Counting terms in the binary lambda calculus - Aofa 2014
Counting terms in the binary lambda calculus
Katarzyna Grygiel
Theoretical Computer Science Department Jagiellonian University in Kraków
Pierre Lescanne
Laboratoire de l'Informatique du Parallélisme École Normale Supérieure de Lyon
Analysis of Algorithms Paris, June 2014
Contents
1. Open problem: enumeration of lambda calculus terms 2. Attempts to solve the enumeration problem 3. Motivation 4. Binary lambda calculus 5. Enumerating binary lambda terms
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
1 / 28
The problem Terms in untyped lambda calculus
Lambda terms: t ::= x | λx.t | (t t)
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
2 / 28
The problem Terms in untyped lambda calculus
Lambda terms: t ::= x | λx.t | (t t)
Two alternative notions of size: |x| = 0 |λx.t| = 1 + |t| |t s| = 1 + |t| + |s|
K. Grygiel & P. Lescanne
|x| = 1 |λx.t| = 1 + |t| |t s| = 1 + |t| + |s|
Counting binary lambda terms
AofA 2014
2 / 28
The problem It's nicer to look at trees
Combinatorial interpretation by lambda trees: λx λy
@ @
λz
x z
@
λw
y x
The lambda tree corresponding to the lambda term λx.(λz.z)x λy .y (λw .x) K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
3 / 28
The problem Open problem
How many closed lambda terms of a given size are there?
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
4 / 28
Attempts Attempts to solve the enumeration problem
• Studying lambda polynomials and their coe cients • Estimating the number of simpler structures • Applying the theory of generating functions • Analyzing lambda terms of bounded unary height • Enumerating BCI/BCK terms
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
5 / 28
Attempts (|x|
Canonical formula
= 0)
Let Tn,m denote the number of terms of size n with occurrences of at most m distinct free variables: T0,m = m Tn+1,m = Tn,m+1 +
K. Grygiel & P. Lescanne
n X
Ti,m Tn−i,m
i=0
Counting binary lambda terms
AofA 2014
6 / 28
Attempts (|x|
Canonical formula
= 0)
Let Tn,m denote the number of terms of size n with occurrences of at most m distinct free variables: T0,m = m Tn+1,m = Tn,m+1 +
n X
Ti,m Tn−i,m
i=0
The sequence (Tn,0 )n≥0 (A220894 in Sloane's OEIS) enumerates closed terms. Its rst values are: 0, 1, 3, 14, 82, 579, 4741, 43977, 454283, 5159441, 63782411.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
6 / 28
Attempts Studying lambda polynomials and their coe cients (|x|
= 0)
For every n ≥ 0 we associate with Tn,m a polynomial Pn (m) in m. P0 (m) = m Pn+1 (m) = Pn (m + 1) +
n X
Pi (m)Pn−i (m)
i=0
For every n ≥ 0 we have Pn (0) = Tn,0 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
7 / 28
Attempts Studying lambda polynomials and their coe cients (|x|
= 0)
For every n ≥ 0 we associate with Tn,m a polynomial Pn (m) in m. P0 (m) = m Pn+1 (m) = Pn (m + 1) +
n X
Pi (m)Pn−i (m)
i=0
For every n ≥ 0 we have Pn (0) = Tn,0 . For k > 0 and n ≥ 0, let pn[k] denote the k -th leading coe cient of the polynomial Pn . KG, PL; 2013 [k]
pn
=
1
√
2k−1 (k − 1)! π
K. Grygiel & P. Lescanne
4n n(2k−5)/2 1 +
1 (2k − 3)(2k − 5) +O 4 . 8n n
Counting binary lambda terms
AofA 2014
7 / 28
Attempts Estimating the number of simpler structures
(|x|
= 0)
David, KG, Kozik, Raalli, Theyssier, Zaionc; 2011 The number of closed lambda terms of size n is bounded: n (4 − ε)n n− ln n • from below by , ln n n (12 + ε)n n− 3 ln n • from above by . ln n
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
8 / 28
Attempts Estimating the number of simpler structures
(|x|
= 0)
David, KG, Kozik, Raalli, Theyssier, Zaionc; 2011 The number of closed lambda terms of size n is bounded: n (4 − ε)n n− ln n • from below by , ln n n (12 + ε)n n− 3 ln n • from above by . ln n Therefore
K. Grygiel & P. Lescanne
Tn,0 ∼ nn (ln n)−n O(e n ).
Counting binary lambda terms
AofA 2014
8 / 28
Attempts Applying generating functions
(|x|
= 1)
Let Ln,k denote the number of lambda P terms of size n and with at most k distinct free variables and let `(z, f ) = n,k≥0 z n f k . Hence, `(z, 0) is the generating function for the sequence enumerating closed lambda terms.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
9 / 28
Attempts (|x|
Applying generating functions
= 1)
Let Ln,k denote the number of lambda P terms of size n and with at most k distinct free variables and let `(z, f ) = n,k≥0 z n f k . Hence, `(z, 0) is the generating function for the sequence enumerating closed lambda terms. Bodini, Gardy, Gittenberger; 2011 The function `(z, f ) satis es the functional equation `(z, f ) = fz + z`2 (z, f ) + z`(z, f + 1).
This leads to `(z, f ) =
Therefore, r `(z, 0) =
1−
1 − 2z + 2z
K. Grygiel & P. Lescanne
1−
p
1 − 4z 2 f − 4z 2 `(z, f + 1) . 2z
q p √ 1 − 4z 2 − 2z + 2z 1 − 8z 2 − 2z + 2z 1 − . . . .
2z
Counting binary lambda terms
AofA 2014
9 / 28
Attempts Analyzing terms of bounded unary height
(|x|
= 1)
Bodini, Gardy, Gittenberger; 2011 Consider closed terms with a limited (given) number of unary nodes on each branch.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
10 / 28
Attempts Analyzing terms of bounded unary height
(|x|
= 1)
Bodini, Gardy, Gittenberger; 2011 Consider closed terms with a limited (given) number of unary nodes on each branch. The generating function for the sequence enumerating such terms is the truncated version of `(z, 0).
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
10 / 28
Attempts Analyzing terms of bounded unary height
(|x|
= 1)
Bodini, Gardy, Gittenberger; 2011 Consider closed terms with a limited (given) number of unary nodes on each branch. The generating function for the sequence enumerating such terms is the truncated version of `(z, 0). One observes the intriguing behavior of dominant singularities of such generating functions. Some singularities cancel one radicand, some cancel two. This leads to two dierent asymptotic behaviors, dependent on the value of unary height.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
10 / 28
Attempts Enumerating BCI/BCK terms
(|x|
= 1)
Special subclasses of lambda terms: BCI /BCK and their generalized versions: BCI (p)/BCK (p)
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
11 / 28
Attempts (|x|
Enumerating BCI/BCK terms
= 1)
Special subclasses of lambda terms: BCI /BCK and their generalized versions: BCI (p)/BCK (p) Bodini, Gardy, Gittenberger, Jacquot; 2013 The number of BCI (p)-terms of size (2p + 1)n − 1 is asymptotically 2p+1
2p − 2 (1.0844 . . .) 2 p p−1
K. Grygiel & P. Lescanne
(4p + 2)p p!
n−1
p 2 −2p n 2p+1 (n − 1)!p ,
Counting binary lambda terms
p ≥ 2.
AofA 2014
11 / 28
Attempts (|x|
Enumerating BCI/BCK terms
= 1)
Special subclasses of lambda terms: BCI /BCK and their generalized versions: BCI (p)/BCK (p) Bodini, Gardy, Gittenberger, Jacquot; 2013 The number of BCI (p)-terms of size (2p + 1)n − 1 is asymptotically 2p+1
2p − 2 (1.0844 . . .) 2 p p−1
(4p + 2)p p!
n−1
p 2 −2p n 2p+1 (n − 1)!p ,
p ≥ 2.
Bodini, Gittenberger; 2014 The number of BCK (2)-terms of size n is asymptotically 7 · 24/5 3/5 17 · 21/5 2/5 41 · 23/5 1/5 −3/5 An2n/5 2n/5 e −2n/5 exp 2−8/5 n4/5 + n − n − n n , 15 75 500
where A is some positive constant. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
11 / 28
Motivation Motivation
• Lambda terms as programs → research on random programs and their behavior
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
12 / 28
Motivation Motivation
• Lambda terms as programs → research on random programs and their behavior • Typed lambda terms as proofs of logical formulae → research on logical entities and their properties
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
12 / 28
Motivation Motivation
• Lambda terms as programs → research on random programs and their behavior • Typed lambda terms as proofs of logical formulae → research on logical entities and their properties • Lambda calculus as the basis for functional programming → testing compilers by generating random lambda terms
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
12 / 28
Binary λ-calculus Lambda terms and de Bruijn indices
Let's eliminate names of variables from the notation of a λ-term: variables x, y , z, . . . λx.M
de Bruijn indices 1, 2, 3, . . . λM, where M is the result of substituting each x by n, where n is the number of λ's
above the given occurrence of x MN
K. Grygiel & P. Lescanne
MN
Counting binary lambda terms
AofA 2014
13 / 28
Binary λ-calculus Lambda terms and de Bruijn indices
Regular lambda term λx λy
@ @
λz
x z
@
λw
y x
The lambda term λx.(λz.z)x λy .y (λw .x)
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
14 / 28
Binary λ-calculus Lambda terms and de Bruijn indices
Lambda term in the de Bruijn version λ
λ
@ @
λ
1 1
@
λ
1 3
The de Bruijn term λ(λ1)1 λ1(λ3)
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
15 / 28
Binary λ-calculus Binary representation of lambda terms
Following John Tromp, we de ne the binary representation of de Bruijn indices in the following way: d b, λ M = 00M dN = 01M bN b, M i bi = 1 0.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
16 / 28
Binary λ-calculus Binary representation of lambda terms
Following John Tromp, we de ne the binary representation of de Bruijn indices in the following way: d b, λ M = 00M dN = 01M bN b, M i bi = 1 0.
Given a λ-term, we de ne its size as the length of the corresponding binary sequence, i.e., |i| = i + 1, |λM| = |M| + 2, |M N| = |M| + |N| + 2. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
16 / 28
Enumerating binary terms How many. . . ?
Let Sm,n denote the number of binary lambda terms of size n with at most m distinct free indices.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
17 / 28
Enumerating binary terms How many. . . ?
Let Sm,n denote the number of binary lambda terms of size n with at most m distinct free indices. Sm,0 = Sm,1 = 0, Sm,n+2 = [m ≥ n + 1] + Sm+1,n +
n X
Sm,k Sm,n−k .
k=0
The sequence (S0,n )n≥0 (A114852 in Sloane's OEIS) enumerates closed binary lambda terms of size n. Its rst 20 values are as follows: 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 6, 5, 13, 14, 37, 44, 101, 134, 298, 431. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
17 / 28
Enumerating binary terms Generating functions again in action
Now let us de ne the family of generating functions for sequences (Sm,n )n≥0 : sm (z) =
∞ X
Sm,n z n .
n=0
Most of all, we are interested in the generating function for the number of closed terms, i.e., s0 (z) =
K. Grygiel & P. Lescanne
∞ X n=0
S0,n z n .
Counting binary lambda terms
AofA 2014
18 / 28
Enumerating binary terms Recurring problem
Lemma sm (z) =
K. Grygiel & P. Lescanne
z 2 (1 − z m ) + z 2 sm+1 (z) + z 2 sm (z)2 . 1−z
Counting binary lambda terms
AofA 2014
19 / 28
Enumerating binary terms Recurring problem
Lemma sm (z) =
z 2 (1 − z m ) + z 2 sm+1 (z) + z 2 sm (z)2 . 1−z
This gives 1− sm (z) =
r
1 − 4z 4
1−z m + s m+1 (z) 1−z . 2z 2
Again we are stuck with in nitely nested radicals. . .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
19 / 28
Enumerating binary terms Let's count all binary terms
Let S∞,n denote the number of all (not necessarily closed) binary terms of size n. We obtain the following recurrence relation: S∞,0 = S∞,1 = 0, S∞,n+2 = 1 + S∞,n +
n X
S∞,k S∞,n−k .
k=0
The sequence (S∞,n )n≥0 has the entry number A114851 in Sloane's OEIS. Its rst 20 values are: 0, 0, 1, 1, 2, 2, 4, 5, 10, 14, 27, 41, 78, 126, 237, 399, 745, 1292, 2404, 4259. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
20 / 28
Enumerating binary terms No problem this time
Let s∞ (z) denote the generating function for the sequence (S∞,n )n≥0 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
21 / 28
Enumerating binary terms No problem this time
Let s∞ (z) denote the generating function for the sequence (S∞,n )n≥0 . Theorem The number of all binary lambda terms of size n satis es C S∞,n ∼ (1/ρ)n · 3/2 , n
where ρ = 0.509308127 . . . and C = 1.021874073 . . ..
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
21 / 28
Enumerating binary terms No problem this time
Let s∞ (z) denote the generating function for the sequence (S∞,n )n≥0 . Theorem The number of all binary lambda terms of size n satis es C S∞,n ∼ (1/ρ)n · 3/2 , n
where ρ = 0.509308127 . . . and C = 1.021874073 . . .. For m ≥ n − 1 we have Sm,n = S∞,n . Therefore s∞ (z) =
∞ X
Sn,n z n ,
n=1
which yields that = ∞. Furthermore, s∞ (z) = lim sm (z). [z n ]sn
[z n ]s
m→∞
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
21 / 28
Enumerating binary terms Singularities of
sm (z)
Lemma Let ρm denote the dominant singularity of sm (z). Then for every m, ρm = ρ0 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
22 / 28
Enumerating binary terms Singularities of
sm (z)
Lemma Let ρm denote the dominant singularity of sm (z). Then for every m, ρm = ρ0 .
Lemma The dominant singularity of s0 (z) is equal to the dominant singularity of s∞ (z), i.e., ρ0 = ρ = 0.509308127 . . .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
22 / 28
Enumerating binary terms Sketch of the proof
Let us de ne functionals 1 − 4z 4 ( 11−z −z + F ) Φm (F ) = , 2z 2 q 1 + F) 1 − 1 − 4z 4 ( 1−z Φ∞ (F ) = . 2z 2 1−
K. Grygiel & P. Lescanne
q
Counting binary lambda terms
m
AofA 2014
23 / 28
Enumerating binary terms Sketch of the proof
Let us de ne functionals 1 − 4z 4 ( 11−z −z + F ) Φm (F ) = , 2z 2 q 1 + F) 1 − 1 − 4z 4 ( 1−z Φ∞ (F ) = . 2z 2 1−
q
m
Then sm (z) = Φm (sm+1 (z)).
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
23 / 28
Enumerating binary terms Sketch of the proof
Let us de ne functionals 1 − 4z 4 ( 11−z −z + F ) Φm (F ) = , 2z 2 q 1 + F) 1 − 1 − 4z 4 ( 1−z Φ∞ (F ) = . 2z 2 1−
q
m
Then sm (z) = Φm (sm+1 (z)).
For z ∈ [0, ρ] we have Φm (sm (z)) ≤ sm (z) ≤ sm+1 (z) ≤ s∞ (z). K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
23 / 28
Enumerating binary terms Sketch of the proof II
Let sem (z) denote the xed point of Φm , i.e., sem (z) is de ned as the solution of the equation sem (z) = Φm (e sm (z)).
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
24 / 28
Enumerating binary terms Sketch of the proof II
Let sem (z) denote the xed point of Φm , i.e., sem (z) is de ned as the solution of the equation sem (z) = Φm (e sm (z)).
For z ∈ [0, ρ] we have sem (z) ≤ sm (z) ≤ s∞ (z).
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
24 / 28
Enumerating binary terms Sketch of the proof II
Let sem (z) denote the xed point of Φm , i.e., sem (z) is de ned as the solution of the equation sem (z) = Φm (e sm (z)).
For z ∈ [0, ρ] we have sem (z) ≤ sm (z) ≤ s∞ (z).
By the de nition of sem (z), 2 (z) − (1 − z 2 )2 se (z) + z 2 sem m
K. Grygiel & P. Lescanne
z 2 (1 − z m ) = 0. 1−z
Counting binary lambda terms
AofA 2014
24 / 28
Enumerating binary terms Sketch of the proof II
Let sem (z) denote the xed point of Φm , i.e., sem (z) is de ned as the solution of the equation sem (z) = Φm (e sm (z)).
For z ∈ [0, ρ] we have sem (z) ≤ sm (z) ≤ s∞ (z).
By the de nition of sem (z), 2 (z) − (1 − z 2 )2 se (z) + z 2 sem m
z 2 (1 − z m ) = 0. 1−z
Let us denote the main singularity of sem (z) by σm . We have σm ≥ ρm ≥ ρ. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
24 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
K. Grygiel & P. Lescanne
4 z 4 (1 − z m ) . 1−z
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
4 z 4 (1 − z m ) . 1−z
The value of σm is equal to the root of smallest modulus of the polynomial Pm (z) := (z − 1)∆m = 4z 4 (1 − z m ) − (1 − z)3 (1 + z)2 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
4 z 4 (1 − z m ) . 1−z
The value of σm is equal to the root of smallest modulus of the polynomial Pm (z) := (z − 1)∆m = 4z 4 (1 − z m ) − (1 − z)3 (1 + z)2 .
The sequence (σm )m∈N of roots of polynomials Pm (z) is decreasing and it converges to ρ.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
4 z 4 (1 − z m ) . 1−z
The value of σm is equal to the root of smallest modulus of the polynomial Pm (z) := (z − 1)∆m = 4z 4 (1 − z m ) − (1 − z)3 (1 + z)2 .
The sequence (σm )m∈N of roots of polynomials Pm (z) is decreasing and it converges to ρ. By σm ≥ ρm ≥ ρ, we get ρm → ρ, as well.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
4 z 4 (1 − z m ) . 1−z
The value of σm is equal to the root of smallest modulus of the polynomial Pm (z) := (z − 1)∆m = 4z 4 (1 − z m ) − (1 − z)3 (1 + z)2 .
The sequence (σm )m∈N of roots of polynomials Pm (z) is decreasing and it converges to ρ. By σm ≥ ρm ≥ ρ, we get ρm → ρ, as well. Since all the ρm 's are equal, we obtain that ρm = ρ for every natural m. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Number of closed binary terms
Theorem The number of closed binary binary lambda terms of size n is of exponential order S0,n ./ 1.963448 . . .n .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
26 / 28
Enumerating binary terms Conjecture
For every m,
Sm,n ∼ o 1.963448 . . .n · n−3/2 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
27 / 28
Enumerating binary terms Conjecture
For every m,
Sm,n ∼ o 1.963448 . . .n · n−3/2 .
Sm,n n ρ n−3/2
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
27 / 28
References O. Bodini, B. Gittenberger: On the asymptotic number of BCK(2)-terms, in 2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 25-39, 2014 O. Bodini, D. Gardy, B. Gittenberger: Lambda-terms of bounded unary height, 2011 Proceedings of the Eighth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), 2011 O. Bodini, D. Gardy, B. Gittenberger, A. Jacquot: Enumeration of generalized BCI Electronic Journal of Combinatorics, 20(4):P30, 2013
lambda-terms,
R. David, K. Grygiel, J. Kozik, Ch. Raalli, G. Theyssier, M. Zaionc: Asymptotically Logical Methods in Computer Science Vol. 9(1:02) 2013, pp. 1-30 almost all lambda terms are strongly normalizing,
K. Grygiel, P. Lescanne: Counting and generating lambda terms, Journal of Functional Programming Vol. 23(5) 2013, pp. 594-628 K. Grygiel, P. Lescanne:
Counting terms in the binary lambda calculus,
2014
P. Lescanne: On counting untyped lambda terms, Theoretical Computer Science, 474:80-97, 2013 J. Tromp: Binary lambda calculus and combinatory logic. Kolmogorov Complexity and Applications, Vol. 06051 of Dagstuhl Seminar Proceedings, Schloss Dagstuhl, Germany, 2006
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
28 / 28
Katarzyna Grygiel
Theoretical Computer Science Department Jagiellonian University in Kraków
Pierre Lescanne
Laboratoire de l'Informatique du Parallélisme École Normale Supérieure de Lyon
Analysis of Algorithms Paris, June 2014
Contents
1. Open problem: enumeration of lambda calculus terms 2. Attempts to solve the enumeration problem 3. Motivation 4. Binary lambda calculus 5. Enumerating binary lambda terms
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
1 / 28
The problem Terms in untyped lambda calculus
Lambda terms: t ::= x | λx.t | (t t)
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
2 / 28
The problem Terms in untyped lambda calculus
Lambda terms: t ::= x | λx.t | (t t)
Two alternative notions of size: |x| = 0 |λx.t| = 1 + |t| |t s| = 1 + |t| + |s|
K. Grygiel & P. Lescanne
|x| = 1 |λx.t| = 1 + |t| |t s| = 1 + |t| + |s|
Counting binary lambda terms
AofA 2014
2 / 28
The problem It's nicer to look at trees
Combinatorial interpretation by lambda trees: λx λy
@ @
λz
x z
@
λw
y x
The lambda tree corresponding to the lambda term λx.(λz.z)x λy .y (λw .x) K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
3 / 28
The problem Open problem
How many closed lambda terms of a given size are there?
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
4 / 28
Attempts Attempts to solve the enumeration problem
• Studying lambda polynomials and their coe cients • Estimating the number of simpler structures • Applying the theory of generating functions • Analyzing lambda terms of bounded unary height • Enumerating BCI/BCK terms
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
5 / 28
Attempts (|x|
Canonical formula
= 0)
Let Tn,m denote the number of terms of size n with occurrences of at most m distinct free variables: T0,m = m Tn+1,m = Tn,m+1 +
K. Grygiel & P. Lescanne
n X
Ti,m Tn−i,m
i=0
Counting binary lambda terms
AofA 2014
6 / 28
Attempts (|x|
Canonical formula
= 0)
Let Tn,m denote the number of terms of size n with occurrences of at most m distinct free variables: T0,m = m Tn+1,m = Tn,m+1 +
n X
Ti,m Tn−i,m
i=0
The sequence (Tn,0 )n≥0 (A220894 in Sloane's OEIS) enumerates closed terms. Its rst values are: 0, 1, 3, 14, 82, 579, 4741, 43977, 454283, 5159441, 63782411.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
6 / 28
Attempts Studying lambda polynomials and their coe cients (|x|
= 0)
For every n ≥ 0 we associate with Tn,m a polynomial Pn (m) in m. P0 (m) = m Pn+1 (m) = Pn (m + 1) +
n X
Pi (m)Pn−i (m)
i=0
For every n ≥ 0 we have Pn (0) = Tn,0 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
7 / 28
Attempts Studying lambda polynomials and their coe cients (|x|
= 0)
For every n ≥ 0 we associate with Tn,m a polynomial Pn (m) in m. P0 (m) = m Pn+1 (m) = Pn (m + 1) +
n X
Pi (m)Pn−i (m)
i=0
For every n ≥ 0 we have Pn (0) = Tn,0 . For k > 0 and n ≥ 0, let pn[k] denote the k -th leading coe cient of the polynomial Pn . KG, PL; 2013 [k]
pn
=
1
√
2k−1 (k − 1)! π
K. Grygiel & P. Lescanne
4n n(2k−5)/2 1 +
1 (2k − 3)(2k − 5) +O 4 . 8n n
Counting binary lambda terms
AofA 2014
7 / 28
Attempts Estimating the number of simpler structures
(|x|
= 0)
David, KG, Kozik, Raalli, Theyssier, Zaionc; 2011 The number of closed lambda terms of size n is bounded: n (4 − ε)n n− ln n • from below by , ln n n (12 + ε)n n− 3 ln n • from above by . ln n
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
8 / 28
Attempts Estimating the number of simpler structures
(|x|
= 0)
David, KG, Kozik, Raalli, Theyssier, Zaionc; 2011 The number of closed lambda terms of size n is bounded: n (4 − ε)n n− ln n • from below by , ln n n (12 + ε)n n− 3 ln n • from above by . ln n Therefore
K. Grygiel & P. Lescanne
Tn,0 ∼ nn (ln n)−n O(e n ).
Counting binary lambda terms
AofA 2014
8 / 28
Attempts Applying generating functions
(|x|
= 1)
Let Ln,k denote the number of lambda P terms of size n and with at most k distinct free variables and let `(z, f ) = n,k≥0 z n f k . Hence, `(z, 0) is the generating function for the sequence enumerating closed lambda terms.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
9 / 28
Attempts (|x|
Applying generating functions
= 1)
Let Ln,k denote the number of lambda P terms of size n and with at most k distinct free variables and let `(z, f ) = n,k≥0 z n f k . Hence, `(z, 0) is the generating function for the sequence enumerating closed lambda terms. Bodini, Gardy, Gittenberger; 2011 The function `(z, f ) satis es the functional equation `(z, f ) = fz + z`2 (z, f ) + z`(z, f + 1).
This leads to `(z, f ) =
Therefore, r `(z, 0) =
1−
1 − 2z + 2z
K. Grygiel & P. Lescanne
1−
p
1 − 4z 2 f − 4z 2 `(z, f + 1) . 2z
q p √ 1 − 4z 2 − 2z + 2z 1 − 8z 2 − 2z + 2z 1 − . . . .
2z
Counting binary lambda terms
AofA 2014
9 / 28
Attempts Analyzing terms of bounded unary height
(|x|
= 1)
Bodini, Gardy, Gittenberger; 2011 Consider closed terms with a limited (given) number of unary nodes on each branch.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
10 / 28
Attempts Analyzing terms of bounded unary height
(|x|
= 1)
Bodini, Gardy, Gittenberger; 2011 Consider closed terms with a limited (given) number of unary nodes on each branch. The generating function for the sequence enumerating such terms is the truncated version of `(z, 0).
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
10 / 28
Attempts Analyzing terms of bounded unary height
(|x|
= 1)
Bodini, Gardy, Gittenberger; 2011 Consider closed terms with a limited (given) number of unary nodes on each branch. The generating function for the sequence enumerating such terms is the truncated version of `(z, 0). One observes the intriguing behavior of dominant singularities of such generating functions. Some singularities cancel one radicand, some cancel two. This leads to two dierent asymptotic behaviors, dependent on the value of unary height.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
10 / 28
Attempts Enumerating BCI/BCK terms
(|x|
= 1)
Special subclasses of lambda terms: BCI /BCK and their generalized versions: BCI (p)/BCK (p)
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
11 / 28
Attempts (|x|
Enumerating BCI/BCK terms
= 1)
Special subclasses of lambda terms: BCI /BCK and their generalized versions: BCI (p)/BCK (p) Bodini, Gardy, Gittenberger, Jacquot; 2013 The number of BCI (p)-terms of size (2p + 1)n − 1 is asymptotically 2p+1
2p − 2 (1.0844 . . .) 2 p p−1
K. Grygiel & P. Lescanne
(4p + 2)p p!
n−1
p 2 −2p n 2p+1 (n − 1)!p ,
Counting binary lambda terms
p ≥ 2.
AofA 2014
11 / 28
Attempts (|x|
Enumerating BCI/BCK terms
= 1)
Special subclasses of lambda terms: BCI /BCK and their generalized versions: BCI (p)/BCK (p) Bodini, Gardy, Gittenberger, Jacquot; 2013 The number of BCI (p)-terms of size (2p + 1)n − 1 is asymptotically 2p+1
2p − 2 (1.0844 . . .) 2 p p−1
(4p + 2)p p!
n−1
p 2 −2p n 2p+1 (n − 1)!p ,
p ≥ 2.
Bodini, Gittenberger; 2014 The number of BCK (2)-terms of size n is asymptotically 7 · 24/5 3/5 17 · 21/5 2/5 41 · 23/5 1/5 −3/5 An2n/5 2n/5 e −2n/5 exp 2−8/5 n4/5 + n − n − n n , 15 75 500
where A is some positive constant. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
11 / 28
Motivation Motivation
• Lambda terms as programs → research on random programs and their behavior
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
12 / 28
Motivation Motivation
• Lambda terms as programs → research on random programs and their behavior • Typed lambda terms as proofs of logical formulae → research on logical entities and their properties
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
12 / 28
Motivation Motivation
• Lambda terms as programs → research on random programs and their behavior • Typed lambda terms as proofs of logical formulae → research on logical entities and their properties • Lambda calculus as the basis for functional programming → testing compilers by generating random lambda terms
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
12 / 28
Binary λ-calculus Lambda terms and de Bruijn indices
Let's eliminate names of variables from the notation of a λ-term: variables x, y , z, . . . λx.M
de Bruijn indices 1, 2, 3, . . . λM, where M is the result of substituting each x by n, where n is the number of λ's
above the given occurrence of x MN
K. Grygiel & P. Lescanne
MN
Counting binary lambda terms
AofA 2014
13 / 28
Binary λ-calculus Lambda terms and de Bruijn indices
Regular lambda term λx λy
@ @
λz
x z
@
λw
y x
The lambda term λx.(λz.z)x λy .y (λw .x)
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
14 / 28
Binary λ-calculus Lambda terms and de Bruijn indices
Lambda term in the de Bruijn version λ
λ
@ @
λ
1 1
@
λ
1 3
The de Bruijn term λ(λ1)1 λ1(λ3)
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
15 / 28
Binary λ-calculus Binary representation of lambda terms
Following John Tromp, we de ne the binary representation of de Bruijn indices in the following way: d b, λ M = 00M dN = 01M bN b, M i bi = 1 0.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
16 / 28
Binary λ-calculus Binary representation of lambda terms
Following John Tromp, we de ne the binary representation of de Bruijn indices in the following way: d b, λ M = 00M dN = 01M bN b, M i bi = 1 0.
Given a λ-term, we de ne its size as the length of the corresponding binary sequence, i.e., |i| = i + 1, |λM| = |M| + 2, |M N| = |M| + |N| + 2. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
16 / 28
Enumerating binary terms How many. . . ?
Let Sm,n denote the number of binary lambda terms of size n with at most m distinct free indices.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
17 / 28
Enumerating binary terms How many. . . ?
Let Sm,n denote the number of binary lambda terms of size n with at most m distinct free indices. Sm,0 = Sm,1 = 0, Sm,n+2 = [m ≥ n + 1] + Sm+1,n +
n X
Sm,k Sm,n−k .
k=0
The sequence (S0,n )n≥0 (A114852 in Sloane's OEIS) enumerates closed binary lambda terms of size n. Its rst 20 values are as follows: 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 6, 5, 13, 14, 37, 44, 101, 134, 298, 431. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
17 / 28
Enumerating binary terms Generating functions again in action
Now let us de ne the family of generating functions for sequences (Sm,n )n≥0 : sm (z) =
∞ X
Sm,n z n .
n=0
Most of all, we are interested in the generating function for the number of closed terms, i.e., s0 (z) =
K. Grygiel & P. Lescanne
∞ X n=0
S0,n z n .
Counting binary lambda terms
AofA 2014
18 / 28
Enumerating binary terms Recurring problem
Lemma sm (z) =
K. Grygiel & P. Lescanne
z 2 (1 − z m ) + z 2 sm+1 (z) + z 2 sm (z)2 . 1−z
Counting binary lambda terms
AofA 2014
19 / 28
Enumerating binary terms Recurring problem
Lemma sm (z) =
z 2 (1 − z m ) + z 2 sm+1 (z) + z 2 sm (z)2 . 1−z
This gives 1− sm (z) =
r
1 − 4z 4
1−z m + s m+1 (z) 1−z . 2z 2
Again we are stuck with in nitely nested radicals. . .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
19 / 28
Enumerating binary terms Let's count all binary terms
Let S∞,n denote the number of all (not necessarily closed) binary terms of size n. We obtain the following recurrence relation: S∞,0 = S∞,1 = 0, S∞,n+2 = 1 + S∞,n +
n X
S∞,k S∞,n−k .
k=0
The sequence (S∞,n )n≥0 has the entry number A114851 in Sloane's OEIS. Its rst 20 values are: 0, 0, 1, 1, 2, 2, 4, 5, 10, 14, 27, 41, 78, 126, 237, 399, 745, 1292, 2404, 4259. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
20 / 28
Enumerating binary terms No problem this time
Let s∞ (z) denote the generating function for the sequence (S∞,n )n≥0 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
21 / 28
Enumerating binary terms No problem this time
Let s∞ (z) denote the generating function for the sequence (S∞,n )n≥0 . Theorem The number of all binary lambda terms of size n satis es C S∞,n ∼ (1/ρ)n · 3/2 , n
where ρ = 0.509308127 . . . and C = 1.021874073 . . ..
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
21 / 28
Enumerating binary terms No problem this time
Let s∞ (z) denote the generating function for the sequence (S∞,n )n≥0 . Theorem The number of all binary lambda terms of size n satis es C S∞,n ∼ (1/ρ)n · 3/2 , n
where ρ = 0.509308127 . . . and C = 1.021874073 . . .. For m ≥ n − 1 we have Sm,n = S∞,n . Therefore s∞ (z) =
∞ X
Sn,n z n ,
n=1
which yields that = ∞. Furthermore, s∞ (z) = lim sm (z). [z n ]sn
[z n ]s
m→∞
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
21 / 28
Enumerating binary terms Singularities of
sm (z)
Lemma Let ρm denote the dominant singularity of sm (z). Then for every m, ρm = ρ0 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
22 / 28
Enumerating binary terms Singularities of
sm (z)
Lemma Let ρm denote the dominant singularity of sm (z). Then for every m, ρm = ρ0 .
Lemma The dominant singularity of s0 (z) is equal to the dominant singularity of s∞ (z), i.e., ρ0 = ρ = 0.509308127 . . .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
22 / 28
Enumerating binary terms Sketch of the proof
Let us de ne functionals 1 − 4z 4 ( 11−z −z + F ) Φm (F ) = , 2z 2 q 1 + F) 1 − 1 − 4z 4 ( 1−z Φ∞ (F ) = . 2z 2 1−
K. Grygiel & P. Lescanne
q
Counting binary lambda terms
m
AofA 2014
23 / 28
Enumerating binary terms Sketch of the proof
Let us de ne functionals 1 − 4z 4 ( 11−z −z + F ) Φm (F ) = , 2z 2 q 1 + F) 1 − 1 − 4z 4 ( 1−z Φ∞ (F ) = . 2z 2 1−
q
m
Then sm (z) = Φm (sm+1 (z)).
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
23 / 28
Enumerating binary terms Sketch of the proof
Let us de ne functionals 1 − 4z 4 ( 11−z −z + F ) Φm (F ) = , 2z 2 q 1 + F) 1 − 1 − 4z 4 ( 1−z Φ∞ (F ) = . 2z 2 1−
q
m
Then sm (z) = Φm (sm+1 (z)).
For z ∈ [0, ρ] we have Φm (sm (z)) ≤ sm (z) ≤ sm+1 (z) ≤ s∞ (z). K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
23 / 28
Enumerating binary terms Sketch of the proof II
Let sem (z) denote the xed point of Φm , i.e., sem (z) is de ned as the solution of the equation sem (z) = Φm (e sm (z)).
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
24 / 28
Enumerating binary terms Sketch of the proof II
Let sem (z) denote the xed point of Φm , i.e., sem (z) is de ned as the solution of the equation sem (z) = Φm (e sm (z)).
For z ∈ [0, ρ] we have sem (z) ≤ sm (z) ≤ s∞ (z).
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
24 / 28
Enumerating binary terms Sketch of the proof II
Let sem (z) denote the xed point of Φm , i.e., sem (z) is de ned as the solution of the equation sem (z) = Φm (e sm (z)).
For z ∈ [0, ρ] we have sem (z) ≤ sm (z) ≤ s∞ (z).
By the de nition of sem (z), 2 (z) − (1 − z 2 )2 se (z) + z 2 sem m
K. Grygiel & P. Lescanne
z 2 (1 − z m ) = 0. 1−z
Counting binary lambda terms
AofA 2014
24 / 28
Enumerating binary terms Sketch of the proof II
Let sem (z) denote the xed point of Φm , i.e., sem (z) is de ned as the solution of the equation sem (z) = Φm (e sm (z)).
For z ∈ [0, ρ] we have sem (z) ≤ sm (z) ≤ s∞ (z).
By the de nition of sem (z), 2 (z) − (1 − z 2 )2 se (z) + z 2 sem m
z 2 (1 − z m ) = 0. 1−z
Let us denote the main singularity of sem (z) by σm . We have σm ≥ ρm ≥ ρ. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
24 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
K. Grygiel & P. Lescanne
4 z 4 (1 − z m ) . 1−z
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
4 z 4 (1 − z m ) . 1−z
The value of σm is equal to the root of smallest modulus of the polynomial Pm (z) := (z − 1)∆m = 4z 4 (1 − z m ) − (1 − z)3 (1 + z)2 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
4 z 4 (1 − z m ) . 1−z
The value of σm is equal to the root of smallest modulus of the polynomial Pm (z) := (z − 1)∆m = 4z 4 (1 − z m ) − (1 − z)3 (1 + z)2 .
The sequence (σm )m∈N of roots of polynomials Pm (z) is decreasing and it converges to ρ.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
4 z 4 (1 − z m ) . 1−z
The value of σm is equal to the root of smallest modulus of the polynomial Pm (z) := (z − 1)∆m = 4z 4 (1 − z m ) − (1 − z)3 (1 + z)2 .
The sequence (σm )m∈N of roots of polynomials Pm (z) is decreasing and it converges to ρ. By σm ≥ ρm ≥ ρ, we get ρm → ρ, as well.
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Sketch of the proof III
The discriminant of this equation is ∆m = (1 − z 2 )2 −
4 z 4 (1 − z m ) . 1−z
The value of σm is equal to the root of smallest modulus of the polynomial Pm (z) := (z − 1)∆m = 4z 4 (1 − z m ) − (1 − z)3 (1 + z)2 .
The sequence (σm )m∈N of roots of polynomials Pm (z) is decreasing and it converges to ρ. By σm ≥ ρm ≥ ρ, we get ρm → ρ, as well. Since all the ρm 's are equal, we obtain that ρm = ρ for every natural m. K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
25 / 28
Enumerating binary terms Number of closed binary terms
Theorem The number of closed binary binary lambda terms of size n is of exponential order S0,n ./ 1.963448 . . .n .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
26 / 28
Enumerating binary terms Conjecture
For every m,
Sm,n ∼ o 1.963448 . . .n · n−3/2 .
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
27 / 28
Enumerating binary terms Conjecture
For every m,
Sm,n ∼ o 1.963448 . . .n · n−3/2 .
Sm,n n ρ n−3/2
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
27 / 28
References O. Bodini, B. Gittenberger: On the asymptotic number of BCK(2)-terms, in 2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 25-39, 2014 O. Bodini, D. Gardy, B. Gittenberger: Lambda-terms of bounded unary height, 2011 Proceedings of the Eighth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), 2011 O. Bodini, D. Gardy, B. Gittenberger, A. Jacquot: Enumeration of generalized BCI Electronic Journal of Combinatorics, 20(4):P30, 2013
lambda-terms,
R. David, K. Grygiel, J. Kozik, Ch. Raalli, G. Theyssier, M. Zaionc: Asymptotically Logical Methods in Computer Science Vol. 9(1:02) 2013, pp. 1-30 almost all lambda terms are strongly normalizing,
K. Grygiel, P. Lescanne: Counting and generating lambda terms, Journal of Functional Programming Vol. 23(5) 2013, pp. 594-628 K. Grygiel, P. Lescanne:
Counting terms in the binary lambda calculus,
2014
P. Lescanne: On counting untyped lambda terms, Theoretical Computer Science, 474:80-97, 2013 J. Tromp: Binary lambda calculus and combinatory logic. Kolmogorov Complexity and Applications, Vol. 06051 of Dagstuhl Seminar Proceedings, Schloss Dagstuhl, Germany, 2006
K. Grygiel & P. Lescanne
Counting binary lambda terms
AofA 2014
28 / 28
