1. Proofs are Programs: 19th Century Logic and 21st Century Computing
CS 510: Programming Languages
(a talk adapted from Philip Wadler)

2. How to Talk to Aliens

3. Basic natural phenomena
Some mathematical phenomena seem so unavoidable that we must view them as “discovered” rather than “invented”
natural numbers
0 appears first in India (700 CE)
mathematical constants
pi, e
geometry
basic shapes (circle, triangle)
geometric laws (triangle inequality, area and volume)

4. Computer Science Invented or discovered?
Are there fundamental, unavoidable computational phenomena?
laws or concepts with the same unavoidable feeling as Newton’s laws of gravity?
How would we know?
these concepts would appear across time and culture
but computer science is only 50 years old! Or is it?

5. The History of Logic George Boole 1815-1864
inventor (discoverer?) of Boolean algebra
conjectured the existence of fundamental Laws of Thought

6. The History of Logic
George Boole (Laws of Thought, 1854)

7. History of Logic Gotlob Frege (1848-1925)
the first hints of modern logic
formal diagrams to depict mathematical inferences

8. History of Logic
Frege’s inferences (1879)
Look vaguely familiar?

9. History of Logic
Frege continued (1879)

10. History of Logic
Frege in modern notation

11. History of Logic Gerhard Gentzen (1934) Natural Deduction
The definition of logical connectives exclusively in terms of introduction of the connective and elimination of the connective
Prior work specified logic via axioms that related one connective to another

12. History of Logic
Gentzen (1934)

13. History of Logic
Gentzen (1934)

14. History of Logic
Gentzen (1934)
in modern notation:

15. History of Logic
Gentzen (1934)
proof simplification rules:

16. History of Logic
Proof simplification

17. History of Logic
Proof simplification

18. History of Logic Alonzo Church
invented (discovered?) the lambda calculus
the first programming language researcher

19. History of Logic Alonzo Church (1903-1995)
invented (discovered?) the lambda calculus
the first programming language researcher
he didn’t know it at the time (we were still lacking the computers)

20. History of Logic Church (1932)
the introduction of free and bound variables

21. History of Logic Church (1932)
the logic is defined entirely and abstractly by a system of inference rules
symbols do not have pre-conceived meanings
pure logic uninfluenced by culture

22. History of Logic Church (1932) in case you missed the last line
I’ll say.

23. History of Logic Church (1932)
proof terms, bound variables and the notion of well-formed objects

24. History of Logic Computing with Church’s lambda calculus
General evaluation rules:

25. History of Logic
Sample evaluation

26. History of Logic
Church’s typed lambda calculus

27. History of Logic
Simplifying programs

28. History of Logic
Simplifying programs

29. History of Logic Curry-Howard Isomorphism William Howard
Haskell Curry ( )
William Howard

30. History of Logic Curry-Howard Isomorphism Fundamental ideas:
First noticed by Curry in 1960
First published by Howard in 1980
Fundamental ideas:
Proofs are programs
Formulas are types
Proof rules are type checking rules
Proof simplification is operational semantics
Ideas and observations about logic are ideas and observations about programming languages to

31. History of Logic Curry-Howard (intuitionistic logic)
2nd-order intuitionistic logic
(formula variable) a
(implication) A => B
(conjunction) A  B
(disjunction) A  B
(truth) True
(falsehood) False
(universal quant) a.A
(existential quant) a.A
Polymorphic Lambda Calculus
(PolyMinML - recursion)
(type variable) a
(function type) A  B
(pair type) A * B
(sum type) A + B
(unit) unit
(void) void
(universal poly) a.A
(existential poly) a.A

32. History of Logic Curry-Howard (intuitionistic logic)
Note: You don’t have to omit proof terms from logical rules
2nd-order intuitionistic logic
Polymorphic Lambda Calculus
G, A |- B
(=> I)
G |- A => B
G, x:A |- e : B
(Fun)
G |- x:A.e : A => B
G |- A => B G |- A
(=>E)
G |- B
G |- e1 : A => B G |- e2 : A
(App)
G |- e1 e2 : B

33. History of Logic Curry-Howard (intuitionistic logic)
2nd-order intuitionistic logic
Polymorphic Lambda Calculus
G |- A G |- B
( I)
G |- A  B
G |- e1 : A G |- e2 : B
(Pair)
G |- <e1,e2> : A * B
G |- A  B
(E1)
G |- A
G |- A  B
(E2)
G |- B
G |- e : A  B
(E1)
G |- e.1 : A
G |- e : A  B
(E2)
G |- e.2 B

34. History of Logic Other Examples
Hindley/Milner: formula reconstruction/type inference
Girard/Reynolds:
System F/Polymorphic lambda calculus
Linear Logic/Syntactic control of interference, linear types
Murthy: classical logic/callcc
Abramsky: multiplicative vs additive/strict vs lazy
Pfenning and Davies: modal logic/RTCG
Phil Wadler: and dual to or/CBV dual to CBN

35. History of Logic The Curry-Howard isomorphism
Coincidence? Or are aliens trying to tell us something?
Identical concepts arise across time and culture
Certain logical/computational definitions seem unavoidable
Are they inventions or discoveries?
Is this talk about the history of logic or the history of computer programming?
Was this course about logic or computer science?