Church and Curry: Combining Intrinsic and Extrinsic Typing

Transcription

1 Church and Curry: Combining Intrinsic and Extrinsic Typing Frank Pfenning Dedicated to Peter Andrews on the occasion of his retirement Department of Computer Science Carnegie Mellon University April 5, / 24

2 Church s Simple Theory of Types Language and logic for the formalization of mathematics (Church 1940) Stood the test of time (72 years!) HOL, Isabelle/HOL, TPS, LEO, Satallax,... 2 / 24

3 Church s Simple Theory of Types Language and logic for the formalization of mathematics (Church 1940) Stood the test of time (72 years!) HOL, Isabelle/HOL, TPS, LEO, Satallax,... Synthesis, simplification, and generalization of Russell and Whitehead s ramified theory of types Church and Rosser s (untyped) λ-calculus 2 / 24

4 Church s Simple Theory of Types Language and logic for the formalization of mathematics (Church 1940) Stood the test of time (72 years!) HOL, Isabelle/HOL, TPS, LEO, Satallax,... Synthesis, simplification, and generalization of Russell and Whitehead s ramified theory of types Church and Rosser s (untyped) λ-calculus Some objections 2 / 24

5 Church s Simple Theory of Types Language and logic for the formalization of mathematics (Church 1940) Stood the test of time (72 years!) HOL, Isabelle/HOL, TPS, LEO, Satallax,... Synthesis, simplification, and generalization of Russell and Whitehead s ramified theory of types Church and Rosser s (untyped) λ-calculus Some objections As a computer scientist: classical 2 / 24

6 Church s Simple Theory of Types Language and logic for the formalization of mathematics (Church 1940) Stood the test of time (72 years!) HOL, Isabelle/HOL, TPS, LEO, Satallax,... Synthesis, simplification, and generalization of Russell and Whitehead s ramified theory of types Church and Rosser s (untyped) λ-calculus Some objections As a computer scientist: classical As a philosopher: impredicative 2 / 24

7 Church s Simple Theory of Types Language and logic for the formalization of mathematics (Church 1940) Stood the test of time (72 years!) HOL, Isabelle/HOL, TPS, LEO, Satallax,... Synthesis, simplification, and generalization of Russell and Whitehead s ramified theory of types Church and Rosser s (untyped) λ-calculus Some objections As a computer scientist: classical As a philosopher: impredicative Components 2 / 24

8 Church s Simple Theory of Types Language and logic for the formalization of mathematics (Church 1940) Stood the test of time (72 years!) HOL, Isabelle/HOL, TPS, LEO, Satallax,... Synthesis, simplification, and generalization of Russell and Whitehead s ramified theory of types Church and Rosser s (untyped) λ-calculus Some objections As a computer scientist: classical As a philosopher: impredicative Components Simply typed λ-calculus (this talk) Logical axioms and inference rules 2 / 24

9 Simply Typed λ-calculus Church s definitions 3 / 24

10 Simply Typed λ-calculus Church s definitions Types 1 ι and o are types. 2 If α and β are types, then α β is a type. (Church wrote βα) 3 / 24

11 Simply Typed λ-calculus Church s definitions Types 1 ι and o are types. 2 If α and β are types, then α β is a type. (Church wrote βα) Well-formed terms M α of type α 1 Any variable x α or constant c α is a term. 2 If x α is a variable and M β a term then (λx. M) α β is a term. 3 If M α β 1 and M α 2 are terms, then (M 1 M 2 ) β is a term. 3 / 24

12 Intrinsic Typing Every well-formed term has an intrinsic type, including variables 4 / 24

13 Intrinsic Typing Every well-formed term has an intrinsic type, including variables This kind of intrinsic formulation has become rare, but it has a number of advantages 4 / 24

14 Intrinsic Typing Every well-formed term has an intrinsic type, including variables This kind of intrinsic formulation has become rare, but it has a number of advantages Supports conventions, such as In the remainder of this [talk] we assume that all terms are well-formed according to the above definition. 4 / 24

15 Intrinsic Typing Every well-formed term has an intrinsic type, including variables This kind of intrinsic formulation has become rare, but it has a number of advantages Supports conventions, such as In the remainder of this [talk] we assume that all terms are well-formed according to the above definition. In a logical framework tp : type. arrow : tp -> tp -> tp. tm : tp -> type. lam : (tm A -> tm B) -> tm (arrow A B). app : tm (arrow A B) -> tm A -> tm B. 4 / 24

16 Untyped λ-calculus (Church 1932) (Church and Rosser 1936) Terms 1 Any variable x or constant c is a term. 2 If x is a variable and M a term then (λx. M) is a term. 3 If M 1 and M 2 are terms, then (M 1 M 2 ) is a term. 5 / 24

17 Extrinsic Typing (Curry 1934) [for combinators, not λ-terms] Types as properties of terms 6 / 24

18 Extrinsic Typing (Curry 1934) [for combinators, not λ-terms] Types as properties of terms Typing judgments defined by rules x:α Γ Γ x : α c:α Σ Γ c : α Γ, x:α M : β (x dom(γ)) Γ λx. M : α β Γ M : α β Γ N : α Γ M N : β 6 / 24

19 Types as Properties A term can have multiple types λx. x : ι ι λx. x : (ι ι) (ι ι) 7 / 24

20 Types as Properties A term can have multiple types λx. x : ι ι λx. x : (ι ι) (ι ι) Express explicitly in the form of a single type 7 / 24

21 Types as Properties A term can have multiple types λx. x : ι ι λx. x : (ι ι) (ι ι) Express explicitly in the form of a single type Parametric polymorphism (universal types) λx. x : t. t t 7 / 24

22 Types as Properties A term can have multiple types λx. x : ι ι λx. x : (ι ι) (ι ι) Express explicitly in the form of a single type Parametric polymorphism (universal types) λx. x : t. t t Ad hoc polymorphism (intersection types) λx. x : (ι ι) ((ι ι) (ι ι)) 7 / 24

23 Extrinsic Rules Parametric polymorphism Γ M : β (t ftv(γ)) Γ M : t. β I Γ M : t. α Γ M : [β/t]α E 8 / 24

24 Extrinsic Rules Parametric polymorphism Γ M : β (t ftv(γ)) Γ M : t. β Intersection polymorphism I Γ M : t. α Γ M : [β/t]α E Γ M : A Γ M : B Γ M : A B I Γ M : A B Γ M : A E 1 Γ M : A B Γ M : B E 2 8 / 24

25 Types as Properties: The Good News Can type terms more generally λx. x x : ( α. α α) ( β. β β) λx. x x : ((ι ι) ι) ι 9 / 24

26 Types as Properties: The Good News Can type terms more generally λx. x x : ( α. α α) ( β. β β) λx. x x : ((ι ι) ι) ι Can type type terms more accurately λx. s(s(s x)) : (even odd) (odd even) 9 / 24

27 Types as Properties: The Bad News The typing judgment is undecidable 10 / 24

28 Types as Properties: The Bad News The typing judgment is undecidable Challenge: generalize to a complete language Practical Useful Philosophically justified Easy to reason about 10 / 24

29 Layering Type Systems Intrinsic simple types for basic consistency Avoiding Russell s paradox 11 / 24

30 Layering Type Systems Intrinsic simple types for basic consistency Avoiding Russell s paradox Extrinsic sorts on well-formed terms for precision Circumvent problems with general extrinsic types Achieve pragmatic goals 11 / 24

31 Layering Type Systems Intrinsic simple types for basic consistency Avoiding Russell s paradox Extrinsic sorts on well-formed terms for precision Circumvent problems with general extrinsic types Achieve pragmatic goals Instances Intersection types to datasort refinement (this talk) Dependent types to index refinements 11 / 24

32 Layering Type Systems Intrinsic simple types for basic consistency Avoiding Russell s paradox Extrinsic sorts on well-formed terms for precision Circumvent problems with general extrinsic types Achieve pragmatic goals Instances Intersection types to datasort refinement (this talk) Dependent types to index refinements Parametric polymorphism is a different story 11 / 24

33 Representing Data Multiple techniques in Church s Type Theory Church numerals Constants and axioms 12 / 24

34 Representing Data Multiple techniques in Church s Type Theory Church numerals Constants and axioms We have only basic type ι (o is for truth values) 12 / 24

35 Representing Data Multiple techniques in Church s Type Theory Church numerals Constants and axioms We have only basic type ι (o is for truth values) Example: natural numbers Constants z ι and s ι ι (constructors) Constant nat ι o (predicate) Axioms nat(z) x ι. nat(x) nat(s(x)) 12 / 24

36 Representing Data Multiple techniques in Church s Type Theory Church numerals Constants and axioms We have only basic type ι (o is for truth values) Example: natural numbers Constants z ι and s ι ι (constructors) Constant nat ι o (predicate) Axioms nat(z) x ι. nat(x) nat(s(x)) Lists, trees, etc. all have type ι Sort out using sorts 12 / 24

37 Simple Sorts Example nat ι sort z ι : nat s ι ι : nat nat 13 / 24

38 Simple Sorts Example nat ι sort z ι : nat s ι ι : nat nat Define sorts S α refining type α under signature Σ 1 A base sort Q ι declared in Σ is a simple sort. 2 If S α and T β are simple sorts, then (S T ) α β is a simple sort. 13 / 24

39 Sorting Judgment Context Γ consisting of declarations x α : S α Sorting judgment Γ M α : S α 14 / 24

40 Sorting Judgment Context Γ consisting of declarations x α : S α Sorting judgment Γ M α : S α Defined only for terms of intrinsic type α and sort refining the same type α! 14 / 24

41 Sorting Judgment Context Γ consisting of declarations x α : S α Sorting judgment Γ M α : S α Defined only for terms of intrinsic type α and sort refining the same type α! Rules x:s Γ Γ x : S c:s Σ Γ c : S Γ, x:s M : T (x dom(γ)) Γ λx. M : S T Γ M : S T Γ N : S Γ M N : T 14 / 24

42 Subsorts Subsort declarations Q ι 1 Q ι 2 New rules Q Q Q 1 Q 2 Q 2 Q 3 Q 1 Q 3 Γ M : Q Q Q Γ M : Q Defined on base types only Can derive principles for higher types 15 / 24

43 Subsorting Example Refining natural numbers zero nat pos nat z : zero s : nat pos 16 / 24

44 Subsorting Example Refining natural numbers Examples zero nat pos nat z : zero s : nat pos λx. x : nat nat λx. x : zero nat λx. λy. x y : (nat zero) (zero nat) λx. s x : nat pos 16 / 24

45 Combining Properties Want to express and exploit multiple properties of terms 17 / 24

46 Combining Properties Want to express and exploit multiple properties of terms Example: even and odd numbers even nat odd nat z : even s : even odd s : odd even 17 / 24

47 Combining Properties Want to express and exploit multiple properties of terms Example: even and odd numbers even nat odd nat z : even s : even odd s : odd even Have no way to express in one sort: λx ι. s(s(s x)) : even odd λx ι. s(s(s x)) : odd even 17 / 24

48 Intersection Sorts Define sorts S α refining types α (in intrinsic style) 1 A base sort Q ι declared in Σ is a sort. 2 If S α and T β are sorts, then (S T ) α β is a sort. 3 If S α and T α are sorts then (S T ) α is a sort. 4 α is a sort for each type α. 18 / 24

49 Extended Sorting Judgments Recall Γ M α : S α 19 / 24

50 Extended Sorting Judgments Recall Γ M α : S α New typing rules Γ M : S 1 Γ M : S 2 Γ M : S 1 S 2 I Γ M : S 1 S 2 Γ M : S 1 E 1 Γ M : S 1 S 2 Γ M : S 2 E 2 Γ M : I 19 / 24

51 Extended Sorting Judgments Recall Γ M α : S α New typing rules Γ M : S 1 Γ M : S 2 Γ M : S 1 S 2 I Γ M : S 1 S 2 Γ M : S 1 E 1 Γ M : S 1 S 2 Γ M : S 2 E 2 Γ M : I Philosophically justified in the sense of Dummett/Martin-Löf 19 / 24

52 Example Revisited Can now conjoin properties λx ι. s(s(s x)) : (even odd) (odd even) (nat pos) / 24

53 Example Revisited Can now conjoin properties λx ι. s(s(s x)) : (even odd) (odd even) (nat pos)... Every (well-formed) term has a principal sort 20 / 24

54 Some Results Sort checking is decidable Proof: there are effectively only finitely many refinements of a given type 21 / 24

55 Some Results Sort checking is decidable Proof: there are effectively only finitely many refinements of a given type Sorting is closed under β-reduction Proof: standard substitution property 21 / 24

56 Some Results Sort checking is decidable Proof: there are effectively only finitely many refinements of a given type Sorting is closed under β-reduction Proof: standard substitution property Sorting is closed under η-expansion Proof: induction over sorts 21 / 24

57 More Results Define η α (M) as η-long form of M α 22 / 24

58 More Results Define η α (M) as η-long form of M α Define subsorting S α T α at higher types S T : x:s η α (x) : T 22 / 24

59 More Results Define η α (M) as η-long form of M α Define subsorting S α T α at higher types S T : x:s η α (x) : T Extend subsumption rule 22 / 24

60 More Results Define η α (M) as η-long form of M α Define subsorting S α T α at higher types S T : x:s η α (x) : T Extend subsumption rule Sorting is closed under β-expansion Proof: intersect all sorts the abstracted term is used at 22 / 24

61 More Results Define η α (M) as η-long form of M α Define subsorting S α T α at higher types S T : x:s η α (x) : T Extend subsumption rule Sorting is closed under β-expansion Proof: intersect all sorts the abstracted term is used at Sorting is closed under η-reduction Proof: by subsorting at higher types 22 / 24

62 More Results Define η α (M) as η-long form of M α Define subsorting S α T α at higher types S T : x:s η α (x) : T Extend subsumption rule Sorting is closed under β-expansion Proof: intersect all sorts the abstracted term is used at Sorting is closed under η-reduction Proof: by subsorting at higher types Conclusion Extrinsic (Curry) sorting with intersections refining intrinsic (Church) typing is closed under λ-conversion! 22 / 24

63 Related Developments (my students only) Expressive power extends tree automata to higher types Canonical (= β-normal, η-long) terms can by typed bidirectionally In: Festschrift in Honor of Peter B. Andrews on his 70th Birthday, C. Benzmüller, C. Brown, J. Siekmann, and R. Statman, editors Crucial for logical frameworks (Lovas 2010) Refinement type inference for ML (Freeman 1994) Practical refinements type for SML (Davies 2005) Dependent refinements over decidable domains (Xi 1998) Unifying sort and dependent refinements (Dunfield 2007) 23 / 24

64 Conclusion Church s original intrinsic formulation of the simply-typed λ-calculus has fallen into disfavor, perhaps unjustly 24 / 24

65 Conclusion Church s original intrinsic formulation of the simply-typed λ-calculus has fallen into disfavor, perhaps unjustly It suggests an elegant layering with Curry s extrinsic typing judgment (translated to λ-calculus) 24 / 24

66 Conclusion Church s original intrinsic formulation of the simply-typed λ-calculus has fallen into disfavor, perhaps unjustly It suggests an elegant layering with Curry s extrinsic typing judgment (translated to λ-calculus) Can be usefully combined with Coppo et al. s intersection types for high expressiveness, precision, and surprisingly strong metatheoretic results 24 / 24