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

How to Talk to Aliens

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)

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?

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

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

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

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

History of Logic Frege continued (1879)

History of Logic Frege in modern notation

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

History of Logic Gentzen (1934)

History of Logic Gentzen (1934)

History of Logic Gentzen (1934) in modern notation:

History of Logic Gentzen (1934) proof simplification rules:

History of Logic Proof simplification

History of Logic Proof simplification

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

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)

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

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

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

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

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

History of Logic Sample evaluation

History of Logic Church’s typed lambda calculus

History of Logic Simplifying programs

History of Logic Simplifying programs

History of Logic Curry-Howard Isomorphism William Howard Haskell Curry (1900-1982) William Howard

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

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

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

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

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

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?