ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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.

Add/remove noisy count

Example

Assume Countable Choice. Let $U$ be a nonempty record universe and let databases be finite-support count functions $D : U \to N_{0}$ , with neighbors formed by adding or removing one occurrence. Let $P : U \to {0, 1}$ be a predicate satisfied by at least one possible record, and let $f (D)$ count the occurrences satisfying $P$ . Then the global add/remove $L^{1}$ sensitivity of $f$ is $1$ . For $ε > 0$ , releasing $f (D) + Z$ with $Z \sim Lap (1 / ε)$ is $(ε, 0)$ -differentially private.

Example references

Finite-support record-count databases and add/remove adjacency

Global L1 sensitivity

The Laplace mechanism privacy guarantee

Dwork and Roth, The Algorithmic Foundations of Differential Privacy, §3.3, printed pp. 33–37.

Vadhan, The Complexity of Differential Privacy, §1.5, PDF pp. 7–8.

Facts & Assumptions

Given: A predicate $P : U \to {0, 1}$ , a satisfying record $u_{0} \in U$ , and $ε > 0$ .

[A1]

Countable Choice is inherited from the real Laplace law; the count sensitivity proof uses no choice.

[F1]

Neighboring databases differ by adding or removing one record occurrence.

[F2]

Global sensitivity is the attained maximum of adjacent query differences.

[F3]

The scalar Laplace mechanism adds noise with scale $Δ_{1} f / ε$ for $Δ_{1} f > 0$ .

[F4]

Laplace privacy theorem gives $(ε, 0)$ -DP for every finite-dimensional query of finite global $L^{1}$ sensitivity when $ε > 0$ .

Verification

1.1F1F2givenalgebra

If $D^{'}$ is obtained from $D$ by adding one occurrence of $u$ , then $f (D^{'}) - f (D) = P (u) \in {0, 1}$ ; removing one occurrence gives the negative of this change. Thus every adjacent difference has absolute value at most $1$ . The empty database and the database containing only $u_{0}$ are neighbors and their counts differ by $1$ , so the maximum is attained and $Δ_{1} f = 1$ .

2.1A1F3F4step 1.1∎

By [F3], the scalar release uses scale $Δ_{1} f / ε = 1 / ε$ and is exactly the stated noisy count. This query has finite dimension one and sensitivity one, so [F4] proves its $(ε, 0)$ guarantee.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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

Cynthia Dwork and Aaron Roth, The Algorithmic Foundations of Differential Privacy (standard reference, not scraped)
Salil Vadhan, The Complexity of Differential Privacy, 2017 manuscript (standard reference, not scraped)