THE REAL ALGEBRAIC NORMAL FORM: FUZZY BOOLEAN LOGIC SATISFYING Δf = 0

We define real extensions f : ℝⁿ ⊃ [0, 1]ⁿ ⟶ [0, 1] ⊂ ℝ of Boolean functions that satisfy Δf = 0 on the whole domain [0, 1]ⁿ, i.e. potential theory as first principle.

We introduce the Real Algebraic Normal Form RANF ∈ ℤ[x₁,…, x_n] of f, a generalization of the Algebraic Normal Form ANF of , and show how the RANF is related to the Disjunctive Normal Form DNF by the Reed-Muller Transform, or the Inclusion-Exclusion Principle.