

2.1

The same procedure may test exponentially many candidate certificates before success or exhaustion, so its running time can be exponential even while its space stays polynomial. That is exactly the gap displayed by the chain in [L2].

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Michael Sipser, MIT 18.404J Theory of Computation, Lecture 17: Space Complexity, PSPACE, Savitch's Theorem (standard reference, not scraped)