Enumerating (restricted) -terms Motzkin trees -terms with bounded - PowerPoint PPT Presentation

Enumerating (restricted) λ -terms Danièle GARDY Enriched trees Enumerating (restricted) λ -terms Motzkin trees λ -terms with bounded number of Danièle GARDY unary nodes λ -terms of bounded PRiSM, Université Versailles St-Quentin en Yvelines unary height and λ -terms of D.M.G., T.U. Wien fixed arity Concluding remarks In collaboration with O. Bodini, B. Gittenberger and A. Jacquot

Enumerating (restricted) λ -terms Danièle Enriched trees 1 GARDY Enriched trees Motzkin trees Motzkin trees 2 λ -terms with bounded number of unary nodes λ -terms with bounded number of unary nodes 3 λ -terms of bounded unary height λ -terms of bounded unary height 4 λ -terms of fixed arity Concluding remarks 5 λ -terms of fixed arity Concluding remarks 6

Enumerating (restricted) λ -terms Danièle GARDY Enriched trees Motzkin trees λ -terms with bounded number of λ -terms and enriched (Motzkin) unary nodes λ -terms of bounded trees unary height λ -terms of fixed arity Concluding remarks

Enumerating (restricted) Definition of λ -terms λ -terms Danièle GARDY Enriched trees Motzkin trees λ -terms with T ::= a | ( T ∗ T ) | λ a . T bounded number of unary nodes ( T ∗ T ) : application λ a . T : abstraction λ -terms of bounded unary height ( λ x . ( x ∗ x ) ∗ λ y . y ) λ y . ( λ x . x ∗ λ x . y ) λ -terms of fixed arity Concluding remarks y x y x x

Enumerating (restricted) Enriched Motzkin trees λ -terms Danièle GARDY Enriched trees Motzkin trees λ -terms with x y bounded number of unary nodes y λ -terms of bounded unary height x x λ -terms of fixed arity Labelling rules: Concluding remarks • Binary nodes are unlabelled • Unary nodes get distinct labels (colors) • Leaves get the label (color) of one of their unary ancestors

Enumerating (restricted) Free and bound variables λ -terms Danièle GARDY Enriched trees • Here all variables are bound: closed terms Motzkin trees λ -terms with bounded number of unary nodes λ -terms of bounded unary height λ -terms of fixed arity • Some variables may be free Concluding remarks

Enumerating (restricted) Enumeration? λ -terms Danièle GARDY • Recursive definition for λ -terms? Enriched trees • L : class of λ -terms with free variables Motzkin trees • N atomic class of binary node λ -terms with • U atomic class of unary node bounded number of • F atomic class of free leaf unary nodes • B atomic class of bound leaf λ -terms of bounded unary height � N × L 2 � L = F + + ( U × subs ( F → F + B , L )) λ -terms of fixed arity • L ℓ, n number of λ -terms of size n (total number of nodes) Concluding remarks with ℓ free leaves ℓ, n L ℓ, n f ℓ z n satisfies a • Generating function L ( z , f ) = � functional equation L ( z , f ) = fz + z L ( z , f ) 2 + z L ( z , f + 1 ) .

Enumerating (restricted) Analytic combinatorics λ -terms Danièle GARDY Enriched trees n a n z n • Generating function of a sequence a n : A ( z ) = � Motzkin trees • A ( z ) considered as a function of complex variable z : λ -terms with bounded domain of analycity? radius of convergence ρ ? number of unary nodes • Type and location of dominant singularity determine the λ -terms of bounded asymptotic behaviour of the sequence a n unary height ρ ) α ( α �∈ N ) gives • E.g., ρ algebraic of type ( 1 − z λ -terms of fixed arity Concluding [ z n ] A ( z ) ∼ ρ n n − α − 1 remarks Γ( − α ) • Extensions to multivariate cases, asymptotic distributions

Enumerating (restricted) Enumeration??? λ -terms Danièle GARDY Enriched trees • Generating function enumerating closed λ -terms Motzkin trees (without free variables): L ( z , 0) λ -terms with bounded • Generating function enumerating all λ -terms: number of L ( z , 1 ) = 1 unary nodes z L ( z , 0 ) − L ( z , 0 ) 2 λ -terms of � � bounded 1 � • L ( z , 0 ) = 1 − Λ( z ) with Λ( z ) equal to unary height 2 z λ -terms of fixed arity � � 1 − 2 z − 4 nz 2 + 2 z √ ... � Concluding 1 − 2 z − 4 z 2 + 2 z 1 − 2 z + 2 z .... remarks • L ( z , 0 ) has null radius of convergence ⇒ standard tools of analytic combinatorics fail

Enumerating (restricted) What can we do? λ -terms Danièle GARDY • Try to find a way to deal with null radius of Enriched trees convergence? Motzkin trees • Ad hoc methods? λ -terms with bounded number of unary nodes � n ( 1 − 1 / log n ) � n ( 1 − 1 / 3 log n ) � ( 4 − ǫ ) n � ( 12 + ǫ ) n ≤ L n ≤ λ -terms of log n log n bounded unary height λ -terms of [David et al. 10; here leaves have size 0] fixed arity • Consider sub-classes of terms? Concluding remarks • Restrict the total number of abstractions [Bodini-G-Gittenberger’14] • Restrict the number of abstractions in a path from the root towards a leaf: bounded unary height [Bodini-G-Gittenberger’11, Bodini-G-Gittenberger’14] • Restrict the number of pointers from an abstraction to a leaf [Bodini-G-Jacquot’10; Bodini-G-Gittenberger-Jacquot’13, Bodini-Gittenberger’15]

Enumerating (restricted) λ -terms Danièle GARDY Enriched trees Motzkin trees λ -terms with bounded number of unary nodes Motzkin trees λ -terms of bounded unary height λ -terms of fixed arity Concluding remarks

Enumerating (restricted) Motzkin trees λ -terms Danièle GARDY Enriched trees Motzkin trees λ -terms with bounded number of unary nodes λ -terms of bounded unary height M = Z + ( U × M ) + ( Z × M 2 ) λ -terms of fixed arity Concluding M ( z ) = 1 remarks � � � 1 − 2 z − 3 z 2 1 − z − 2 z Dominant singularity at z = 1 / 3 of square-root type 3 n + 1 2 [ z n ] M ( z ) ∼ 2 n √ π n

Enumerating (restricted) q unary nodes λ -terms Danièle GARDY q � M q = U × M q − 1 + A × M ℓ × M q − ℓ . Enriched trees Motzkin trees ℓ = 0 λ -terms with Recurrence equation on the generating functions bounded number of unary nodes zM q − 1 ( z ) + z � 1 ≤ ℓ ≤ q − 1 M ℓ ( z ) M q − ℓ ( z ) M q ( z ) = . λ -terms of 1 − 2 zM 0 ( z ) bounded unary height ⇒ there exist polynomials P q s.t. λ -terms of fixed arity M q ( z ) = z q + 1 P q ( z 2 ) Concluding remarks 2 , ( 1 − 4 z 2 ) q − 1 Straightforward computations give √ 2 P q ( 1 / 4 ) 4 n n q − 3 [ z n ] M q ( z ) ∼ [ z n ] M ≤ q ∼ 2 Γ( q − 1 2 )

Enumerating (restricted) Leaves at same unary height λ -terms Danièle GARDY Enriched trees Motzkin trees λ -terms with bounded number of unary nodes λ -terms of bounded unary height λ -terms of fixed arity Concluding remarks • Tree on the left: all leaves have unary height 1 • Tree on the right: leaves have unary heights 1, 2 and 1

Enumerating (restricted) Leaves at same unary height λ -terms Danièle GARDY MH k = U × MH k − 1 + A × MH 2 Enriched trees k Motzkin trees On generating functions λ -terms with bounded   � number of � unary nodes � MH k = 1 �  1 − 1 − 2 z + 2 z 1 − 2 z + 2 z ... + 2 z 1 − 4 z 2   λ -terms of 2  bounded unary height λ -terms of Two singularities fixed arity Concluding 1 • z = − 1 2 (negligible) 2 of type ( 1 + 2 z ) remarks 1 2 k + 1 (dominant, comes from the • z = 1 2 of type ( 1 + 2 z ) innermost radicand) 1 1 2 k + 1 2 n n − 1 − ⇒ [ z n ] MH k ( z ) ∼ 2 2 k + 1 1 2 k + 1 Γ( 1 − 2 k + 1 )

Enumerating (restricted) Bounded unary height λ -terms Danièle GARDY Here leaves can have different unary height! Enriched trees Motzkin trees MH ≤ k = Z + U × MH ≤ k − 1 + A × MH 2 ≤ k λ -terms with bounded number of Generating function unary nodes λ -terms of  �  � bounded � � 1 � � 1 − 2 z − 4 z 2 + 2 z 1 − 2 z − 4 z 2 + 2 z unary height  � 1 − 4 z 2  MH ≤ k =  1 − ... + 2 z   2  λ -terms of fixed arity Dominant singularity ρ k comes from outermost radicand, Concluding remarks decreases towards 1 3 � 1 + 4 ρ 2 k ⇒ [ z n ] MH ≤ k ∼ n √ π n 4 ρ n + 1 k

Enumerating (restricted) λ -terms Danièle GARDY Enriched trees Motzkin trees λ -terms with bounded λ -terms with bounded number number of unary nodes λ -terms of of unary nodes bounded unary height λ -terms of fixed arity Concluding remarks

Enumerating (restricted) q unary nodes λ -terms Danièle GARDY q Enriched trees � � � S q = U × subs ( F → F + B , S q − 1 ) + ( A , S ℓ , S q − ℓ ) Motzkin trees ℓ = 0 λ -terms with bounded number of Generating function unary nodes λ -terms of bounded q unary height � S q ( z , f ) = zS q − 1 ( z , f + 1 ) + z S ℓ ( z , f ) S q − ℓ ( z , f ) . λ -terms of fixed arity ℓ = 0 Concluding remarks G.F. for closed terms S q ( z , 0 ) ? √ 1 1 − 4 z 2 S 1 ( z , 0 ) = 2 − ; 2 √ 2 z 3 z 1 − 4 z 2 − z 1 − 8 z 2 2 ( 1 − 2 z 2 ) + √ √ S 2 ( z , 0 ) = 1 − 4 z 2 ; 2 (no terms of size n = q mod 2)