Less naive type theory

Transcription

1 Institute of Informatics Warsaw University 26 May 2007

2 Plan 1 Syntax of lambda calculus Why typed lambda calculi? 2 3

3 Syntax of lambda calculus Why typed lambda calculi? origins in 1930s (Church, Curry) alternative foundation of mathematics model of computation basic notion: function function understood in an intensional way

4 Syntax of lambda calculus Syntax of lambda calculus Why typed lambda calculi? Lambda term variables x, y,... are lambda terms if M, N are lambda terms then MN is also a lambda term if x is a variable and M is a lambda term then λx.m is a lambda term Intuitive meaning Lambda term represents a function. variables x, y,... represent some functions MN is an application of a function M to an argument N λx.m is a function with a parameter x and a definition M

5 Syntax of lambda calculus Syntax of lambda calculus Why typed lambda calculi? Lambda term variables x, y,... are lambda terms if M, N are lambda terms then MN is also a lambda term if x is a variable and M is a lambda term then λx.m is a lambda term Intuitive meaning Lambda term represents a function. variables x, y,... represent some functions MN is an application of a function M to an argument N λx.m is a function with a parameter x and a definition M

6 Syntax of lambda calculus Why typed lambda calculi? Problem in lambda calculus Problem We compose arbitrary lambda terms (arbitrary functions). Consider a function f (x) = x 2

7 Syntax of lambda calculus Why typed lambda calculi? Problem in lambda calculus Problem We compose arbitrary lambda terms (arbitrary functions). Consider a function f (x) = x 2

8 Syntax of lambda calculus Why typed lambda calculi? Problem in lambda calculus Problem We compose arbitrary lambda terms (arbitrary functions). Consider a function f (x) = x 2 What is f ( )?

9 Syntax of lambda calculus Why typed lambda calculi? Problem in lambda calculus Problem We compose arbitrary lambda terms (arbitrary functions). Consider a function f (x) = x 2 What is f ( )? Solution with types

10 Type assignment systems Γ M : τ Environment Lambda term Type assigns types object describing to variables domain and codomain of a function Typing rules Rules telling how to type lambda terms.

11 Type assignment systems Γ M : τ Environment Lambda term Type assigns types object describing to variables domain and codomain of a function Typing rules Rules telling how to type lambda terms.

12 Type assignment systems Γ M : τ Environment Lambda term Type assigns types object describing to variables domain and codomain of a function Typing rules Rules telling how to type lambda terms.

13 Important issue Inhabitation problem Is there a lambda term M such that M : τ?? : τ

14 Important issue Inhabitation problem Is there a lambda term M such that M : τ? M : τ Inhabitant Inhabited type

15 Important issue Inhabitation problem Is there a lambda term M such that M : τ? M : τ Inhabitant Inhabited type

16 Simply typed lambda calculus λ Γ, x : τ x : τ Γ, x : τ M : σ Γ λx.m : τ σ Γ M : τ σ Γ N : τ Γ MN : σ

17 Simply typed lambda calculus λ Γ, x : τ x : τ Γ, x : τ M : σ Γ λx.m : τ σ Γ M : τ σ Γ N : τ Γ MN : σ

18 Simply typed lambda calculus λ Γ, τ τ Γ, τ σ Γ τ σ Γ τ σ Γ σ Γ τ

19 Simply typed lambda calculus λ Γ, τ τ Γ, τ σ Γ τ σ Γ τ σ Γ σ Γ τ Looks familiar?

20 simply typed lambda calculus λ Logic minimal propositional (intuitionistic) logic types formulas typing rules proof rules inhabitation provability? : τ τ? inhabited types provable formulas

21 simply typed lambda calculus λ Logic minimal propositional (intuitionistic) logic types formulas typing rules proof rules inhabitation provability? : τ τ? inhabited types provable formulas

22 simply typed lambda calculus λ Logic minimal propositional (intuitionistic) logic types formulas typing rules proof rules inhabitation provability? : τ τ? inhabited types provable formulas

23 simply typed lambda calculus λ Logic minimal propositional (intuitionistic) logic types formulas typing rules proof rules inhabitation provability? : τ τ? inhabited types provable formulas

24 simply typed lambda calculus λ Logic minimal propositional (intuitionistic) logic types formulas typing rules proof rules inhabitation provability? : τ τ? inhabited types provable formulas

25 simply typed lambda calculus λ Logic minimal propositional (intuitionistic) logic types formulas typing rules proof rules inhabitation provability? : τ τ? inhabited types provable formulas

26 Extensions of simple types system λp first-order logic system F second-order propositional logic system F ω higher-order propositional logic calculus of constructions λc higher-order predicate logic other

27 The λ-cube F ω λc F λp2 λω λpω λ λp

28 The λ-cube F ω λc F λp2 Pure Type Systems λω λpω λ λp

29 (PTSs) formalism to talk about type systems parametric representation useful for comparing type systems beyond the cube used for defining new type systems (new logics)

30 Threat with PTSs Example PTS system NTT S =, A = : R = (, ), (, ), (,, ) Inconsistent type system Every type is inhabited everything is provable.

31 Threat with PTSs Example PTS system NTT S =, A = : R = (, ), (, ), (,, ) Inconsistent type system Every type is inhabited everything is provable.

32 S = t, p, t, p A = t : t, p : p R = ( t, t ), ( p, p ), ( t, p ), ( t, p, t ), ( t, t ) Theorem is consistent.

33 S = t, p, t, p A = t : t, p : p R = ( t, t ), ( p, p ), ( t, p ), ( t, p, t ), ( t, t ) Theorem is consistent.