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
And/Or trees and the limiting probability Take a set of Boolean connectors {∧ , ∨} a set of literals { x 1 , ¯ x 1 , . . . , x n , ¯ x n } 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 }
And/Or trees and the limiting probability Take a set of Boolean connectors {∧ , ∨} a set of literals { x 1 , ¯ x 1 , . . . , x n , ¯ x n } 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 T m , n ( f ) P n ( f ) = lim m →∞ P m , n ( f ) = lim (if it exists) T m , n m →∞
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
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´ olya) model: outdegree � = 1, nonplane, stratified (=associative & commutative)
All limiting probabilities exist Lemma In all models, the limiting probability P n ( f ) exists for all f. Proof idea: Choose a model, m ≥ 0 T m , n z m generating function of trees ∗ T ( z ) = � singularity ρ n m ≥ 0 T m , n ( f ) z n GF of trees computing f ∗ T f ( z ) = � system of equations for T f ( z ) ⇒ Drmota-Lalley-Woods theorem: same singularity ⇒ transfer lemma.
Example: Associative plane trees A ( z ) = ˆ A ( z ) + ˇ A ( z ) − 2 nz ˆ A 2 ( z ) A ( z ) k ˇ A =ˆ A ˆ � ˇ A ( z ) = 2 nz + = 2 nz + 1 − ˆ A ( z ) k ≥ 2 ⇒ A ( z )= 1 � � � 1 − 12 nz + 4 n 2 z 2 1 − 2 nz − 2 √ singularity: ρ n = 3 − 2 2 . 2 n ∞ ˆ � � A g 1 ( z ) · · · ˇ ˇ A f ( z ) = z l 1 { f lit. } + A g i ( z ) g 1 ,..., g i , i = 2 g 1 ∧···∧ g i = f ∞ ˇ � � A g 1 ( z ) · · · ˆ ˆ A f ( z ) = z l 1 { f lit. } + A g i ( z ) . g 1 ,..., g i , i = 2 g 1 ∨···∨ g i = f
Limiting probabilities Theorem (Kozik, 2008) Let T be the family of binary planar And/Or trees. Then λ f P n ( f ) ∼ n L ( f )+ 1 , as n → ∞ . Theorem (Fournier, Gardy, Genitrini, G., 2008) Let T be the family of binary planar implication trees. Then ˜ λ f P n ( f ) ∼ n L ( f )+ 1 , as n → ∞ .
The limiting probabilities of constant functions Theorem The limiting probability of constant functions in the different models is binary plane: associative: √ � 1 � 1 3 P n ( True ) = 51 − 36 2 P n ( True ) = � � 4 n + O + O n 2 n n 2 binary commutative: general: � 1 � 1 P n ( True ) = ( 2 ln 2 − 1 ) 2 641 P n ( True ) = � � 1024 n + O + O n 2 4 n n 2
Limiting distribution of general functions Theorem For all models M of And/Or trees studied, and for all Boolean functions f, P n ( f ) ∼ λ M ( f ) n L ( f )+ 1 , as n → ∞ where λ M ( f ) is related to the # of possible expansions of a minimal tree of f.
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)
Sketch of proof: simple tautologies simple tautology (realized by x ): x ∨ ¯ x ∨ f ∨ ∨ ∨ x ¯ x ∨ ∨ ∨ ∨ ∨ ∧ x ¯ x
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 ∈ { x 1 , ¯ x 1 , . . . , x n , ¯ x n } .
Example L [ T ] N = •| N ∨ N | N ∧ � ∨ ∨ ∧ ∨ • • � • ∧ • �
Example L [ T ] N = •| N ∨ N | N ∧ � ∨ ∨ ∧ ¯ x 4 ∨ x 1 ∧ x 1 x 5 ¯ ∧ x 3 x 2 x 3
Example L [ T ] N = •| N ∨ N | N ∧ � ∨ ∨ ∧ ¯ x 4 ∨ x 1 ∧ x 1 x 5 ¯ ∧ x 3 x 2 x 3
Pattern languages for associative trees R = { ˆ N , ˇ N } ˆ ˇ N = •| N ∧ � | N ∧ � ∧ � | · · · 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 .
Simple tautologies Proposition If L is subcritical w.r.t. T then � 1 L [ T ] ( k ) � m = O . lim n k T m m →∞ Lemma All tautologies with 1 L-restriction are simple. Proposition Asymptotically almost all tautologies are simple (and realized by exactly 1 variable), when m → ∞ .
Limiting probability of simple tautologies ST x ( z ) : gf of simple tautologies realized by x . ST ( z ) = nST x ( z ) . ST x = {∨} × { x } × { ¯ x } × set ( T \ { x , ¯ x } ) ST x ( z ) = z 2 × � ℓ ( ℓ − 1 )( T ( z ) − 2 z ) ℓ − 2 ℓ ≥ 2
Limiting probability of simple tautologies ST x ( z ) : gf of simple tautologies realized by x . ST ( z ) = nST x ( z ) . ST x = {∨} × { x } × { ¯ x } × set ( T \ { x , ¯ x } ) ST x ( z ) = z 2 × � ℓ ( ℓ − 1 )( T ( z ) − 2 z ) ℓ − 2 ℓ ≥ 2 [ z m ] ST ( z ) ST ′ ( z ) P ( True ) = lim [ z m ] T ( z ) ∼ lim T ′ ( z ) . m →∞ z → ρ
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!
Mobiles N = •| N ∨ N | N ∧ � ∨ ∨ ∧ • • ∨ � • ∧ • �
Mobiles N = •| N ∨ N | N ∧ � ∨ ∨ ∧ x 1 x 4 ¯ ∨ ∧ x 5 � ¯ x 1 ∧ x 3 x 2 x 3
Mobiles N = •| N ∨ N | N ∧ � ∨ ∨ ∧ � x 1 x 4 ¯ ∨ � � ∧ x 5 x 1 � � ¯ ∧ x 3 x 2 x 3 �
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)
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!
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