linearly definable classes of boolean functions...zhegalkin polynomial of f : f0;1gn!f0;1g x s2mf x...

Linearly definable classes of Boolean functions

Erkko Lehtonen

Centro de Matemática e Aplicações, Faculdade de Ciências e TecnologiaUniversidade Nova de Lisboa

joint work with Miguel Couceiro (Loria)

ALGOS 2020

26–28 August 2020

E. Lehtonen (CMA/UNL) Linearly definable classes ALGOS 2020 1 / 13

Composition of functions and function classes

composition of functionsf : Bn → C, g1, . . . ,gn : Am → Bf (g1, . . . ,gn) : Am → Cf (g1, . . . ,gn)(a) := f (g1(a), . . . ,gn(a)) for all a ∈ Am

composition of function classesC ⊆ FBC , K ⊆ FAB

CK := f (g1, . . . ,gn) | f ∈ C(n), g1, . . . ,gn ∈ K (m), n,m ∈ N+


Clones, stability

A class C ⊆ OA is a clone if IA ⊆ C and CC ⊆ C.

Let K ⊆ FAB, C1 ⊆ OB, C2 ⊆ OA.

K is stable under left composition with C1 if C1K ⊆ K .

K is stable under right composition with C2 if KC2 ⊆ K .

If C1K ⊆ K and KC2 ⊆ K , we say that K is (C1,C2)-stable.

C-stable is short for (C,C)-stable.


Stability

Special cases of (C1,C2)-stability (C1 and C2 clones):– both C1 and C2 are clones of projections: minor-closed classes– C1 is the clone of projections on B: C2-minor-closed classes– C2 is the clone of projections on A: clonoids

N. PIPPENGER, Galois theory for minors of finite functions, Discrete Math. 254 (2002) 405–419.

E. AICHINGER, P. MAYR, Finitely generated equational classes, J. Pure Appl. Algebra 220 (2016)2816–2827.

A. SPARKS, On the number of clonoids, Algebra Universalis 80(4) (2019) Paper No. 53, 10 pp.E. Lehtonen (CMA/UNL) Linearly definable classes ALGOS 2020 4 / 13

Stability

Theorem (Pippenger (2002), Couceiro, Foldes (2005))Let A and B be arbitary nonempty sets, and let F ⊆ FAB. Then F islocally closed and closed under minors if and only if F is definable bysome set of relation pairs.

Theorem (Couceiro, Foldes (2009))Let A and B be arbitrary nonempty sets, and let C1 and C2 be cloneson B and A, respectively. Let F ⊆ FAB. Then F is locally closed and(C1,C2)-stable if and only if F is definable by some set of relation pairs(R,S) where R is invariant under C2 and S is invariant under C1.

N. PIPPENGER, Galois theory for minors of finite functions, Discrete Math. 254 (2002) 405–419.

M. COUCEIRO, S. FOLDES, On closed sets of relational constraints and classes of functionsclosed under variable substitutions, Algebra Universalis 54 (2005) 149–165.

M. COUCEIRO, S. FOLDES, Function classes and relational constraints stable undercompositions with clones, Discuss. Math. Gen. Algebra Appl. 29 (2009) 109–121.



functional equation

h1(f(g1(v1, . . . ,vp)), . . . , f(gm(v1, . . . ,vp)))

= h2(f(g′1(v1, . . . ,vp)), . . . , f(g′

t (v1, . . . ,vp)))

linearly definable classdefinable by a functional equation in which all functions hi , gj , g′

k arelinear

Theorem (Couceiro, Foldes (2004))A class of Boolean functions is linearly definable if and only if it isLc-stable.

O. EKIN, S. FOLDES, P. L. HAMMER, L. HELLERSTEIN, Equational characterizations of Booleanfunction classes, Discrete Math. 211 (2000) 27–51.

M. COUCEIRO, S. FOLDES, Definability of Boolean function classes by linear equations overGF(2), Discret Appl. Math. 142 (2004) 29–34.


Post’s lattice

Ic

I∗

I0I1

Lc

LS L0L1

LSM

Sc

S

M

˜cVc

TcT0T1

Ω



How many linearly definable, or, equivalently, Lc-stable, classes ofBoolean functions are there? What are they?

Theorem (Sparks (2019))Let A be a finite set with |A| > 1, and let B := 0,1. Let C be a cloneon B, and denote by JA the clone of projections on A. Then thefollowing statements hold.

1 L(C,JA) is finite if and only if C contains a near-unanimity operation.2 L(C,JA) is countably infinite if and only if C contains a Mal’cev

operation but no majority operation.3 L(C,JA) has the cardinality of the continuum if and only if C

contains neither a near-unanimity operation nor a Mal’cevoperation.

A. SPARKS, On the number of clonoids, Algebra Universalis 80(4) (2019) Paper No. 53, 10 pp.E. Lehtonen (CMA/UNL) Linearly definable classes ALGOS 2020 8 / 13

Lc-stable classes

Ω

P0 C0 E0 E1 C1 P1

C0 ∩ E0 C1 ∩ E1 C0 ∩ E1 C1 ∩ E0

Ω = all Boolean functionsCa = f ∈ Ω | f (0, . . . ,0) = a Ea = f ∈ Ω | f (1, . . . ,1) = a P0 = f ∈ Ω | f (0, . . . ,0) = f (1, . . . ,1) P1 = f ∈ Ω | f (0, . . . ,0) 6= f (1, . . . ,1)


Lc-stable classes

Zhegalkin polynomial of f : 0,1n → 0,1∑S∈Mf

xS xS :=∏i∈S

xi

degree of f : deg(f ) := maxS∈Mf

|S|

characteristic of A ⊆ [n] in f : ch(A, f ) := |S ∈ Mf | A ( S | mod 2

characteristic rank of f :χ(f ) := the smallest integer m such that ch(A, f ) = 0 for all subsetsA ⊆ [n] with |A| ≥ m

For i , j ∈ N: Di := f ∈ Ω | deg(f ) ≤ i Xj := f ∈ Ω | χ(f ) ≤ j


Lc-stable classes

D1

D2 ∩ X1

D2 D3 ∩ X1

D3 ∩ X2

D3

X1

X2

X3

Ω

P0 C0 E0 E1 C1 P1

Tc

X0 S

Sc

L0 L1 LS

Lc

D0

∅


Conclusive remarks and related work

One can quite easily determine from the current results also the(C1,C2)-stable classes of Boolean functions where C1 and C2 containLc, and in fact, those where C1 contains Lc and C2 is arbitrary.

Future work(C1,C2)-stability for other pairs of clones C1, C2

Related studiesS. FIORAVANTI, Closed sets of finitary functions between finite fields ofcoprime order, arXiv:1910.11759.S. KREINECKER, Closed function sets on groups of prime order, J.Mult.-Valued Logic Soft Comput. 33 (2019) 51–74.


The End

Thank you for your attention.


linearly definable classes of boolean functions...zhegalkin polynomial of f : f0;1gn!f0;1g x s2mf x...

Documents