Let \(\rho _1,\rho _2,\sigma \) be restrictions on \(n\) variables, and let \(\rho _1 \ast \rho _2\) denote the composition of \(\rho _1\) and \(\rho _2\), whose value at a coordinate \(i\) is the value fixed by \(\rho _1\) at \(i\) if \(\rho _1\) fixes it and otherwise the value \(\rho _2\) assigns at \(i\). Then \(\rho _1 \ast \rho _2 = \sigma \) if and only if, for every coordinate \(i \in \mathrm{Fin}\, n\), the value of \(\rho _1 \ast \rho _2\) at \(i\) equals \(\sigma (i)\).

Let \(\rho \) be a restriction on \(n\) variables, and let \(\varnothing \) denote the empty restriction that leaves every variable free. Then the composition of \(\rho \) with \(\varnothing \), applying \(\rho \) first and then \(\varnothing \) on the variables \(\rho \) leaves free, equals \(\rho \).

Let \(p\) be a real parameter and let \(\rho \) be a restriction on \(n\) variables, assigning each coordinate either a fixed Boolean value or leaving it free. Writing \(k\) for the number of free coordinates, the Bernoulli weight of \(\rho \) equals the product over all \(n\) coordinates of the per-variable weights:

where each factor \(w_p(\rho (i))\) equals \(p\) when coordinate \(i\) is free and \((1-p)/2\) when it is fixed.

Regard each coordinate of a random restriction as being in one of three states: *free* (written \(\ast \)) or *fixed* to one of the two Boolean values. For a parameter \(p \in \mathbb {R}\), let \(w_p\) be the per-variable weight assigning \(w_p(\ast ) = p\) to the free state and \((1-p)/2\) to each of the two fixed states, and define the composite \(a \circ b\) of two states to equal \(a\) when \(a\) is fixed and \(b\) when \(a\) is free. Then for all \(p, q \in \mathbb {R}\) and every state \(c\),

where both sums range over the three states and \([\, \cdot \, ]\) equals \(1\) when the enclosed equation holds and \(0\) otherwise.

Fix parameters \(p, q \in \mathbb {R}\) and a restriction \(\sigma \) on \(n\) variables. Summing the product of the Bernoulli weight of \(\rho _1\) at parameter \(p\) and the Bernoulli weight of \(\rho _2\) at parameter \(q\) over all pairs of restrictions \((\rho _1, \rho _2)\) on \(n\) variables whose composition equals \(\sigma \) yields the Bernoulli weight of \(\sigma \) at parameter \(pq\):

where \(w_r(\rho )\) denotes the Bernoulli weight of \(\rho \) at parameter \(r\), \(\rho _1 \circ \rho _2\) is the restriction that fixes each coordinate to the value given by \(\rho _1\) where defined and to that of \(\rho _2\) otherwise, and \([\, \cdot \, ]\) is the indicator taking the value \(1\) when its condition holds and \(0\) otherwise.