[Paper Review] On a quasi-ordering on Boolean functions

Miguel Couceiro, Maurice Pouzet|ArXiv.org|Jan 10, 2006

Advanced Algebra and Logic3 references4 citations

TL;DR

This paper establishes that the poset of equivalence classes of Boolean functions under a quasi-ordering based on variable operations (identification, permutation, addition) embeds both into and from the poset of finite subsets of natural numbers ordered by inclusion. It proves that equational classes of Boolean functions—defined by finitely many functional equations—coincide with initial segments of this quasi-ordered set, and provides an explicit equation characterizing linear functions with at most k essential variables.

ABSTRACT

It was proved few years ago that classes of Boolean functions definable by means of functional equations \cite{EFHH}, or equivalently, by means of relational constraints \cite{Pi2}, coincide with initial segments of the quasi-ordered set $(Ω, \leq)$ made of the set $Ω$ of Boolean functions, suitably quasi-ordered. The resulting ordered set $(Ω/\equiv, \sqsubseteq)$ embeds into $([ω]^{

Motivation & Objective

• To establish a deep structural connection between the quasi-ordering of Boolean functions and the inclusion order on finite subsets of ω.
• To characterize equational classes of Boolean functions as initial segments of the quasi-ordered set (Ω, ≤), and show they coincide with classes defined by finitely many obstructions.
• To provide a concrete equational characterization of the class of linear Boolean functions with at most k essential variables.
• To demonstrate that the poset (Ω/≡, ⊑) is both embeddable into and embeds the poset ([ω]^{<ω}, ⊆), revealing a rich combinatorial structure.
• To show that classes of Boolean functions with bounded essential variables are finitely axiomatizable via functional equations.

Proposed method

• Define a quasi-order ≤ on Boolean functions where g ≤ f if g can be obtained from f by variable identification, permutation, or addition.
• Define an equivalence relation ≡ on Ω by f ≡ g iff f ≤ g and g ≤ f, leading to the quotient poset (Ω/≡, ⊑).
• Prove that each initial segment ↓x in (Ω/≡, ⊑) is finite, enabling a level-wise decomposition of the poset into finite levels Ω/≡_n.
• Use the duality between initial segments and antichains: every initial segment I corresponds to a unique minimal antichain A such that I = Forbid(A).
• Construct an explicit order-embedding φ: ([ω]^{<ω}, ⊆) → (Ω/≡, ⊑) by mapping each finite subset I to a function g_I' derived from a construction involving variable substitution.
• Use linear algebra over the 2-element field to derive a functional equation that characterizes linear functions with at most k essential variables, based on parity and intersection properties of vectors.

Experimental results

Research questions

• RQ1 Can the poset (Ω/≡, ⊑) be embedded into ([ω]^{<ω}, ⊆), and does it also embed ([ω]^{<ω}, ⊆) back into it?
• RQ2 Are equational classes of Boolean functions—defined by functional equations—precisely the initial segments of the quasi-ordered set (Ω, ≤)?
• RQ3 Is the class of linear Boolean functions with at most k essential variables definable by a single functional equation?
• RQ4 What is the relationship between finite antichains of obstructions and finitely axiomatizable equational classes?
• RQ5 Can the structure of the poset (Ω/≡, ⊑) be fully characterized via level decomposition and finite initial segments?

Key findings

• The poset (Ω/≡, ⊑) embeds both into and from the poset ([ω]^{<ω}, ⊆), establishing a bi-embeddability between the two structures.
• Every initial segment of (Ω, ≤) that is definable by finitely many obstructions is also definable by a single functional equation, confirming a strong equivalence between obstruction-based and equation-based definitions.
• The class of linear functions with at most k essential variables is axiomatized by a single equation involving the sum of variables and parity conditions on their products.
• The equation characterizing L^k (linear functions with ≤k essential variables) is: (f(0) ∧ ⋀_{1≤i≤k+1} f(x_i) → ⋁_{j<l} f(x_j x_l)) ∨ (f(0) ∧ ⋀_{1≤i≤k+1} (f(x_i)+1) → ⋁_{j<l} (f(x_j x_l)+1)) = 1.
• A key lemma is proven: if k+1 vectors of odd weight are given in an n-dimensional space over F_2 with n ≤ k, then at least one pair has an odd-weight component-wise product, which underpins the correctness of the equation for L^k.
• The class of Boolean functions with at most k essential variables is finitely axiomatizable, and this holds in particular for linear functions.