Essential arity gap of Boolean functions

We investigate the Boolean functions with essential arity gap 2. We use Full Conjunctive Normal Forms instead of Zhegalkin’s polynomials, which allow us to simplify the proofs and to obtain several combinatorial results, concerning the Boolean functions with a given arity gap.

💡 Research Summary

The paper investigates Boolean functions whose essential arity gap (the difference between the number of essential variables of a function and the maximal number of essential variables among its identification minors) equals two. The authors replace the traditional Zhegalkin polynomial representation with the Full Conjunctive Normal Form (FCNF), which expresses a Boolean function as a XOR‑sum of conjunctions that involve all variables. This change simplifies the proofs and makes the combinatorial structure of the functions more transparent.

After defining essential variables, fictitious (non‑essential) variables, and the identification minor operation fᵢ←ⱼ (obtained by equating two variables), the arity gap is formally introduced as gap(f)=ess(f)−max_{g∈M(f)}ess(g). It is known that for Boolean functions gap(f)≤2, and the paper focuses on the case where the bound is attained.

Two key lemmas are proved: (1) if two functions have the same identification minor for a pair of variables and the corresponding binary strings agree in the positions of those variables, then the coefficients of the matching monomials in their FCNF are equal; (2) when n≥3, if every pairwise identification minor of two functions coincides, then the functions themselves are identical. These lemmas allow the authors to reason about the structure of functions with gap = 2 by examining their FCNF coefficients.

Proposition 3.1 establishes that the arity gap is invariant under complementing all inputs, complementing the output, and permuting variables. This symmetry lets the authors classify functions up to these transformations.

The main constructive result (Proposition 3.2) shows that for any n≥4, the functions obtained by XOR‑summing all monomials whose binary index has an odd number of 1’s (the set Oddₙ) or an even number of 1’s (the set Evenₙ) have arity gap exactly two. In other words, f = ⊕{α∈Oddₙ} x{α₁}…x_{αₙ} and f = ⊕{α∈Evenₙ} x{α₁}…x_{αₙ} belong to the class G_{n,2}.

The paper then gives a complete enumeration for small arities. For n=2, Theorem 3.1 characterises G_{2,2} as the six functions of the form
a₀·(x₀₁x₀₂⊕x₁₁x₁₂)⊕a₁·x₀₁x₁₂⊕a₂·x₁₁x₀₂ with a₁≠a₀ or a₂≠a₀.
For n=3, Theorem 3.2 shows that any function with gap = 2 can be written, after a suitable permutation of variables, in one of two canonical forms:

1. f = x_{α₃}(x₀₁x₁₂⊕x₁₁x₀₂)⊕x_{β₁}x_{β₂}, or
2. f = x_{α₃}(x₀₁x₀₂⊕x₁₁x₁₂)⊕x_{¬α₃}(x₀₁x₁₂⊕x₁₁x₀₂),

where α,β∈{0,1}. Counting permutations and the two possible choices of α yields exactly ten non‑equivalent functions in G_{3,2}.

For n=4 and higher, Lemma 3.1 proves that if a function f = x₀₄·g(x₁,x₂,x₃)⊕x₁₄·h(x₁,x₂,x₃) belongs to G_{4,2}, then every identification minor of the three‑variable subfunctions g and h must have fewer than two essential variables. This condition forces g and h to be of the Odd₃ or Even₃ type, and consequently f reduces to the general Oddₙ/Evenₙ construction described earlier.

The authors also provide explicit counts: |G_{2,2}|=6 and |G_{3,2}|=10, confirming that the combinatorial classification is exhaustive.

Overall, the paper demonstrates that using FCNF yields cleaner proofs and a transparent combinatorial picture of Boolean functions with maximal arity gap. The results have practical implications: functions with a large gap admit simpler circuit realizations because identifying variables reduces the essential arity by two in a single step. This insight can be leveraged in logic synthesis, optimization of switching circuits, and the study of functional schemas in theoretical computer science.