Untyped Lambda Calculus

An introduction to the Untyped Lambda Calculus.

Topics:
- Structure and Evaluation
- Higher Order Functions, Currying
- Church Encodings (Booleans, Pairs, Numerals)

This talk was given at the Lambda Pi MeetUp in Hamburg (https://github.com/lambda-pi-meetup/talks).

Transcript

1. Untyped Lambda Calculus Structure and Evaluation, Currying, Church Encodings Sven Tennie August 9, 2018 Dream IT https://dreamit.de
2. Introduction
3. Lambda Calculus • Invented by Alonzo Church (1920s) • Equally expressive to the Turing Machine(s) • Formal Language • Computational Model • Lisp (1950s) • ML • Haskell • "Lambda Expressions" in almost every modern programming language 1
4. Why should I care? • Simple Computational Model • to describe structure and behaviour (E.g. Operational Semantics, Type Systems) • to reason and prove 2
5. Why should I care? • Simple Computational Model • to describe structure and behaviour (E.g. Operational Semantics, Type Systems) • to reason and prove • Explains why things in FP are like they are • Pure Functions • Higher-Order Functions • Currying • Lazy Evaluation 2
6. Why should I care? • Simple Computational Model • to describe structure and behaviour (E.g. Operational Semantics, Type Systems) • to reason and prove • Explains why things in FP are like they are • Pure Functions • Higher-Order Functions • Currying • Lazy Evaluation • Understand FP Compilers • Introduce FP stuff into other languages • Write your own compiler • GHC uses an enriched Lambda Calculus internally 2
7. Basics
8. Untyped Lambda Calculus t ::= Terms: x Variable λx.t Abstraction t t Application 3
9. Untyped Lambda Calculus t ::= Terms: x Variable λx.t Abstraction t t Application Example - Identity Lambda Calculus λx.x Abstraction y Variable Application → y 3
10. Untyped Lambda Calculus t ::= Terms: x Variable λx.t Abstraction t t Application Example - Identity Lambda Calculus λx.x Abstraction y Variable Application → y Javascript (function (x){return x; } Abstraction ) ( y Variable ) Application 3
11. Example - (λx.λy.x y) a b Abstractions Think: Function Definitions (λx.λy.x y) a b 4
12. Example - (λx.λy.x y) a b Abstractions Think: Function Definitions (λx.λy.x y) a b Variables Think: Parameters (λx.λy.x y) a b 4
13. Example - (λx.λy.x y) a b Abstractions Think: Function Definitions (λx.λy.x y) a b Variables Think: Parameters (λx.λy.x y) a b Applications Think: Function Calls (λx.λy.x y) a b 4
14. Example - (λx.λy.x y) a b (λx. λy.x y) a b 5
15. Example - (λx.λy.x y) a b (λx. λy.x y) a b Substitute x → a 5
16. Example - (λx.λy.x y) a b (λx. λy.x y) a b Substitute x → a 5
17. Example - (λx.λy.x y) a b (λx. λy.x y) a b Substitute x → a → (λy.a y) b 5
18. Example - (λx.λy.x y) a b (λx. λy.x y) a b Substitute x → a → (λy.a y) b Substitute y → b 5
19. Example - (λx.λy.x y) a b (λx. λy.x y) a b Substitute x → a → (λy.a y) b Substitute y → b 5
20. Example - (λx.λy.x y) a b (λx. λy.x y) a b Substitute x → a → (λy.a y) b Substitute y → b → a b 5
21. Notational Conventions • We use parentheses to clearify what’s meant • Applications associate to the left s t u ≡ (s t) u • Abstractions expand as much to the right as possible λx.λy.x y x ≡ λx.(λy.(x y x)) 6
22. Scope λx.λy.x y z Bound and Free λy y is bound, x and z are free λx x and y are bound, z is free λx, λy binders 7
23. Scope λx.λy.x y z Bound and Free λy y is bound, x and z are free λx x and y are bound, z is free λx, λy binders A term with no free variables is closed • A combinator • id ≡ λx.x • Y, S, K, I . . . 7
24. Higher Order Functions • Functions that take or return functions • Are there "by definition" λx.x Abstraction λy.y Abstraction Application → λy.y Abstraction 8
25. Currying Idea • Take a function with n arguments • Create a function that takes one argument and returns a function with n − 1 arguments 9
26. Currying Idea • Take a function with n arguments • Create a function that takes one argument and returns a function with n − 1 arguments Example • (+1) Section in Haskell • (λx.λy. + x y) 1 → λy. + 1 y 9
27. Currying Idea • Take a function with n arguments • Create a function that takes one argument and returns a function with n − 1 arguments Example • (+1) Section in Haskell • (λx.λy. + x y) 1 → λy. + 1 y • Partial Function Application is there "by definition" • You can use this stunt to "curry" in every language that supports "Lambda Expressions" 9
28. Reductions and Conversions Alpha Conversion λx.x →α λy.y 10
29. Reductions and Conversions Alpha Conversion λx.x →α λy.y Beta Reduction (λx.x) y →β y 10
30. Reductions and Conversions Alpha Conversion λx.x →α λy.y Beta Reduction (λx.x) y →β y Eta Conversion Iff (if and only if) x is not free in f : λx.f x →η f (λx. (λy.y) f x) a →η (λy.y) f a 10
31. Reductions and Conversions Alpha Conversion λx.x →α λy.y Beta Reduction (λx.x) y →β y Eta Conversion Iff (if and only if) x is not free in f : λx.f x →η f (λx. (λy.y) f x) a →η (λy.y) f a If x is free in f , η conversion not possible: λx. (λy.y Bound ↓ x ) f x →η (λy.y Free?! ↓ x ) 10
32. Remarks • Everything (Term) is an Expression • No statements • No "destructive" Assignments • The reason why FP Languages promote pure functions • But you could invent a built-in function to manipulate "state". . . 11
33. Evaluation
34. Operational Semantics • We learned how to write down and talk about Lambda Calculus Terms • How to evaluate them? • Different Strategies • Interesting outcomes 12
35. Full Beta-Reduction • RedEx • Reducible Expression • Always an Application (λx.x) ((λy.y) (λz. (λa.a) z RedEx ) RedEx ) RedEx 13
36. Full Beta-Reduction • RedEx • Reducible Expression • Always an Application (λx.x) ((λy.y) (λz. (λa.a) z RedEx ) RedEx ) RedEx Full Beta-Reduction • Any RedEx, Any Time • Like in Arithmetics • Too vague for programming. . . 13
37. Normal Order Reduction (λx.x) ((λy.y) (λz.(λa.a) z)) Normal Order Reduction • Left-most, Outer-most RedEx 14
38. Normal Order Reduction (λx.x) ((λy.y) (λz.(λa.a) z)) Normal Order Reduction • Left-most, Outer-most RedEx 14
39. Normal Order Reduction (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) Normal Order Reduction • Left-most, Outer-most RedEx 14
40. Normal Order Reduction (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) Normal Order Reduction • Left-most, Outer-most RedEx 14
41. Normal Order Reduction (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) →λz.(λa.a) z Normal Order Reduction • Left-most, Outer-most RedEx 14
42. Normal Order Reduction (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) →λz.(λa.a) z Normal Order Reduction • Left-most, Outer-most RedEx 14
43. Normal Order Reduction (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) →λz.(λa.a) z →λz.z Normal Order Reduction • Left-most, Outer-most RedEx 14
44. Call-by-Name (λx.x) ((λy.y) (λz.(λa.a) z)) Call-by-Name • like Normal Order Reduction, but no reductions inside Abstractions • Abstractions are values • lazy, non-strict • Parameters are not evaluated before they are used • Optimization: Save results → Call-by-Need 15
45. Call-by-Name (λx.x) ((λy.y) (λz.(λa.a) z)) Call-by-Name • like Normal Order Reduction, but no reductions inside Abstractions • Abstractions are values • lazy, non-strict • Parameters are not evaluated before they are used • Optimization: Save results → Call-by-Need 15
46. Call-by-Name (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) Call-by-Name • like Normal Order Reduction, but no reductions inside Abstractions • Abstractions are values • lazy, non-strict • Parameters are not evaluated before they are used • Optimization: Save results → Call-by-Need 15
47. Call-by-Name (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) Call-by-Name • like Normal Order Reduction, but no reductions inside Abstractions • Abstractions are values • lazy, non-strict • Parameters are not evaluated before they are used • Optimization: Save results → Call-by-Need 15
48. Call-by-Name (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) →λz.(λa.a) z Call-by-Name • like Normal Order Reduction, but no reductions inside Abstractions • Abstractions are values • lazy, non-strict • Parameters are not evaluated before they are used • Optimization: Save results → Call-by-Need 15
49. Call-by-Name (λx.x) ((λy.y) (λz.(λa.a) z)) →(λy.y) (λz.(λa.a) z) →λz.(λa.a) z → Call-by-Name • like Normal Order Reduction, but no reductions inside Abstractions • Abstractions are values • lazy, non-strict • Parameters are not evaluated before they are used • Optimization: Save results → Call-by-Need 15
50. Call-by-Value (λx.x) ((λy.y) (λz.(λa.a) z)) Call-by-Value • Outer-most, only if right-hand side was reduced to a value • No reductions inside Abstractions • Abstractions are values • eager, strict • Parameters are evaluated before they are used 16
51. Call-by-Value (λx.x) ((λy.y) (λz.(λa.a) z)) Call-by-Value • Outer-most, only if right-hand side was reduced to a value • No reductions inside Abstractions • Abstractions are values • eager, strict • Parameters are evaluated before they are used 16
52. Call-by-Value (λx.x) ((λy.y) (λz.(λa.a) z)) →(λx.x) (λz.(λa.a) z) Call-by-Value • Outer-most, only if right-hand side was reduced to a value • No reductions inside Abstractions • Abstractions are values • eager, strict • Parameters are evaluated before they are used 16
53. Call-by-Value (λx.x) ((λy.y) (λz.(λa.a) z)) →(λx.x) (λz.(λa.a) z) Call-by-Value • Outer-most, only if right-hand side was reduced to a value • No reductions inside Abstractions • Abstractions are values • eager, strict • Parameters are evaluated before they are used 16
54. Call-by-Value (λx.x) ((λy.y) (λz.(λa.a) z)) →(λx.x) (λz.(λa.a) z) →λz.(λa.a) z Call-by-Value • Outer-most, only if right-hand side was reduced to a value • No reductions inside Abstractions • Abstractions are values • eager, strict • Parameters are evaluated before they are used 16
55. Call-by-Value (λx.x) ((λy.y) (λz.(λa.a) z)) →(λx.x) (λz.(λa.a) z) →λz.(λa.a) z → Call-by-Value • Outer-most, only if right-hand side was reduced to a value • No reductions inside Abstractions • Abstractions are values • eager, strict • Parameters are evaluated before they are used 16
56. Church Encodings
57. Church Encodings • Encode Data into the Lambda Calculus • To simplify our formulas, let’s say that we have declarations id ≡ λx.x id y → y 17
58. Booleans true ≡ λt.λf .t false ≡ λt.λf .f 18
59. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f 18
60. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b 18
61. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b 18
62. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b 18
63. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b 18
64. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b 18
65. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b 18
66. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b 18
67. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b 18
68. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b →true a b 18
69. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b →true a b 18
70. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b →true a b ≡(λt.λf .t) a b 18
71. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b →true a b ≡(λt.λf .t) a b 18
72. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b →true a b ≡(λt.λf .t) a b →(λf .a) b 18
73. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b →true a b ≡(λt.λf .t) a b →(λf .a) b 18
74. Booleans true ≡ λt.λf .t false ≡ λt.λf .f test ≡ λc.λt.λf .c t f test true a b ≡ (λc.λt.λf .c t f ) true a b → (λt.λf .true t f ) a b → (λf .true a f ) b →true a b ≡(λt.λf .t) a b →(λf .a) b →a 18
75. And true ≡ λt.λf .t false ≡ λt.λf .f 19
76. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p 19
77. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false 19
78. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false 19
79. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false 19
80. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false 19
81. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false 19
82. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false 19
83. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false →true false true 19
84. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false →true false true 19
85. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false →true false true ≡(λt.λf .t) false true 19
86. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false →true false true ≡(λt.λf .t) false true 19
87. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false →true false true ≡(λt.λf .t) false true →(λf .false) true 19
88. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false →true false true ≡(λt.λf .t) false true →(λf .false) true 19
89. And true ≡ λt.λf .t false ≡ λt.λf .f and ≡ λp.λq.p q p and true false ≡(λp.λq.p q p) true false →(λq.true q true) false →true false true ≡(λt.λf .t) false true →(λf .false) true →false 19
90. Or λp.λq.p p q 20
91. Pairs pair ≡ λx.λy.λz.z x y 21
92. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y 21
93. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y pairAB ≡ pair a b 21
94. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y pairAB ≡ pair a b 21
95. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y pairAB ≡ pair a b ≡ (λx.λy.λz.z x y) a b 21
96. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y pairAB ≡ pair a b ≡ (λx.λy.λz.z x y) a b 21
97. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y pairAB ≡ pair a b ≡ (λx.λy.λz.z x y) a b → (λy.λz.z a y) b 21
98. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y pairAB ≡ pair a b ≡ (λx.λy.λz.z x y) a b → (λy.λz.z a y) b 21
99. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y pairAB ≡ pair a b ≡ (λx.λy.λz.z x y) a b → (λy.λz.z a y) b → λz.z a b 21
100. Pairs pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x second ≡ (λp.p) λx.λy.y pairAB ≡ pair a b ≡ (λx.λy.λz.z x y) a b → (λy.λz.z a y) b → λz.z a b ≡ pairab 21
101. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b 22
102. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab 22
103. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab 22
104. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab 22
105. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab 22
106. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) 22
107. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) 22
108. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) ≡ (λz.z a b) (λx.λy.x) 22
109. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) ≡ (λz.z a b) (λx.λy.x) 22
110. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) ≡ (λz.z a b) (λx.λy.x) → (λx.λy.x) a b 22
111. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) ≡ (λz.z a b) (λx.λy.x) → (λx.λy.x) a b 22
112. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) ≡ (λz.z a b) (λx.λy.x) → (λx.λy.x) a b → (λy.a) b 22
113. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) ≡ (λz.z a b) (λx.λy.x) → (λx.λy.x) a b → (λy.a) b 22
114. Pairs (continued) pair ≡ λx.λy.λz.z x y first ≡ (λp.p) λx.λy.x pairab ≡ λz.z a b first pairab ≡ (λp.p) (λx.λy.x) pairab → pairab (λx.λy.x) ≡ (λz.z a b) (λx.λy.x) → (λx.λy.x) a b → (λy.a) b → a 22
115. Numerals Peano Axioms Every natural number can be defined with 0 and a successor function 0 ≡ λf .λx.x 1 ≡ λf .λx.f x 2 ≡ λf .λx.f (f x) 3 ≡ λf .λx.f (f (f x)) Meaning 0 f is evaluated 0 times 1 f is evaluated once x can be every lambda term 23
116. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x 24
117. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) 24
118. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 24
119. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 24
120. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 24
121. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 24
122. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 → λf .λx.f (1 f x) 24
123. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 → λf .λx.f (1 f x) Note We use Normal Order Reduction to reduce inside abstractions! 24
124. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 → λf .λx.f (1 f x) ≡ λf .λx.f ((λf .λx.f x) f x) Note We use Normal Order Reduction to reduce inside abstractions! 24
125. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 → λf .λx.f (1 f x) ≡ λf .λx.f ((λf .λx.f x) f x) Note We use Normal Order Reduction to reduce inside abstractions! 24
126. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 → λf .λx.f (1 f x) ≡ λf .λx.f ((λf .λx.f x) f x) → λf .λx.f ((λx.f x) x) Note We use Normal Order Reduction to reduce inside abstractions! 24
127. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 → λf .λx.f (1 f x) ≡ λf .λx.f ((λf .λx.f x) f x) → λf .λx.f ((λx.f x) x) Note We use Normal Order Reduction to reduce inside abstractions! 24
128. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 → λf .λx.f (1 f x) ≡ λf .λx.f ((λf .λx.f x) f x) → λf .λx.f ((λx.f x) x) → λf .λx.f (f x) Note We use Normal Order Reduction to reduce inside abstractions! 24
129. Numerals Example - Successor 0 ≡ λf .λx.x 1 ≡ λf .λx.f x successor ≡ λn.λf .λx.f (n f x) successor 1 ≡ (λn.λf .λx.f (n f x)) 1 → λf .λx.f (1 f x) ≡ λf .λx.f ((λf .λx.f x) f x) → λf .λx.f ((λx.f x) x) → λf .λx.f (f x) ≡ 2 Note We use Normal Order Reduction to reduce inside abstractions! 24
130. Numerals Example - 0 + 0 0 ≡ λf .λx.x 25
131. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) 25
132. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 25
133. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 25
134. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 25
135. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 25
136. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 25
137. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 25
138. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) 25
139. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) 25
140. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) 25
141. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) 25
142. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) 25
143. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) 25
144. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) → λf .λx.0 f x 25
145. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) → λf .λx.0 f x 25
146. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) → λf .λx.0 f x ≡ λf .λx.(λf .λx.x) f x 25
147. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) → λf .λx.0 f x ≡ λf .λx.(λf .λx.x) f x 25
148. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) → λf .λx.0 f x ≡ λf .λx.(λf .λx.x) f x → λf .λx.(λx.x) x 25
149. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) → λf .λx.0 f x ≡ λf .λx.(λf .λx.x) f x → λf .λx.(λx.x) x 25
150. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) → λf .λx.0 f x ≡ λf .λx.(λf .λx.x) f x → λf .λx.(λx.x) x → λf .λx.x 25
151. Numerals Example - 0 + 0 0 ≡ λf .λx.x plus ≡ λm.λn.λf .λx.m f (n f x) plus 0 0 ≡ (λm.λn.λf .λx.m f (n f x)) 0 0 → (λn.λf .λx.0 f (n f x)) 0 → λf .λx.0 f (0 f x) ≡ λf .λx.(λf .λx.x) f (0 f x) → λf .λx.(λx.x) (0 f x) → λf .λx.0 f x ≡ λf .λx.(λf .λx.x) f x → λf .λx.(λx.x) x → λf .λx.x ≡ 0 25
152. End
153. Thanks • Hope you enjoyed this talk and learned something new. • Hope it wasn’t too much math and dusty formulas . . . :) 26
154. Good Math "A Geek’s Guide to the Beauty of Numbers, Logic, and Computation" • Easy to understand
155. The Implementation of Functional Programming Languages • How to compile to the Lambda Calculus? • Out-of-print, but freely available • https://www.microsoft.com/en- us/research/publication/the- implementation-of-functional- programming-languages/
156. Types and Programming Languages • Types systems explained by building interpreters and proving properties • Very "mathematical", but very complete and self-contained