Statistical properties of random lambda-terms in de-Bruijn notation - - PowerPoint PPT Presentation

Problem and Motivation Statistics of lambda-terms Open problems

Statistical properties of random lambda-terms in de-Bruijn notation∗

Maciej Bendkowski3 Olivier Bodini1 Sergey Dovgal1,2,4

1Université Paris-13, 2Université Paris-Diderot 3Jagiellonian University 4Moscow

Institute of Physics and Technology

CLA-2017, Götenburg, Sweden

∗ in progress

Bendkowski, Bodini, D. Statistics of lambda-terms 1 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

1

Problem and Motivation

2

Statistics of lambda-terms

3

Open problems

Bendkowski, Bodini, D. Statistics of lambda-terms 2 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Outline

1

Problem and Motivation

2

Statistics of lambda-terms

3

Open problems

Bendkowski, Bodini, D. Statistics of lambda-terms 2 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Example of lambda-term in de-Bruijn notation

λ λ λ @ @ 2 @ 1 A closed lambda-term λx.λy.λz.xz(yz)

Bendkowski, Bodini, D. Statistics of lambda-terms 3 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Grammar of plain lambda-terms

L = λ L + @ L L + N L — plain lambda-term λ — abstraction @ — application N — variable; de Bruijn index ∈ {0, 1, 2, . . .}

Bendkowski, Bodini, D. Statistics of lambda-terms 4 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Redex and beta-reduction

@ λ L L (λn.n × 2)7

β

→ 7 × 2 Ω = (λx.xx)(λx.xx) Ω

β

→ Ω

Bendkowski, Bodini, D. Statistics of lambda-terms 5 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Redex and beta-reduction

@ λ L ⊙ ⊙ ⊙ L L L

β

− →

Bendkowski, Bodini, D. Statistics of lambda-terms 6 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Main question

Investigate statistical properties of random plain / closed lambda-terms in de-Bruijn notation:

• number of lambdas, variables, abstractions,. . .
• length to the lefmost outermost redex,
• unary height, longest lambda-run
• . . .

Statistical properties Random generation ⇒ Property-based testing

Bendkowski, Bodini, D. Statistics of lambda-terms 7 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Size notion of lambda-terms

[Bodini, Gardy, Gitenberger, Jacquot ’13] Closed lambda-terms with variable size = 1. [David, Grygiel, Kozik, Raffalli, Theyssier, Zaionc ’13] Closed lambda-terms with variable size = 0. [Gitenberger, Gołe ¸biewski ’16] Natural counting of lambda-terms. |0| = a, |S| = b, |λ| = d, |@| = d

Bendkowski, Bodini, D. Statistics of lambda-terms 8 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Size notion of lambda-terms

[Bodini, Gardy, Gitenberger, Jacquot ’13] Closed lambda-terms with variable size = 1. [David, Grygiel, Kozik, Raffalli, Theyssier, Zaionc ’13] Closed lambda-terms with variable size = 0. [Gitenberger, Gołe ¸biewski ’16] Natural counting of lambda-terms. |0| = a, |S| = b, |λ| = d, |@| = d

Bendkowski, Bodini, D. Statistics of lambda-terms 8 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Size notion of lambda-terms

[Bodini, Gardy, Gitenberger, Jacquot ’13] Closed lambda-terms with variable size = 1. [David, Grygiel, Kozik, Raffalli, Theyssier, Zaionc ’13] Closed lambda-terms with variable size = 0. [Gitenberger, Gołe ¸biewski ’16] Natural counting of lambda-terms. |0| = a, |S| = b, |λ| = d, |@| = d

Bendkowski, Bodini, D. Statistics of lambda-terms 8 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

What is “random”?

Thanks to previous talks

Natural size notion of lambda-term.

Stay tuned for the definition and comparison to other models.

Sample lambda-terms of size n uniformly

(leitmotif of this talk, but not the only possibility)

Sample lambda-terms of size n and parameter value k uniformly.

Bivariate generating function + tuning of Boltzmann sampler.

Choose any subset of parameters, fix their values and sample at uniform from the desired set.

[Bodini, Ponty ’10]: Newton iteration or asymptotic approximations [Bodini, D. ’17]: Fast exact tuning in O

• log2(size) · #vars7 · (#vars+#eqs)
• .

Bendkowski, Bodini, D. Statistics of lambda-terms 9 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

What is “random”?

Thanks to previous talks

Natural size notion of lambda-term.

Stay tuned for the definition and comparison to other models.

Sample lambda-terms of size n uniformly

(leitmotif of this talk, but not the only possibility)

Sample lambda-terms of size n and parameter value k uniformly.

Bivariate generating function + tuning of Boltzmann sampler.

Choose any subset of parameters, fix their values and sample at uniform from the desired set.

[Bodini, Ponty ’10]: Newton iteration or asymptotic approximations [Bodini, D. ’17]: Fast exact tuning in O

• log2(size) · #vars7 · (#vars+#eqs)
• .

Bendkowski, Bodini, D. Statistics of lambda-terms 9 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

What is “random”?

Thanks to previous talks

Natural size notion of lambda-term.

Stay tuned for the definition and comparison to other models.

Sample lambda-terms of size n uniformly

(leitmotif of this talk, but not the only possibility)

Sample lambda-terms of size n and parameter value k uniformly.

Bivariate generating function + tuning of Boltzmann sampler.

Choose any subset of parameters, fix their values and sample at uniform from the desired set.

[Bodini, Ponty ’10]: Newton iteration or asymptotic approximations [Bodini, D. ’17]: Fast exact tuning in O

• log2(size) · #vars7 · (#vars+#eqs)
• .

Bendkowski, Bodini, D. Statistics of lambda-terms 9 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

What is “random”?

Thanks to previous talks

Natural size notion of lambda-term.

Stay tuned for the definition and comparison to other models.

Sample lambda-terms of size n uniformly

(leitmotif of this talk, but not the only possibility)

Sample lambda-terms of size n and parameter value k uniformly.

Bivariate generating function + tuning of Boltzmann sampler.

Choose any subset of parameters, fix their values and sample at uniform from the desired set.

[Bodini, Ponty ’10]: Newton iteration or asymptotic approximations [Bodini, D. ’17]: Fast exact tuning in O

• log2(size) · #vars7 · (#vars+#eqs)
• .

Bendkowski, Bodini, D. Statistics of lambda-terms 9 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Closed lambda-terms?

λ λ λ @ @ 3 @ 1 ? The value of each index shouldn’t exceed maximal unary distance to parent lambda.

Bendkowski, Bodini, D. Statistics of lambda-terms 10 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

m-open lambda-terms

• Def. A lambda-term T is m-open if λmT is closed.
• Observation. m = 0 corresponds to closed terms
• Def. Lm — class of m-open lambda-terms.
• Def. L∞ — class of plain lambda-terms.
• Observation. L0 ⊂ L1 ⊂ L2 ⊂ . . . ⊂ L∞

Bendkowski, Bodini, D. Statistics of lambda-terms 11 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

m-open lambda-terms

• Def. A lambda-term T is m-open if λmT is closed.
• Observation. m = 0 corresponds to closed terms
• Def. Lm — class of m-open lambda-terms.
• Def. L∞ — class of plain lambda-terms.
• Observation. L0 ⊂ L1 ⊂ L2 ⊂ . . . ⊂ L∞

Bendkowski, Bodini, D. Statistics of lambda-terms 11 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

m-open lambda-terms

• Def. A lambda-term T is m-open if λmT is closed.
• Observation. m = 0 corresponds to closed terms
• Def. Lm — class of m-open lambda-terms.
• Def. L∞ — class of plain lambda-terms.
• Observation. L0 ⊂ L1 ⊂ L2 ⊂ . . . ⊂ L∞

Bendkowski, Bodini, D. Statistics of lambda-terms 11 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

m-open lambda-terms

• Def. A lambda-term T is m-open if λmT is closed.
• Observation. m = 0 corresponds to closed terms
• Def. Lm — class of m-open lambda-terms.
• Def. L∞ — class of plain lambda-terms.
• Observation. L0 ⊂ L1 ⊂ L2 ⊂ . . . ⊂ L∞

Bendkowski, Bodini, D. Statistics of lambda-terms 11 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

m-open lambda-terms

• Def. A lambda-term T is m-open if λmT is closed.
• Observation. m = 0 corresponds to closed terms
• Def. Lm — class of m-open lambda-terms.
• Def. L∞ — class of plain lambda-terms.
• Observation. L0 ⊂ L1 ⊂ L2 ⊂ . . . ⊂ L∞

Bendkowski, Bodini, D. Statistics of lambda-terms 11 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Closed terms specification

L0 = λ L1 + @ L0 L0 L1 = λ L2 + @ L1 L1 + 1 Lh = λ Lh

+ 1

+ @ Lh Lh + 1 + 2 + · · · +

h

Bendkowski, Bodini, D. Statistics of lambda-terms 12 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Asymptotic number of plain lambda-terms

Theorem. As n → ∞, the number of plain lambda-terms of size n is asymptotically b∞n−3/2 2√π 1 ρ n , (1 − ρ)3 = 4ρ2 . ρ ≈ 0.29559

Bendkowski, Bodini, D. Statistics of lambda-terms 13 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Asymptotic number of m-open lambda-terms

Theorem. As n → ∞, the number of plain lambda-terms of size n is asymptotically b∞n−3/2 2√π 1 ρ n , (1 − ρ)3 = 4ρ2 . Theorem. The asymptotic probability that a random plain lambda-term of size n is m-open tends to some positive constant pm as n → ∞. This distribution is computable.

Bendkowski, Bodini, D. Statistics of lambda-terms 14 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Asymptotic number of m-open lambda-terms

Theorem. The asymptotic probability that a random plain lambda-term of size n is m-open tends to some positive constant pm as n → ∞. This distribution is computable. Open question. What is the behaviour of the sequence (pk)∞

k=0

which is a cumulative distribution function? We only know that pm+1 ≥ pm and pm → 1 as m → ∞.

Bendkowski, Bodini, D. Statistics of lambda-terms 14 / 31

Problem and Motivation Statistics of lambda-terms Open problems Plain and closed lambda-terms

Asymptotic number of m-open lambda-terms

Theorem. The asymptotic probability that a random plain lambda-term of size n is m-open tends to some positive constant pm as n → ∞. This distribution is computable. Open question. What is the behaviour of the sequence (pk)∞

k=0

which is a cumulative distribution function? We only know that pm+1 ≥ pm and pm → 1 as m → ∞. The reccurence can be considered either forward or backwards:        am+1 = am/ρ − a2

m − 1 − ρm

1 − ρ , bm+1 = bm/ρ − 2ambm, pm = bm/b∞

Bendkowski, Bodini, D. Statistics of lambda-terms 14 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Outline

1

Problem and Motivation

2

Statistics of lambda-terms

3

Open problems

Bendkowski, Bodini, D. Statistics of lambda-terms 15 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Zoo of different statistics

Marking techniques Number of lambdas Number of variables Number of abstractions Number of redexes Value of de-Bruijn index Number of head abstractions Extremal techniques Maximal de-Bruijn index value Unary height of a random term Longest lambda-run Advanced marking Number of free variables Number of closed subterms Expected search time for β-reduction Number of variables bound to top lambda

Bendkowski, Bodini, D. Statistics of lambda-terms 16 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Marking techniques

Marking techniques Number of lambdas Number of variables Number of abstractions Number of redexes Number of head abstractions Value of de-Bruijn index

Bendkowski, Bodini, D. Statistics of lambda-terms 17 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Statistics with gaussian distribution

Theorem. In random plain lambda-term of size n En(# lambdas) = Cλ · n + O(n1/2) , En(# applications) = C@ · n + O(n1/2) , En(# variables) = CN · n + O(n1/2) , En(# redexes) = Credex · n + O(n1/2) , The distribution is asymptotically Gaussian, i.e. # − E# V#

d

→ N(0, 1)

Bendkowski, Bodini, D. Statistics of lambda-terms 18 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Statistics with gaussian distribution

Theorem. In random closed lambda-term of size n En(# lambdas) = Cλ · n + O(n1/2) , En(# applications) = C@ · n + O(n1/2) , En(# variables) = CN · n + O(n1/2) , En(# redexes) = Credex · n + O(n1/2) , The constants Cλ, C@, CN , Credex are the same as in the plain case. # − E# V#

d

→ N(0, 1)

Bendkowski, Bodini, D. Statistics of lambda-terms 19 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Head abstractions

Theorem. The number of head abstractions in random plain lambda-term has a limiting distribution Geom(ρ), i.e. P(# head abstractions ≤ m) = 1 − ρm+1 The number of head abstractions in random closed lambda-term has cumulative distribution function P(# head abstractions ≤ m) = 1 − ρm+1 pm+1 p0

Bendkowski, Bodini, D. Statistics of lambda-terms 20 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Head abstractions

Theorem. The number of head abstractions in random plain lambda-term has a limiting distribution Geom(ρ), i.e. P(# head abstractions ≤ m) = 1 − ρm+1 The number of head abstractions in random closed lambda-term has cumulative distribution function P(# head abstractions ≤ m) = 1 − ρm+1 pm+1 p0

Bendkowski, Bodini, D. Statistics of lambda-terms 20 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Distribution of de-Bruijn index

Theorem. In large random plain lambda-terms, the value of de Bruijn index has a limiting distribution which is Geom(ρ). In large random closed lambda-terms, the value of de Bruijn index has a computable limiting distribution. Pm ∼ ρ2m (1 − 2ρa0) . . . (1 − 2ρam−1)

Bendkowski, Bodini, D. Statistics of lambda-terms 21 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Distribution of de-Bruijn index

Theorem. In large random plain lambda-terms, the value of de Bruijn index has a limiting distribution which is Geom(ρ). In large random closed lambda-terms, the value of de Bruijn index has a computable limiting distribution. Pm ∼ ρ2m (1 − 2ρa0) . . . (1 − 2ρam−1)

Bendkowski, Bodini, D. Statistics of lambda-terms 21 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Advanced marking

Advanced marking Number of free variables Number of closed subterms Expected search time for β-reduction Number of variables bound to head lambda

Bendkowski, Bodini, D. Statistics of lambda-terms 22 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Number of free variables

Theorem. Inside large plain lambda-terms the number of free variables has a computable discrete limiting distribution, in particular, the average number of free variables is a constant En ∼ 2 (1 − ρ)3

Bendkowski, Bodini, D. Statistics of lambda-terms 23 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Number of free variables

Theorem. Inside large plain lambda-terms the number of free variables has a computable discrete limiting distribution, in particular, the average number of free variables is a constant “Paradox” Almost all the variables are bounded but not all the terms are closed!

Bendkowski, Bodini, D. Statistics of lambda-terms 23 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Number of free variables

Theorem. Inside large plain lambda-terms the number of free variables has a computable discrete limiting distribution, in particular, the average number of free variables is a constant “Paradox” Almost all the variables are bounded but not all the terms are closed! Intuition. Distribution of db-index is geometric, but unary height is O(√n). A small proportion of variables makes the term

• pen with positive probability.

Bendkowski, Bodini, D. Statistics of lambda-terms 23 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Number of closed subterms

Theorem. Inside large plain (also closed) lambda-terms the number of closed subterms satisfies En ∼ Θ(n), Vn ∼ Θ(1) Open question. What is the distribution?

Bendkowski, Bodini, D. Statistics of lambda-terms 24 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Expected search time for β-reduction

Theorem. Inside large plain lambda-terms the number of steps until first redex discovery has a computable discrete limiting distribution, in particular, the expected time is a computable constant.

Bendkowski, Bodini, D. Statistics of lambda-terms 25 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Average number of variables bound to top lambda

Theorem. Inside large plain lambda-terms choose an abstraction uniformly at random among abstractions at unary height 1. E (number of vars bound by top lambda) ∼ C The same holds for lambda at fixed unary height m. The constant is not necessary the same. Open question 1. Limiting discrete distribution? The same holds for closed lambda-terms? Open question 2.∗ Number of binding lambdas?

Bendkowski, Bodini, D. Statistics of lambda-terms 26 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Extremal techniques Maximal de-Bruijn index value Unary height of a random term Longest lambda-run

Bendkowski, Bodini, D. Statistics of lambda-terms 27 / 31

Problem and Motivation Statistics of lambda-terms Open problems Zoo of different statistics Marking techniques

Extremal statistics

Theorem. In random plain lambda-term of size n En(longest lambda-run) ∼ log n log(1/ρ) + O(log log n), En(maximal de Bruijn index) ∼ log n log(1/ρ) + O(log log n), En(unary height) ∼ Θ(√n). Conjecture. The same is true for closed lambda-terms.

Bendkowski, Bodini, D. Statistics of lambda-terms 28 / 31

Problem and Motivation Statistics of lambda-terms Open problems

Outline

1

Problem and Motivation

2

Statistics of lambda-terms

3

Open problems

Bendkowski, Bodini, D. Statistics of lambda-terms 29 / 31

Problem and Motivation Statistics of lambda-terms Open problems

Open problems

Supposedly hard

Combinatorics of lambda-term afer beta-reduction procedure Closed BCI, BCK, λ-I terms. Each lambda binds (·) variables

≤ 1 BCI = 1 λ-I ≥ 1 BCK

Riccati PDE, multivariate saddle-point, etc. [Lescanne ’17] SwissCheese: keeping a large vector of information to track the number of variables on every level

Number of binding lambdas Phase transitions with respect to abcd size notion

Bendkowski, Bodini, D. Statistics of lambda-terms 30 / 31

Problem and Motivation Statistics of lambda-terms Open problems

Open problems

Supposedly hard

Combinatorics of lambda-term afer beta-reduction procedure Closed BCI, BCK, λ-I terms. Each lambda binds (·) variables

≤ 1 BCI = 1 λ-I ≥ 1 BCK

Riccati PDE, multivariate saddle-point, etc. [Lescanne ’17] SwissCheese: keeping a large vector of information to track the number of variables on every level

Number of binding lambdas Phase transitions with respect to abcd size notion

Bendkowski, Bodini, D. Statistics of lambda-terms 30 / 31

Problem and Motivation Statistics of lambda-terms Open problems

Open problems

Supposedly hard

Combinatorics of lambda-term afer beta-reduction procedure Closed BCI, BCK, λ-I terms. Each lambda binds (·) variables

≤ 1 BCI = 1 λ-I ≥ 1 BCK

Riccati PDE, multivariate saddle-point, etc. [Lescanne ’17] SwissCheese: keeping a large vector of information to track the number of variables on every level

Number of binding lambdas Phase transitions with respect to abcd size notion

Bendkowski, Bodini, D. Statistics of lambda-terms 30 / 31

Problem and Motivation Statistics of lambda-terms Open problems

Open problems

Supposedly hard

Combinatorics of lambda-term afer beta-reduction procedure Closed BCI, BCK, λ-I terms. Each lambda binds (·) variables

≤ 1 BCI = 1 λ-I ≥ 1 BCK

Riccati PDE, multivariate saddle-point, etc. [Lescanne ’17] SwissCheese: keeping a large vector of information to track the number of variables on every level

Number of binding lambdas Phase transitions with respect to abcd size notion

Bendkowski, Bodini, D. Statistics of lambda-terms 30 / 31

Problem and Motivation Statistics of lambda-terms Open problems

That’s all!

Thank you for your atention!

Bendkowski, Bodini, D. Statistics of lambda-terms 31 / 31