SLIDE 1 Associative and commutative tree representations for Boolean functions
Bernhard Gittenberger ∗
Joint work with Antoine Genitrini, Veronika Kraus and C´ ecile Mailler
Institute for Discrete Mathematics and Geometry Vienna University of Technology
AofA 2013, Menorca, May 29, 2013
∗Supported by the Austrian Science Foundation FWF , SFB F50
SLIDE 2
And/Or trees and the limiting probability
Take a set of Boolean connectors {∧, ∨} a set of literals {x1, ¯ x1, . . . , xn, ¯ xn} a family T of unlabelled trees, size m = # leaves And/Or tree: element t ∈ T ∗ internal vertices labelled with connectors ∗ leaves labelled with literals ⇒ represents a Boolean function f : {0, 1}n → {0, 1}
SLIDE 3
And/Or trees and the limiting probability
Take a set of Boolean connectors {∧, ∨} a set of literals {x1, ¯ x1, . . . , xn, ¯ xn} a family T of unlabelled trees, size m = # leaves And/Or tree: element t ∈ T ∗ internal vertices labelled with connectors ∗ leaves labelled with literals ⇒ represents a Boolean function f : {0, 1}n → {0, 1} limiting probability of f Pn(f) = lim
m→∞ Pm,n(f) = lim m→∞
Tm,n(f) Tm,n (if it exists)
SLIDE 4
Previous work
Catalan model (And/Or)
Lefmann, Savick´ y ’97 Chauvin, Gardy, Flajolet, G. ’04 Gardy ’06 Kozik ’08
Catalan model (implication)
Fournier, Gardy, Genitrini, G. ’08, ’12 Genitrini, G., Kraus, Mailler ’12
SLIDE 5 Models considered
classical (Catalan) model: binary, planar associative model: outdegree = 1, planar ”stratified”: neighbours cannot have the same label commutative model: binary, non-plane general (P´
- lya) model: outdegree = 1, nonplane, stratified
(=associative & commutative)
SLIDE 6
All limiting probabilities exist
Lemma In all models, the limiting probability Pn(f) exists for all f. Proof idea: Choose a model, ∗ T(z) =
m≥0 Tm,nzm generating function of trees
singularity ρn ∗ Tf(z) =
m≥0 Tm,n(f)zn GF of trees computing f
system of equations for Tf(z) ⇒ Drmota-Lalley-Woods theorem: same singularity ⇒ transfer lemma.
SLIDE 7 Example: Associative plane trees
A(z) = ˆ A(z) + ˇ A(z) − 2nz ˆ A(z) = 2nz +
ˇ A(z)k ˇ
A=ˆ A
= 2nz + ˆ A2(z) 1 − ˆ A(z) ⇒ A(z)= 1 2
- 1 − 2nz −
- 1 − 12nz + 4n2z2
- singularity: ρn = 3−2
√ 2 2n
. ˆ Af(z) = z l 1{f lit.} +
∞
g1∧···∧gi=f
ˇ Ag1(z) · · · ˇ Agi(z) ˇ Af(z) = z l 1{f lit.} +
∞
g1∨···∨gi=f
ˆ Ag1(z) · · · ˆ Agi(z).
SLIDE 8
Limiting probabilities
Theorem (Kozik, 2008) Let T be the family of binary planar And/Or trees. Then Pn(f) ∼ λf nL(f)+1 , as n → ∞. Theorem (Fournier, Gardy, Genitrini, G., 2008) Let T be the family of binary planar implication trees. Then Pn(f) ∼ ˜ λf nL(f)+1 , as n → ∞.
SLIDE 9 The limiting probabilities of constant functions
Theorem The limiting probability of constant functions in the different models is binary plane: associative: Pn(True) =
3 4n + O
1
n2
√ 2 n
+ O 1
n2
general: Pn(True) =
641 1024n + O
1
n2
4n
+ O 1
n2
SLIDE 10
Limiting distribution of general functions
Theorem For all models M of And/Or trees studied, and for all Boolean functions f, Pn(f) ∼ λM(f) nL(f)+1 , as n → ∞ where λM(f) is related to the # of possible expansions of a minimal tree of f.
SLIDE 11
Sketch of proof:
Ingredients: Simple tautologies Representation of trees by pattern languages Expansions by tautologies and literals are enough Asymptotically almost every tree computing f is a minimal tree expanded once. (Kozik ’08 for the Catalan model)
SLIDE 12 Sketch of proof: simple tautologies
simple tautology (realized by x): x ∨ ¯ x ∨ f
∨ ∨ ∨ ∨ ∨ ∨ x ∨ ∧ ¯ x
∨ x ¯ x
SLIDE 13
Sketch of proof: pattern languages
pattern language L: ∗ planar tree family, ∗ internal nodes labelled with connectors, ∗ leaves labelled by {•, }. together with T ⇒ L[T ]: ∗ ← element from T , ∗ • ← label ∈ {x1, ¯ x1, . . . , xn, ¯ xn}.
SLIDE 14 Example L[T ]
N = •|N ∨ N|N ∧
∨ ∧ ∨
SLIDE 15
Example L[T ]
N = •|N ∨ N|N ∧
∨ ∧ ∨ ∨ x1 ¯ x1 x4 x3 ∧ x2 ∧ x5 ¯ x3
SLIDE 16
Example L[T ]
N = •|N ∨ N|N ∧
∨ ∧ ∨ ∨ x1 ¯ x1 x4 ∧ x3 x2 ∧ x5 ¯ x3
SLIDE 17
Pattern languages for associative trees
R = { ˆ N, ˇ N} ˆ N = •|N ∧ |N ∧ ∧ | · · · ˇ N = •|N ∨ N|N ∨ N ∨ N| · · · x essential variable of f: f depends on x # repetitions = # pattern leaves −# distinct variables # of restrictions = # repetitions+# of essential variables property of N: set all pattern leaves to false ⇒ whole tree computes false. ⇒ ∃ repetition x/¯ x in tree computing True.
SLIDE 18 Simple tautologies
Proposition If L is subcritical w.r.t. T then lim
m→∞
L[T ](k)
m
Tm = O 1 nk
Lemma All tautologies with 1 L-restriction are simple. Proposition Asymptotically almost all tautologies are simple (and realized by exactly 1 variable), when m → ∞.
SLIDE 19 Limiting probability of simple tautologies
STx(z): gf of simple tautologies realized by x. ST(z) = nSTx(z). STx = {∨} × {x} × {¯ x} × set(T \ {x, ¯ x}) STx(z) = z2 ×
ℓ(ℓ − 1)(T(z) − 2z)ℓ−2
SLIDE 20 Limiting probability of simple tautologies
STx(z): gf of simple tautologies realized by x. ST(z) = nSTx(z). STx = {∨} × {x} × {¯ x} × set(T \ {x, ¯ x}) STx(z) = z2 ×
ℓ(ℓ − 1)(T(z) − 2z)ℓ−2 P(True) = lim
m→∞
[zm]ST(z) [zm]T(z) ∼ lim
z→ρ
ST ′(z) T ′(z) .
SLIDE 21
Commutative trees
∗ ∄ unambiguous pattern with T → L[T] → T. ∗ “mobiles”: halfembedding of T : at each pattern node choose left-right order ⇒ injection T → L[T ] ∗ minimal embedding: choose one with # restrictions = min. ∗ No subcriticality any more!
SLIDE 22 Mobiles
N = •|N ∨ N|N ∧
∨ ∧ ∨
SLIDE 23
Mobiles
N = •|N ∨ N|N ∧
∨ ∧ ∨ ∨ x1 ¯ x1 x4 x3 ∧ x2 ∧ x5 ¯ x3
SLIDE 24 Mobiles
N = •|N ∨ N|N ∧
∨ ∧ ∨
x1 x4 ∧ x3 x2
x5 ¯ x3
SLIDE 25
Outlook
No Shannon effect
disproved for Galton-Watson And/Or trees (Lefmann, Savick´ y, 1997) disproved for implication trees (Genitrini, G., 2010)
Characterize the class of functions which yields the total mass define size = # all vertices in associative models (Work in progress)
SLIDE 26
Outlook
No Shannon effect
disproved for Galton-Watson And/Or trees (Lefmann, Savick´ y, 1997) disproved for implication trees (Genitrini, G., 2010)
Characterize the class of functions which yields the total mass define size = # all vertices in associative models (Work in progress)
Muchas Gracias!
