Counting terms in the binary lambda calculus

Transcription

1 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

2 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 / 28

3 The problem Terms in untyped lambda calculus Lambda terms: t ::= x λx.t (t t) K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

4 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 x = 1 λx.t = 1 + t t s = 1 + t + s K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

5 The problem It's nicer to look at trees Combinatorial interpretation by lambda trees: λy λz x x λw 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 / 28

6 The problem Open problem How many closed lambda terms of a given size are there? K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

7 Attempts Attempts to solve the enumeration problem Studying lambda polynomials and their coecients 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 / 28

8 Attempts Canonical formula ( x = 0) Let T n,m denote the number of terms of size n with occurrences of at most m distinct free variables: T 0,m = m T n+1,m = T n,m+1 + n T i,m T n i,m i=0 K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

9 Attempts Canonical formula ( x = 0) Let T n,m denote the number of terms of size n with occurrences of at most m distinct free variables: T 0,m = m T n+1,m = T n,m+1 + n T i,m T n i,m The sequence (T n,0 ) n 0 (A in Sloane's OEIS) enumerates closed terms. Its rst values are: i=0 0, 1, 3, 14, 82, 579, 4741, 43977, , , K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

10 Attempts Studying lambda polynomials and their coecients ( x = 0) For every n 0 we associate with T n,m a polynomial P n (m) in m. P 0 (m) = m P n+1 (m) = P n (m + 1) + For every n 0 we have P n (0) = T n,0. n P i (m)p n i (m) i=0 K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

11 Attempts Studying lambda polynomials and their coecients ( x = 0) For every n 0 we associate with T n,m a polynomial P n (m) in m. P 0 (m) = m P n+1 (m) = P n (m + 1) + For every n 0 we have P n (0) = T n,0. n P i (m)p n i (m) i=0 For k > 0 and n 0, let p n [k] polynomial P n. denote the k-th leading coecient of the KG, PL; 2013 p [k] n = 1 ( 2 k 1 (k 1)! π 4n n (2k 5)/2 (2k 3)(2k 5) ( 1 )) O 8n n 4. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

12 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: ( ) (4 ε)n n n ln n from below by, ln n ( ) (12 + ε)n n n 3 ln n from above by. ln n K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

13 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: ( ) (4 ε)n n n ln n from below by, ln n ( ) (12 + ε)n n n 3 ln n from above by. ln n Therefore T n,0 n n (ln n) n O(e n ). K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

14 Attempts Applying generating functions ( x = 1) Let L n,k denote the number of lambda terms of size n and with at most k distinct free variables and let l(z, f ) = n,k 0 zn f k. Hence, l(z, 0) is the generating function for the sequence enumerating closed lambda terms. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

15 Attempts Applying generating functions ( x = 1) Let L n,k denote the number of lambda terms of size n and with at most k distinct free variables and let l(z, f ) = n,k 0 zn f k. Hence, l(z, 0) is the generating function for the sequence enumerating closed lambda terms. Bodini, Gardy, Gittenberger; 2011 The function l(z, f ) satises the functional equation This leads to Therefore, l(z, 0) = 1 l(z, f ) = fz + zl 2 (z, f ) + zl(z, f + 1). l(z, f ) = 1 1 4z 2 f 4z 2 l(z, f + 1). 2z 1 2z + 2z 1 4z 2 2z + 2z 1 8z 2 2z + 2z z K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

16 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 / 28

17 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 l(z, 0). K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

18 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 l(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 / 28

19 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 / 28

20 Attempts Enumerating BCI/BCK terms ( x = 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 ( ) 2 p 1 p ( (4p + 2) p p! ) n 1 n p2 2p 2p+1 (n 1)! p, p 2. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

21 Attempts Enumerating BCI/BCK terms ( x = 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 ( ) 2 p 1 p ( (4p + 2) p p! ) n 1 n p2 2p 2p+1 (n 1)! p, p 2. Bodini, Gittenberger; 2014 The number of BCK(2)-terms of size n is asymptotically An 2n/5 n/5 2 e 2n/5 exp (2 8/5 n 4/ /5 n 3/5 where A is some positive constant /5 n 2/ / n 1/5 ) n 3/5, K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

22 Motivation Motivation Lambda terms as programs research on random programs and their behavior K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

23 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 / 28

24 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 / 28

25 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,... de Bruijn indices 1, 2, 3,... λx.m 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 MN K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

26 Binary λ-calculus Lambda terms and de Bruijn indices Regular lambda term λy λz x x λw The lambda term λx.(λz.z)x ( λy.y(λw.x) ) K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

27 Binary λ-calculus Lambda terms and de Bruijn indices Lambda term in the de Bruijn version λ λ The de Bruijn term λ(λ1)1 ( λ1(λ3) ) K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

28 Binary λ-calculus Binary representation of lambda terms Following John Tromp, we dene the binary representation of de Bruijn indices in the following way: λm = 00 M, M N = 01 M N, î = 1 i 0. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

29 Binary λ-calculus Binary representation of lambda terms Following John Tromp, we dene the binary representation of de Bruijn indices in the following way: λm = 00 M, M N = 01 M N, î = 1 i 0. Given a λ-term, we dene 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 / 28

30 Enumerating binary terms How many...? Let S m,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 / 28

31 Enumerating binary terms How many...? Let S m,n denote the number of binary lambda terms of size n with at most m distinct free indices. S m,0 = S m,1 = 0, S m,n+2 = [m n + 1] + S m+1,n + n S m,k S m,n k. The sequence (S 0,n ) n 0 (A 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=0 K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

32 Enumerating binary terms Generating functions again in action Now let us dene the family of generating functions for sequences (S m,n ) n 0 : s m (z) = S m,n z n. n=0 Most of all, we are interested in the generating function for the number of closed terms, i.e., s 0 (z) = S 0,n z n. n=0 K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

33 Enumerating binary terms Recurring problem Lemma s m (z) = z2 (1 z m ) 1 z + z 2 s m+1 (z) + z 2 s m (z) 2. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

34 Enumerating binary terms Recurring problem Lemma This gives s m (z) = z2 (1 z m ) 1 z s m (z) = 1 + z 2 s m+1 (z) + z 2 s m (z) 2. ( ) 1 4z 4 1 z m 1 z + s m+1(z) 2z 2. Again we are stuck with innitely nested radicals... K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

35 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 S,k S,n k. The sequence (S,n ) n 0 has the entry number A 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, k=0 K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

36 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 / 28

37 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 satises S,n (1/ρ) n C n 3/2, where ρ = and C = K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

38 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 satises S,n (1/ρ) n C n 3/2, where ρ = and C = For m n 1 we have S m,n = S,n. Therefore s (z) = which yields that [z n ]s n = [z n ]s. Furthermore, s (z) = lim m s m(z). S n,n z n, n=1 K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

39 Enumerating binary terms Singularities of s m (z) Lemma Let ρ m denote the dominant singularity of s m (z). Then for every m, ρ m = ρ 0. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

40 Enumerating binary terms Singularities of s m (z) Lemma Let ρ m denote the dominant singularity of s m (z). Then for every m, ρ m = ρ 0. Lemma The dominant singularity of s 0 (z) is equal to the dominant singularity of s (z), i.e., ρ 0 = ρ = K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

41 Enumerating binary terms Sketch of the proof Let us dene functionals Φ m (F ) = Φ (F ) = 1 1 4z 4 ( 1 zm 1 z + F ) 2z 2, 1 1 4z 4 ( 1 1 z + F ) 2z 2. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

42 Enumerating binary terms Sketch of the proof Let us dene functionals Φ m (F ) = Φ (F ) = 1 1 4z 4 ( 1 zm 1 z + F ) 2z 2, 1 1 4z 4 ( 1 1 z + F ) 2z 2. Then s m (z) = Φ m (s m+1 (z)). K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

43 Enumerating binary terms Sketch of the proof Let us dene functionals Φ m (F ) = Φ (F ) = 1 1 4z 4 ( 1 zm 1 z + F ) 2z 2, 1 1 4z 4 ( 1 1 z + F ) 2z 2. Then s m (z) = Φ m (s m+1 (z)). For z [0, ρ] we have Φ m (s m (z)) s m (z) s m+1 (z) s (z). K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

44 Enumerating binary terms Sketch of the proof II Let s m (z) denote the xed point of Φ m, i.e., s m (z) is dened as the solution of the equation s m (z) = Φ m ( s m (z)). K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

45 Enumerating binary terms Sketch of the proof II Let s m (z) denote the xed point of Φ m, i.e., s m (z) is dened as the solution of the equation For z [0, ρ] we have s m (z) = Φ m ( s m (z)). s m (z) s m (z) s (z). K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

46 Enumerating binary terms Sketch of the proof II Let s m (z) denote the xed point of Φ m, i.e., s m (z) is dened as the solution of the equation For z [0, ρ] we have By the denition of s m (z), s m (z) = Φ m ( s m (z)). s m (z) s m (z) s (z). z 2 s 2 m(z) (1 z 2 ) 2 s m (z) + z2 (1 z m ) 1 z = 0. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

47 Enumerating binary terms Sketch of the proof II Let s m (z) denote the xed point of Φ m, i.e., s m (z) is dened as the solution of the equation For z [0, ρ] we have By the denition of s m (z), s m (z) = Φ m ( s m (z)). s m (z) s m (z) s (z). z 2 s 2 m(z) (1 z 2 ) 2 s m (z) + z2 (1 z m ) 1 z = 0. Let us denote the main singularity of s m (z) by σ m. We have σ m ρ m ρ. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

48 Enumerating binary terms Sketch of the proof III The discriminant of this equation is m = (1 z 2 ) 2 4z4 (1 z m ). 1 z K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

49 Enumerating binary terms Sketch of the proof III The discriminant of this equation is m = (1 z 2 ) 2 4z4 (1 z m ). 1 z The value of σ m is equal to the root of smallest modulus of the polynomial P m (z) := (z 1) m = 4z 4 (1 z m ) (1 z) 3 (1 + z) 2. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

50 Enumerating binary terms Sketch of the proof III The discriminant of this equation is m = (1 z 2 ) 2 4z4 (1 z m ). 1 z The value of σ m is equal to the root of smallest modulus of the polynomial P m (z) := (z 1) m = 4z 4 (1 z m ) (1 z) 3 (1 + z) 2. The sequence (σ m ) m N of roots of polynomials P m (z) is decreasing and it converges to ρ. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

51 Enumerating binary terms Sketch of the proof III The discriminant of this equation is m = (1 z 2 ) 2 4z4 (1 z m ). 1 z The value of σ m is equal to the root of smallest modulus of the polynomial P m (z) := (z 1) m = 4z 4 (1 z m ) (1 z) 3 (1 + z) 2. The sequence (σ m ) m N of roots of polynomials P m (z) is decreasing and it converges to ρ. By σ m ρ m ρ, we get ρ m ρ, as well. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

52 Enumerating binary terms Sketch of the proof III The discriminant of this equation is m = (1 z 2 ) 2 4z4 (1 z m ). 1 z The value of σ m is equal to the root of smallest modulus of the polynomial P m (z) := (z 1) m = 4z 4 (1 z m ) (1 z) 3 (1 + z) 2. The sequence (σ m ) m N of roots of polynomials P m (z) is decreasing and it converges to ρ. By σ m ρ m ρ, we get ρ m ρ, as well. Since all the ρ m 's are equal, we obtain that ρ m = ρ for every natural m. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

53 Enumerating binary terms Number of closed binary terms Theorem The number of closed binary binary lambda terms of size n is of exponential order S 0,n n. K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

54 Enumerating binary terms Conjecture For every m, S m,n o ( n n 3/2). K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

55 Enumerating binary terms Conjecture For every m, S m,n o ( n n 3/2). S m,n ( ρ n n 3/2) K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28

56 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 , 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 lambda-terms, Electronic Journal of Combinatorics, 20(4):P30, 2013 R. David, K. Grygiel, J. Kozik, Ch. Raalli, G. Theyssier, M. Zaionc: Asymptotically almost all lambda terms are strongly normalizing, Logical Methods in Computer Science Vol. 9(1:02) 2013, pp K. Grygiel, P. Lescanne: Counting and generating lambda terms, Journal of Functional Programming Vol. 23(5) 2013, pp 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 of Dagstuhl Seminar Proceedings, Schloss Dagstuhl, Germany, 2006 K. Grygiel & P. Lescanne Counting binary lambda terms AofA / 28