Statistical properties of lambda terms Maciej Bendkowski Olivier - - PowerPoint PPT Presentation

▶
statistical properties of lambda terms
SMART_READER_LITE
LIVE PREVIEW

Statistical properties of lambda terms Maciej Bendkowski Olivier - - PowerPoint PPT Presentation

Dec 15, 2022 673 likes •2.12k views

Lambda terms Statistical properties of lambda terms Maciej Bendkowski Olivier Bodini Sergey Dovgal CLA, Jussieu, Paris, 24/05/2018 Lambda terms Chapter 1. Historical overview Lambda terms Historical overview Lambda terms Historical

slide-1
SLIDE 1

Lambda terms

Statistical properties

  • f lambda terms

Maciej Bendkowski Olivier Bodini Sergey Dovgal CLA, Jussieu, Paris, 24/05/2018

slide-2
SLIDE 2

Lambda terms

Chapter 1. Historical overview

slide-3
SLIDE 3

Lambda terms Historical overview

slide-4
SLIDE 4

Lambda terms Historical overview

What is a closed lambda term?

slide-5
SLIDE 5

Lambda terms Historical overview

slide-6
SLIDE 6

Lambda terms Historical overview

Expression such as λx.λy.(λz.(λx.zx)(λy.zy))(xy)

slide-7
SLIDE 7

Lambda terms Historical overview

Expression such as λx.λy.(λz.(λx.zx)(λy.zy))(xy) λ — abstraction

slide-8
SLIDE 8

Lambda terms Historical overview

Expression such as λx.λy.(λz.(λx.z@x)(λy.z@y))@(x@y) λ — abstraction @ — application

slide-9
SLIDE 9

Lambda terms Historical overview

Expression such as λx.λy.(λz.(λx.zx)(λy.zy))(xy) λ — abstraction @ — application x, y, z — variables

slide-10
SLIDE 10

Lambda terms Historical overview

also lambda terms are..

slide-11
SLIDE 11

Lambda terms Historical overview

Unary-binary trees with links

slide-12
SLIDE 12

Lambda terms Historical overview

Unary-binary trees with links λ @ λ @ @

  • @
slide-13
SLIDE 13

Lambda terms Historical overview

Unary-binary trees with links λ @ λ @ @

  • @
  • λ

are abstractions @ are applications

  • are variables
slide-14
SLIDE 14

Lambda terms Historical overview

Local summary

Closed lambda terms are:

slide-15
SLIDE 15

Lambda terms Historical overview

Local summary

Closed lambda terms are: Expressions such as λx.λy.(λz.(λx.zx)(λy.zy))(xy) Unary-binary trees with links TODO: What are plain lambda terms? TODO: What is the difference between closed and plain lambda terms?

slide-16
SLIDE 16

Lambda terms Historical overview

Local summary

Closed lambda terms are: Expressions such as λx.λy.(λz.(λx.zx)(λy.zy))(xy) Unary-binary trees with links TODO: What are plain lambda terms? TODO: What is the difference between closed and plain lambda terms?

slide-17
SLIDE 17

Lambda terms Historical overview

Local summary

Closed lambda terms are: Expressions such as λx.λy.(λz.(λx.zx)(λy.zy))(xy) Unary-binary trees with links TODO: What are plain lambda terms? TODO: What is the difference between closed and plain lambda terms?

slide-18
SLIDE 18

Lambda terms Historical overview

Local summary

Closed lambda terms are: Expressions such as λx.λy.(λz.(λx.zx)(λy.zy))(xy) Unary-binary trees with links TODO: What are plain lambda terms? TODO: What is the difference between closed and plain lambda terms?

slide-19
SLIDE 19

Lambda terms Historical overview

slide-20
SLIDE 20

Lambda terms Historical overview

Motivation

slide-21
SLIDE 21

Lambda terms Historical overview

slide-22
SLIDE 22

Lambda terms Historical overview

First application. Sofware testing techniques.

slide-23
SLIDE 23

Lambda terms Historical overview

First application. Sofware testing techniques.

slide-24
SLIDE 24

Lambda terms Historical overview

Second motivation. Relation between (linear) lambda terms and (trivalent) maps.

slide-25
SLIDE 25

Lambda terms Historical overview

Third motivation. Development of analytic combinatorics.

slide-26
SLIDE 26

Lambda terms Historical overview

Motivation

Summary

Sofware testing techniques Lambda terms vs. maps Development of analytic combinatorics

slide-27
SLIDE 27

Lambda terms Historical overview

slide-28
SLIDE 28

Lambda terms Historical overview

Plain and closed lambda terms

slide-29
SLIDE 29

Lambda terms Historical overview

slide-30
SLIDE 30

Lambda terms Historical overview

Closed terms: unary-binary trees with links between variables and abstractions Plain terms: unary-binary trees with some variables unlinked

slide-31
SLIDE 31

Lambda terms Historical overview

Example.

slide-32
SLIDE 32

Lambda terms Historical overview

Example. λz.(λy.zy) is a closed term

slide-33
SLIDE 33

Lambda terms Historical overview

Example. λz.(λy.zy) is a closed term λy.zy is a plain term

slide-34
SLIDE 34

Lambda terms Historical overview

Example. λz.(λy.zy) is a closed term λy.zy is a plain term (λx.x)(λy.xy) ?

slide-35
SLIDE 35

Lambda terms Historical overview

slide-36
SLIDE 36

Lambda terms Historical overview

Size notion of a lambda term

slide-37
SLIDE 37

Lambda terms Historical overview

slide-38
SLIDE 38

Lambda terms Historical overview

What is the size of λ @ λ @ @

  • @
slide-39
SLIDE 39

Lambda terms Historical overview

First approach. Total number of nodes. λ @ λ @ @

  • @
  • 5 •

+ 4 @ + 2 λ = 11

slide-40
SLIDE 40

Lambda terms Historical overview

Second approach. Only abstractions and applications. λ @ λ @ @

  • @
  • 4 @ + 2 λ

= 6

slide-41
SLIDE 41

Lambda terms Historical overview

Third approach. Natural counting. λ @ λ @ @ 1 @ 4 @ + 2 λ + (2+1+1+1+1) • = 12

slide-42
SLIDE 42

Lambda terms Historical overview

Fourth approach. Generalised natural counting. λ @ λ @ @ 1 @ 4a @ +2b λ +5c • + (1+0+0+0+0)d • = 4a + 2b + 5c + d

slide-43
SLIDE 43

Lambda terms Historical overview

Size notions, recap

Variable size = 1 Variable size = 0 Variable size 1 (Natural counting) Generalised natural counting

slide-44
SLIDE 44

Lambda terms Historical overview

Size notions, recap

Variable size = 1 Variable size = 0 Variable size 1 (Natural counting) Generalised natural counting

slide-45
SLIDE 45

Lambda terms Historical overview

Size notions, recap

Variable size = 1 Variable size = 0 Variable size 1 (Natural counting) Generalised natural counting

slide-46
SLIDE 46

Lambda terms Historical overview

Size notions, recap

Variable size = 1 Variable size = 0 Variable size 1 (Natural counting) Generalised natural counting

slide-47
SLIDE 47

Lambda terms Historical overview

In-detail analysis of size notions

slide-48
SLIDE 48

Lambda terms Historical overview

  • 1. Variable size equal 1
slide-49
SLIDE 49

Lambda terms Historical overview

Variable size equal 1

[Bodini, Gardy, Gitenberger, Jacquot ’11+] Linear closed lambda terms Upper and lower bounds for asymptotics of all closed terms New bijections Continued [Zeilberger, Giorgeti + ’15+] :( Formal power series is divergent

slide-50
SLIDE 50

Lambda terms Historical overview

Variable size equal 1

[Bodini, Gardy, Gitenberger, Jacquot ’11+] Linear closed lambda terms Upper and lower bounds for asymptotics of all closed terms New bijections Continued [Zeilberger, Giorgeti + ’15+] :( Formal power series is divergent

slide-51
SLIDE 51

Lambda terms Historical overview

Variable size equal 1

[Bodini, Gardy, Gitenberger, Jacquot ’11+] Linear closed lambda terms Upper and lower bounds for asymptotics of all closed terms New bijections Continued [Zeilberger, Giorgeti + ’15+] :( Formal power series is divergent

slide-52
SLIDE 52

Lambda terms Historical overview

Variable size equal 1

[Bodini, Gardy, Gitenberger, Jacquot ’11+] Linear closed lambda terms Upper and lower bounds for asymptotics of all closed terms New bijections Continued [Zeilberger, Giorgeti + ’15+] :( Formal power series is divergent

slide-53
SLIDE 53

Lambda terms Historical overview

Variable size equal 1

[Bodini, Gardy, Gitenberger, Jacquot ’11+] Linear closed lambda terms Upper and lower bounds for asymptotics of all closed terms New bijections Continued [Zeilberger, Giorgeti + ’15+] :( Formal power series is divergent

slide-54
SLIDE 54

Lambda terms Historical overview

Variable size equal 1

[Bodini, Gardy, Gitenberger, Jacquot ’11+] Linear closed lambda terms Upper and lower bounds for asymptotics of all closed terms New bijections Continued [Zeilberger, Giorgeti + ’15+] :( Formal power series is divergent

slide-55
SLIDE 55

Lambda terms Historical overview

  • 2. Variable size equal 0
slide-56
SLIDE 56

Lambda terms Historical overview

Variable size equal 0

[Grygiel, Lescanne ’12] Enumeration and random generation [David, Grygiel, Kozik, Raffalli, Theyssier, Zaionc ’13] Θ(

  • n/ log n) head abstractions

:( Formal power series is divergent

slide-57
SLIDE 57

Lambda terms Historical overview

Variable size equal 0

[Grygiel, Lescanne ’12] Enumeration and random generation [David, Grygiel, Kozik, Raffalli, Theyssier, Zaionc ’13] Θ(

  • n/ log n) head abstractions

:( Formal power series is divergent

slide-58
SLIDE 58

Lambda terms Historical overview

Variable size equal 0

[Grygiel, Lescanne ’12] Enumeration and random generation [David, Grygiel, Kozik, Raffalli, Theyssier, Zaionc ’13] Θ(

  • n/ log n) head abstractions

:( Formal power series is divergent

slide-59
SLIDE 59

Lambda terms Historical overview

Variable size equal 0

[Grygiel, Lescanne ’12] Enumeration and random generation [David, Grygiel, Kozik, Raffalli, Theyssier, Zaionc ’13] Θ(

  • n/ log n) head abstractions

:( Formal power series is divergent

slide-60
SLIDE 60

Lambda terms Historical overview

Variable size equal 0

[Grygiel, Lescanne ’12] Enumeration and random generation [David, Grygiel, Kozik, Raffalli, Theyssier, Zaionc ’13] Θ(

  • n/ log n) head abstractions

:( Formal power series is divergent

slide-61
SLIDE 61

Lambda terms Historical overview

  • 3. Natural counting
slide-62
SLIDE 62

Lambda terms Historical overview

Natural counting

Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

slide-63
SLIDE 63

Lambda terms Historical overview

Natural counting

Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

slide-64
SLIDE 64

Lambda terms Historical overview

Natural counting

Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

slide-65
SLIDE 65

Lambda terms Historical overview

Natural counting

Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

slide-66
SLIDE 66

Lambda terms Historical overview

Natural counting

Variable size = unary distance to binding λ [Bendkowski, Grygiel, Lescanne, Zaionc ’12] [Bodini, Gitenberger, Gołe ¸biewski ’16] Enumeration and asymptotics :) Formal power series is convergent Uniform generation by Boltzmann sampling

slide-67
SLIDE 67

Lambda terms Historical overview

Why consider a simpler model?

slide-68
SLIDE 68

Lambda terms Historical overview

Why consider a simpler model?

Main reason: advantages in random generation

slide-69
SLIDE 69

Lambda terms Historical overview

Why consider a simpler model?

Marking variables imply control over expectations

slide-70
SLIDE 70

Lambda terms Historical overview

Why consider a simpler model?

Another reason: parameter study

slide-71
SLIDE 71

Lambda terms Historical overview

Why consider a simpler model?

Marking variables allow to study distributions

slide-72
SLIDE 72

Lambda terms Statistical properties

Chapter 2. Statistical properties

slide-73
SLIDE 73

Lambda terms Statistical properties

slide-74
SLIDE 74

Lambda terms Statistical properties

Plain lambda terms

slide-75
SLIDE 75

Lambda terms Statistical properties

Generating function for plain lambda terms

L = λ L + @ L L + D L(z) = zL(z) + zL2(z) + z 1 − z

slide-76
SLIDE 76

Lambda terms Statistical properties

Abstractions in plain terms?

L = λ L + @ L L + D L(z, u) = zuL(z, u) + zL(z, u) + z 1 − z L(z, u) ∼ a(z, u) − b(z, u)

  • 1 −

z ρ(u)

slide-77
SLIDE 77

Lambda terms Statistical properties

Multivariate Central Limit Theorem

slide-78
SLIDE 78

Lambda terms Statistical properties

Step 1. Extract coefficient of L(z, u)

slide-79
SLIDE 79

Lambda terms Statistical properties

L(z, u) ∼ a(z, u) − b(z, u)

  • 1 −

z ρ(u)

slide-80
SLIDE 80

Lambda terms Statistical properties

Step 2. Asymptotic behaviour of probability generating function

slide-81
SLIDE 81

Lambda terms Statistical properties

pn(u) ∼ A(u)B(u)n

slide-82
SLIDE 82

Lambda terms Statistical properties

Step 3. Gaussian approximation from A(u)B(u)n.

slide-83
SLIDE 83

Lambda terms Statistical properties

Applying multivariate CLT

Theorem Joint Gaussian distribution for number of abstractions number of variables number of redexes in plain lambda terms

slide-84
SLIDE 84

Lambda terms Statistical properties

Discrete distributions in plain terms

slide-85
SLIDE 85

Lambda terms Statistical properties

Head abstractions

L∞ = H D + H @

L∞|u=1 L∞|u=1

L(z, u) = 1 1 − zu

  • z

1 − z + zL(z, 1)2

slide-86
SLIDE 86

Lambda terms Statistical properties

Theorem The following statistics follow discrete (geometric) limiting distributions The number of head abstractions Randomly chosed value of de Bruijn index

slide-87
SLIDE 87

Lambda terms Infinite systems

Chapter 3. Infinite systems

slide-88
SLIDE 88

Lambda terms Infinite systems

Drmota–Lalley–Woods theorem

slide-89
SLIDE 89

Lambda terms Infinite systems

Drmota–Lalley–Woods theorem

Let F(z) be a generating function Suppose it satisfies F(z) = Φ(F(z), z) with Φ having combinatorial origin Then, F(z) ∼ a − b

  • 1 − z

ρ

slide-90
SLIDE 90

Lambda terms Infinite systems

Drmota–Lalley–Woods theorem

Let F(z) be a generating function Suppose it satisfies F(z) = Φ(F(z), z) with Φ having combinatorial origin Then, F(z) ∼ a − b

  • 1 − z

ρ

slide-91
SLIDE 91

Lambda terms Infinite systems

Drmota–Lalley–Woods theorem

Let F(z) be a generating function Suppose it satisfies F(z) = Φ(F(z), z) with Φ having combinatorial origin Then, F(z) ∼ a − b

  • 1 − z

ρ

slide-92
SLIDE 92

Lambda terms Infinite systems

What happens with infinite systems?

slide-93
SLIDE 93

Lambda terms Infinite systems

[Drmota, Gitenberger, Morgenbesser ’12] If Jacobian is a sum of identity matrix and a compact operator And specification is strongly connected Then Infinite-dimensional version holds

slide-94
SLIDE 94

Lambda terms Infinite systems

Closed lambda terms satisfy an infinite system

slide-95
SLIDE 95

Lambda terms Infinite systems

L0(z) = zL1(z) + zL0(z)2 , L1(z) = zL2(z) + zL1(z)2 + z , L2(z) = zL3(z) + zL2(z)2 + z + z2 , . . . L∞(z) = zL∞(z) + zL∞(z) + z 1 − z .

slide-96
SLIDE 96

Lambda terms Infinite systems

Why?

slide-97
SLIDE 97

Lambda terms Infinite systems

Adding m abstractions on the top of Lm makes the term closed

slide-98
SLIDE 98

Lambda terms Infinite systems

Example.

slide-99
SLIDE 99

Lambda terms Infinite systems

Example. λ @ λ @ @ 1 @ 3

slide-100
SLIDE 100

Lambda terms Infinite systems

Lm = λ Lm+1 + @ Lm Lm + Dm Lm(z) = zLm+1(z) + zL2

m(z) + z 1 − zm

1 − z

slide-101
SLIDE 101

Lambda terms Infinite systems

However

slide-102
SLIDE 102

Lambda terms Infinite systems

This system

slide-103
SLIDE 103

Lambda terms Infinite systems

L0(z) = zL1(z) + zL0(z)2 , L1(z) = zL2(z) + zL1(z)2 + z , L2(z) = zL3(z) + zL2(z)2 + z + z2 , . . . L∞(z) = zL∞(z) + zL∞(z) + z 1 − z .

slide-104
SLIDE 104

Lambda terms Infinite systems

Doesn’t satisfy infinite DLW theorem

slide-105
SLIDE 105

Lambda terms Infinite systems

slide-106
SLIDE 106

Lambda terms Infinite systems

New “master theorem”

slide-107
SLIDE 107

Lambda terms Infinite systems

Master theorem

Motivation

slide-108
SLIDE 108

Lambda terms Infinite systems

Master theorem

Motivation Lm(z) = zLm+1(z) + zLm(z)2 + z 1 − zm 1 − z

Dm

, L∞(z) = zL∞(z) + zL∞(z)2 + z 1 1 − z

D∞

. The difference D∞ − Dm is exponentially small

slide-109
SLIDE 109

Lambda terms Infinite systems

Master theorem

More generally. Lm(z) = Km(Lm, Lm+1, z, u) , L∞(z) = K∞(L∞, L∞, z, u) , K∞ − Km is exponentially small specified at L∞ Lm, Km, u are vectors

slide-110
SLIDE 110

Lambda terms Infinite systems

Master theorem

Assumptions

1 The limiting system has Puiseux expansion 2 Limiting system dominates m-th system 3 The difference is exponentially small

Statement Each Lm also has Puiseux expansion∗

slide-111
SLIDE 111

Lambda terms Infinite systems

Each Lm also has Puiseux expansion∗

Two possible variants:

1 (Weak)

[zn]Lm(z, u) ∼ [zn]

  • a(u) − b(u)
  • 1 −

z ρ(u)

  • 2 (Strong)

The analytic continuation of Lm(z, u) in a delta-domain satisfies Lm(z, u) ∼ a(u) − b(u)

  • 1 −

z ρ(u)

slide-112
SLIDE 112

Lambda terms Infinite systems

slide-113
SLIDE 113

Lambda terms Infinite systems

Applications of “master theorem”

slide-114
SLIDE 114

Lambda terms Infinite systems

slide-115
SLIDE 115

Lambda terms Infinite systems

Theorem Joint Gaussian distribution for number of abstractions number of variables number of successors number of redexes in closed lambda terms

slide-116
SLIDE 116

Lambda terms Infinite systems

Theorem Discrete distributions for number of head abstractions randomly chosed index value redex search time free variables missing top abstractions in closed lambda terms

slide-117
SLIDE 117

Lambda terms Infinite systems

Theorem Rayleigh distributions for unary height profile natural height profile in plain and closed lambda terms for variables, abstractions and applications

slide-118
SLIDE 118

Lambda terms Infinite systems

Summary for statistics

Normal limit laws New discrete limit laws Rayleigh limit laws

slide-119
SLIDE 119

Lambda terms Infinite systems

Summary for statistics

Normal limit laws New discrete limit laws Rayleigh limit laws

slide-120
SLIDE 120

Lambda terms Infinite systems

Summary for statistics

Normal limit laws New discrete limit laws Rayleigh limit laws

slide-121
SLIDE 121

Lambda terms Infinite systems

Grand summary for statistics

slide-122
SLIDE 122

Lambda terms Open problems and experiments

  • Epilogue. Open problems and experiments
slide-123
SLIDE 123

Lambda terms Open problems and experiments

Redex search time distribution

2 4 6 8 10 12 0.0 0.1 0.2 0.3 0.4 0.5

plain terms

2 4 6 8 10 12 0.00 0.05 0.10 0.15 0.20 0.25 0.30

closed terms

slide-124
SLIDE 124

Lambda terms Open problems and experiments

Head abstractions

1 2 3 4 5 6 7 8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

plain terms

1 2 3 4 5 6 7 8 0.0 0.1 0.2 0.3 0.4 0.5

closed terms

slide-125
SLIDE 125

Lambda terms Open problems and experiments

Free variables

2 4 6 8 10 12 0.0 0.1 0.2 0.3 0.4 0.5

slide-126
SLIDE 126

Lambda terms Open problems and experiments

Maximum number of variables bound to a single abstraction

16 21 26 31 36 41 46 51 56 61 66 71 77 500 1000 1500 2000 2500 3000 3500

plain terms

16 21 26 31 36 41 46 51 56 61 66 71 76 84 500 1000 1500 2000 2500 3000 3500

closed terms

slide-127
SLIDE 127

Lambda terms Open problems and experiments

Generalised m-openness

  • 7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7 8 9 10 11 12 2000 4000 6000 8000 10000 12000 14000

slide-128
SLIDE 128

Lambda terms Open problems and experiments

Number of binding abstractions

0.61 0.62 0.63 0.64 0.65 200 400 600 800 1000 1200 1400 1600

plain terms

0.61 0.62 0.63 0.64 0.65 200 400 600 800 1000 1200 1400

closed terms

slide-129
SLIDE 129

Lambda terms Open problems and experiments

Open questions

Maximum number of variables bound to a single abstraction Number of binding abstractions Generalised m-openness Number of BCI, BCK terms in natural size notion Average shape of a random lambda term afer k β-reductions?

slide-130
SLIDE 130

Lambda terms Open problems and experiments

Open questions

Maximum number of variables bound to a single abstraction Number of binding abstractions Generalised m-openness Number of BCI, BCK terms in natural size notion Average shape of a random lambda term afer k β-reductions?

slide-131
SLIDE 131

Lambda terms Open problems and experiments

Open questions

Maximum number of variables bound to a single abstraction Number of binding abstractions Generalised m-openness Number of BCI, BCK terms in natural size notion Average shape of a random lambda term afer k β-reductions?

slide-132
SLIDE 132

Lambda terms Open problems and experiments

Open questions

Maximum number of variables bound to a single abstraction Number of binding abstractions Generalised m-openness Number of BCI, BCK terms in natural size notion Average shape of a random lambda term afer k β-reductions?

slide-133
SLIDE 133

Lambda terms Open problems and experiments

Open questions

Maximum number of variables bound to a single abstraction Number of binding abstractions Generalised m-openness Number of BCI, BCK terms in natural size notion Average shape of a random lambda term afer k β-reductions?

slide-134
SLIDE 134

Lambda terms Conclusions

Conclusions

slide-135
SLIDE 135

Lambda terms Conclusions

First.

slide-136
SLIDE 136

Lambda terms Conclusions

  • First. Many tractable parameters in closed lambda

terms in de Bruijn size notion

slide-137
SLIDE 137

Lambda terms Conclusions

Second.

slide-138
SLIDE 138

Lambda terms Conclusions

  • Second. Efficient multiparametric Boltzmann

sampling, tweaking the desired parameters

slide-139
SLIDE 139

Lambda terms Conclusions

  • Second. Efficient multiparametric Boltzmann

sampling, tweaking the desired parameters L = λ L

30%

+ @ L L + D

slide-140
SLIDE 140

Lambda terms Conclusions

Third.

slide-141
SLIDE 141

Lambda terms Conclusions

  • Third. Infinite systems of algebraic equations∗
slide-142
SLIDE 142

Lambda terms Conclusions

  • Third. Infinite systems of algebraic equations∗

Conjecture Premises of DLW Theorem for infinite strongly connected systems are not sufficient.

slide-143
SLIDE 143

Lambda terms Conclusions

T h a n k y

  • u

!