Symmetric Boolean Function

In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones (or zeros) in the input.^[1] For this reason they are also known as Boolean counting functions.^[2] There are 2ⁿ⁺¹ symmetric n-ary Boolean functions. Instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an n-variable symmetric Boolean function: the (n + 1)-vector, whose i-th entry (i = 0, ..., n) is the value of the function on an input vector with i ones. Mathematically, the symmetric Boolean functions correspond one-to-one with the functions that map n+1 elements to two elements, [math]\displaystyle{ f: \{0, 1, ..., n\} \rightarrow \{0, 1\} }[/math].

Symmetric Boolean functions are used to classify Boolean satisfiability problems.

Special cases

A number of special cases are recognized:^[1]

Majority function: their value is 1 on input vectors with more than n/2 ones
Threshold functions: their value is 1 on input vectors with k or more ones for a fixed k
All-equal and not-all-equal function: their values is 1 when the inputs do (not) all have the same value
Exact-count functions: their value is 1 on input vectors with k ones for a fixed k
- One-hot or 1-in-n function: their value is 1 on input vectors with exactly one one
- One-cold function: their value is 1 on input vectors with exactly one zero
Congruence functions: their value is 1 on input vectors with the number of ones congruent to k mod m for fixed k, m
Parity function: their value is 1 if the input vector has odd number of ones

The n-ary versions of AND, OR, XOR, NAND, NOR and XNOR are also symmetric Boolean functions.

Properties

In the following, [math]\displaystyle{ f_k }[/math] denotes the value of the function [math]\displaystyle{ f: \{0, 1\}^n \rightarrow \{0, 1\} }[/math] when applied to an input vector of weight [math]\displaystyle{ k }[/math].

Weight

The weight of the function can be calculated from its value vector:

[math]\displaystyle{ |f| = \sum_{k=0}^n \binom{n}{k}f_k }[/math]

Algebraic normal form

The algebraic normal form either contains all monomials of certain order [math]\displaystyle{ m }[/math], or none of them; i.e. the Möbius transform [math]\displaystyle{ \hat f }[/math] of the function is also a symmetric function. It can thus also be described by a simple (n+1) bit vector, the ANF vector [math]\displaystyle{ \hat f_m }[/math]. The ANF and value vectors are related by a Möbius relation:[math]\displaystyle{ \hat f_m = \bigoplus_{k_2 \subseteq m_2} f_k }[/math]where [math]\displaystyle{ k_2 \subseteq m_2 }[/math] denotes all the weights k whose base-2 representation is covered by the base-2 representation of m (a consequence of Lucas’ theorem).^[3] Effectively, an n-variable symmetric Boolean function corresponds to a log(n)-variable ordinary Boolean function acting on the base-2 representation of the input weight.

For example, for three-variable functions:

[math]\displaystyle{ \begin{array}{lcl}\hat f_0 & = & f_0 \\ \hat f_1 & = & f_0 \oplus f_1 \\ \hat f_2 & = & f_0 \oplus f_2 \\ \hat f_3 & = & f_0 \oplus f_1 \oplus f_2 \oplus f_3 \end{array} }[/math]

So the three variable majority function with value vector (0, 0, 1, 1) has ANF vector (0, 0, 1, 0), i.e.:[math]\displaystyle{ \text{Maj}(x, y, z) = xy \oplus xz \oplus yz }[/math]

Unit hypercube polynomial

The coefficients of the real polynomial agreeing with the function on [math]\displaystyle{ \{0, 1\}^n }[/math] are given by:[math]\displaystyle{ f^*_m = \sum_{k = 0}^m (-1)^{|k|+|m|} \binom{m}{k} f_k }[/math]For example, the three variable majority function polynomial has coefficients (0, 0, 1, -2):[math]\displaystyle{ \text{Maj}(x, y, z) = (xy + xz + yz) -2(xyz) }[/math]

Examples

The 16 symmetric Boolean functions of three variables
Function value				Value vector	Weight	Name	Colloquial description	ANF vector
0	1	2	3	Value vector	Weight	Name	Colloquial description	ANF vector
F	F	F	F	(0, 0, 0, 0)	0	Constant false	"never"	(0, 0, 0, 0)
F	F	F	T	(0, 0, 0, 1)	1	Three-way AND, Threshold(3), Count(3)	"all three"	(0, 0, 0, 1)
F	F	T	F	(0, 0, 1, 0)	3	Count(2), One-cold	"exactly two"	(0, 0, 1, 1)
F	F	T	T	(0, 0, 1, 1)	4	Majority, Threshold(2)	"most", "at least two"	(0, 0, 1, 0)
F	T	F	F	(0, 1, 0, 0)	3	Count(1), One-hot	"exactly one"	(0, 1, 0, 1)
F	T	F	T	(0, 1, 0, 1)	4	Three-way XOR, (odd) parity	"one or three"	(0, 1, 0, 0)
F	T	T	F	(0, 1, 1, 0)	6	Not-all-equal	"one or two"	(0, 1, 1, 0)
F	T	T	T	(0, 1, 1, 1)	7	Three-way OR, Threshold(1)	"any", "at least one"	(0, 1, 1, 1)
T	F	F	F	(1, 0, 0, 0)	1	Three-way NOR, Count(0)	"none"	(1, 1, 1, 1)
T	F	F	T	(1, 0, 0, 1)	2	All-equal	"all or none"	(1, 1, 1, 0)
T	F	T	F	(1, 0, 1, 0)	4	Three-way XNOR, even parity	"none or two"	(1, 1, 0, 0)
T	F	T	T	(1, 0, 1, 1)	5		"not exactly one"	(1, 1, 0, 1)
T	T	F	F	(1, 1, 0, 0)	4	(Horn clause)	"at most one"	(1, 0, 1, 0)
T	T	F	T	(1, 1, 0, 1)	5		"not exactly two"	(1, 0, 1, 1)
T	T	T	F	(1, 1, 1, 0)	7	Three-way NAND	"at most two"	(1, 0, 0, 1)
T	T	T	T	(1, 1, 1, 1)	8	Constant true	"always"	(1, 0, 0, 0)

References

↑ ^1.0 ^1.1 Ingo Wegener, "The Complexity of Symmetric Boolean Functions", in: Computation Theory and Logic, Lecture Notes in Computer Science, vol. 270, 1987, pp. 433–442
↑ "BooleanCountingFunction—Wolfram Language Documentation". https://reference.wolfram.com/language/ref/BooleanCountingFunction.html.en.
↑ Canteaut, A.; Videau, M. (2005). "Symmetric Boolean functions". IEEE Transactions on Information Theory 51 (8): 2791–2811. doi:10.1109/TIT.2005.851743. ISSN 1557-9654. https://hal.inria.fr/inria-00001148/document.