Statement

A statement is a logical expression that contains one or more variables and that can be either true or false. A statement is abstract in the sense that the statement alone do not have any value, but specify a set of logic rules that can be used to determine if the statement is true or false for given values of the variables.

Statement can often be expressed in natural language, for example "Over 18 years old" is a statement that contains the variable age, and that is true if and only if age > 18.

Formalisation

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Let consider $s$ a statement as a function defined on $n$ real variables:

We can assign the values for each variables of a statement as a vector $v = (v_1, v_2, ..., v_n)$, with $v_i$ the value of the $i$-th variable. We note $V$ the set of all possible vectors possible, $V = \mathbb{R}^n$. So $\forall v, v \in V$.

The logic rules are represented as a set $\cal R$ of logical propositions. Each rule $r_i$ can involves any variables $v_i$ (atleast one), and can only be expressed using logical operators ($\land$, $\lor$, $\lnot$) and/or comparison operators ($=$, $>$, $<$, $\geq$, $\leq$).

The function $s$ can be defined as the following:

Example:

A statement is analogous to a predicate in logic.

Some observations:

• If there is two rules that are mutually exclusive, then $s$ cannot be true for any $v$.
• If there's a variable $v_i$ that is not used in any rule, then $s$ is equivalent to the statement that does not contain $v_i$ but has the same set of rules.

Zero-knowledge statement

We can divide the set $V$ as the union of two distinct sets: $V_{private}$ and $V_{public}$, such that $V_{private} \cap V_{public} = \emptyset$ and $V_{private} \cup V_{public} = V$.

We redefine the statement function $s$ as the prover that returns a new proof $p \in P$:

$\large s: (V_{private}, V_{public}) \rightarrow P$, with $P$ the set of all possible proofs of the statement $s$.

We say that $p$ is an Attestation proof.

Example: We can rewrite the previous statement as