Alexander J Summers Department of Computing Imperial College London

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1 Alexander J Summers Department of Computing Imperial College London
Natural Delimited Control A Curry-Howard Correspondence for a Canonical Classical Natural Deduction Alexander J Summers Department of Computing Imperial College London Set context of work/intro

2 Overview Interested in the extension of the Curry-Howard Correspondence to Classical Logics Talk roughly in three parts... 1. Brief introduction to Control Operators 2. Definition of a programming calculus based on classical natural deduction 3. How are the two related? Talk is semi-informal: focus on intuition/explanation Feel free to ask questions..

3 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> n*(f rest))

4 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> n*(f rest)) 3 4 1 7 9

5 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> n*(f rest)) 3 4 1 7 9

6 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> n*(f rest))  3 4 1 7 9

7 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> if n=0 then 0 else n*(f rest))  3 4 1 7 9

8 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> if n=0 then 0 else n*(f rest))  3 4 1 7 9

9 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> if n=0 then 0 else n*(f rest)) 3 4 1 7 9

10 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> if n=0 then 0 else n*(f rest))  3 4 1 7 9

11 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> if n=0 then 0 else n*(f rest))  3 4 1 7 9

12 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> if n=0 then 0 else n*(f rest))  3 4 1 7 9

13 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> if n=0 then 0 else n*(f rest))  3 4 1 7 9

14 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l (Fix f. λy.match y with [] > 1 n:rest --> if n=0 then 0 else n*(f rest)) ?  3 4 1 7 9

15 Contexts and Continuations
Control operators give control over the context in which an execution takes place term context

16 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context

17 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context

18 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context

19 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context We write contexts as E{●} where ● is the “hole”

20 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context We write contexts as E{●} where ● is the “hole” E{●} = print (1 * (2 * (●)))

21 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context We write contexts as E{●} where ● is the “hole” E{●} = print (1 * (2 * (●))) E{3 * 4} = print (1 * (2 * (3 * 4)))

22 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context We write contexts as E{●} where ● is the “hole” E{●} = print (1 * (2 * (●))) E{3 * 4} = print (1 * (2 * (3 * 4))) A continuation is a term representation of a context A term with a ‘hole’ for another term

23 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context We write contexts as E{●} where ● is the “hole” E{●} = print (1 * (2 * (●))) E{3 * 4} = print (1 * (2 * (3 * 4))) A continuation is a term representation of a context A term with a ‘hole’ for another term We represent this with a binder ν over the ‘hole’

24 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context We write contexts as E{●} where ● is the “hole” E{●} = print (1 * (2 * (●))) E{3 * 4} = print (1 * (2 * (3 * 4))) A continuation is a term representation of a context A term with a ‘hole’ for another term We represent this with a binder ν over the ‘hole’ νx.print (1 * (2 * (●)))

25 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context We write contexts as E{●} where ● is the “hole” E{●} = print (1 * (2 * (●))) E{3 * 4} = print (1 * (2 * (3 * 4))) A continuation is a term representation of a context A term with a ‘hole’ for another term We represent this with a binder ν over the ‘hole’ νx.print (1 * (2 * (x)))

26 Contexts and Continuations
Control operators give control over the context in which an execution takes place print (1 * (2 * (3 * 4))) term context We write contexts as E{●} where ● is the “hole” E{●} = print (1 * (2 * (●))) E{3 * 4} = print (1 * (2 * (3 * 4))) A continuation is a term representation of a context A term with a ‘hole’ for another term We represent this with a binder ν over the ‘hole’ νx.print (1 * (2 * (x)))

27 Control Operators Programming constructs for functional languages
Allow the expression of “non-functional” behaviour e.g., jumps, exceptions, loops, ... Simplest example: A (“abort”) Defined by: E{A M} → M Completely discards the surrounding context print (1 * (2 * (A (3 * 4))))

28 Control Operators Programming constructs for functional languages
Allow the expression of “non-functional” behaviour e.g., jumps, exceptions, loops, ... Simplest example: A (“abort”) Defined by: E{A M} → M Completely discards the surrounding context print (1 * (2 * (A (3 * 4)))) → print (1 * (2 * (A 12)))

29 Control Operators Programming constructs for functional languages
Allow the expression of “non-functional” behaviour e.g., jumps, exceptions, loops, ... Simplest example: A (“abort”) Defined by: E{A M} → M Completely discards the surrounding context print (1 * (2 * (A (3 * 4)))) → print (1 * (2 * (A 12)))

30 Control Operators More interesting example: C (“control”)
C M gives M explicit control over the context Stores the context in an “escape procedure” E{C M} → M λx.(A E{x}) The escape procedure is passed to M If M calls procedure with argument N, resulting term A E{N} aborts current execution and restores the context E{N} In this way, M can “throw” values back to the context

31 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l C λk.k(Fix f. λy.match y with [] --> 1 n:rest --> if n=0 then 0 else n*(f rest)) ?  3 4 1 7 9

32 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l C λk.k(Fix f. λy.match y with [] --> 1 n:rest --> if n=0 then 0 else n*(f rest)) ?  3 4 1 7 9

33 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l C λk.k(Fix f. λy.match y with [] --> 1 n:rest --> if n=0 then 0 else n*(f rest)) ?  3 4 1 7 9

34 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l C λk.k(Fix f. λy.match y with [] --> 1 n:rest --> if n=0 then 0 else n*(f rest)) ?  3 4 1 7 9

35 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l C λk.k(Fix f. λy.match y with [] --> 1 n:rest --> if n=0 then k 0 else n*(f rest)) ?  3 4 1 7 9

36 Multiplying lists of numbers..
Exercise: write a recursive function / functional program to calculate the product of a list of numbers Lists are defined by: l ::= [] | n:l C λk.k(Fix f. λy.match y with [] --> 1 n:rest --> if n=0 then k 0 else n*(f rest))  3 4 1 7 9

37 Types for Control Operators
What about types for these operators? e.g., abort Consider the reduction rule: E{A M} → M (A M) should have the type of the “hole” in E M can have any type? But, for the rule to be sound, M must have the type of E{A M} Griffin (‘90): introduce special  type for “top-level” The type of “finished computation” - M must have it Γ ⊢ M : B (A) Γ ⊢ (A M) : C

38 Types for Control Operators
What about types for these operators? e.g., abort Consider the reduction rule: E{A M} → M (A M) should have the type of the “hole” in E M can have any type? But, for the rule to be sound, M must have the type of E{A M} Griffin (‘90): introduce special  type for “top-level” The type of “finished computation” - M must have it Now A makes logical sense in terms of types too corresponds with -elimination from intuitionistic logic Γ ⊢ M :  (A) Γ ⊢ (A M) : C

39 Types for Control Operators
What about types for these operators? e.g., abort Consider the reduction rule: E{A M} → M (A M) should have the type of the “hole” in E M can have any type? But, for the rule to be sound, M must have the type of E{A M} Griffin (‘90): introduce special  type for “top-level” The type of “finished computation” - M must have it Now A makes logical sense in terms of types too corresponds with -elimination from intuitionistic logic Γ ⊢ M :  (A) Γ ⊢ (A M) : C

40 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x})

41 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) 

42 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) A 

43 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) A  B

44 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) A  B

45 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) A  B A→B

46 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) A  B A→B

47 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) A  B (A→B)→ A→B

48 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) A  B (A→B)→ A→B

49 Control Operators - types
What about the more powerful C control operator? Recall the reduction rule: E{C M} → M λx.(A E{x}) Observation (Griffin): if we set B= then C can be typed as double-negation-elimination: ((A→)→)→A ≡ ¬ ¬ A→A Led to research into computational interpretations of classical logics.. including this talk! A  B ((A→B)→)→A (A→B)→ A→B

50 Delimited Control Operators
So far, control operators capture entire context A refinement to the previous ideas: delimited control “markers” # can be placed around any subterm: # M Only capture the context up to the enclosing # Refines reduction behaviours:

51 Delimited Control Operators
So far, control operators capture entire context A refinement to the previous ideas: delimited control “markers” # can be placed around any subterm: # M Only capture the context up to the enclosing # Refines reduction behaviours: E{# E'{A M}} → E{# M}

52 Delimited Control Operators
So far, control operators capture entire context A refinement to the previous ideas: delimited control “markers” # can be placed around any subterm: # M Only capture the context up to the enclosing # Refines reduction behaviours: E{# E'{A M}} → E{# M}

53 Delimited Control Operators
So far, control operators capture entire context A refinement to the previous ideas: delimited control “markers” # can be placed around any subterm: # M Only capture the context up to the enclosing # Refines reduction behaviours: E{# E'{A M}} → E{# M}

54 Delimited Control Operators
So far, control operators capture entire context A refinement to the previous ideas: delimited control “markers” # can be placed around any subterm: # M Only capture the context up to the enclosing # Refines reduction behaviours: E{# E'{A M}} → E{# M} E{# E'{C M}} → E{# (M λx.(A E'{x}))}

55 Delimited Control Operators
So far, control operators capture entire context A refinement to the previous ideas: delimited control “markers” # can be placed around any subterm: # M Only capture the context up to the enclosing # Refines reduction behaviours: E{# E'{A M}} → E{# M} E{# E'{C M}} → E{# (M λx.(A E'{x}))}

56 Delimited Control Operators
So far, control operators capture entire context A refinement to the previous ideas: delimited control “markers” # can be placed around any subterm: # M Only capture the context up to the enclosing # Refines reduction behaviours: E{# E'{A M}} → E{# M} E{# E'{C M}} → E{# (M λx.(A E'{x}))} Can express more complex/interesting behaviour Nicer semantics – compositional with E{}

57 Ok... now, for the logic bit. What we’re going to talk about
the effect of different connectives on term calculi Write this at the end, when all other slides are done.#

58 Curry-Howard Correspondence
Historically relates minimal logic and λ-calculus Formulas relate to types Proofs relate to terms Proof reductions relate to β reductions Logic and calculus were invented independently We can borrow the idea of the correspondence.. Existing calculus ⇒ logic-based type system Existing logic ⇒ new programming calculus Extract the ‘computational content’ of the logic What we’re going to talk about the effect of different connectives on term calculi Write this at the end, when all other slides are done.#

59 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it)

60 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it)

61 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) (Ax) Γ, x:A ⊢ x : A

62 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B

63 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

64 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

65 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

66 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

67 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

68 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

69 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

70 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

71 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

72 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

73 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

74 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

75 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

76 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

77 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

78 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

79 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

80 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

81 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

82 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

83 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

84 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

85 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

86 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

87 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

88 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

89 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

90 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

91 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

92 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

93 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

94 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ A→B Γ ⊢ A (→E) Γ ⊢ B

95 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ A (→E) Γ ⊢ B

96 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ A (→E) Γ ⊢ B

97 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ A (→E) Γ ⊢ B

98 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ A (→E) Γ ⊢ B

99 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ A (→E) Γ ⊢ B

100 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ A (→E) Γ ⊢ B

101 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ A (→E) Γ ⊢ B

102 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

103 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

104 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

105 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

106 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

107 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

108 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

109 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

110 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

111 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

112 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

113 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

114 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ B

115 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

116 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

117 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

118 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

119 Curry-Howard Correspondence
Minimal Natural Deduction (If Lambda Calculus did not exist, it would be necessary to invent it) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

120 Curry-Howard Correspondence
Minimal Natural Deduction Reductions? Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

121 Curry-Howard Correspondence
Minimal Natural Deduction A notion of proof reduction exists (Gentzen/Prawitz) Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

122 Curry-Howard Correspondence
Minimal Natural Deduction A notion of proof reduction exists (Gentzen/Prawitz) It gives rise to the standard rule λx.M N → M< N \ x > Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

123 Curry-Howard Correspondence
Minimal Natural Deduction We can regard λ-calculus as computational content of minimal natural deduction.. Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

124 Curry-Howard Correspondence
Minimal Natural Deduction We can regard λ-calculus as computational content of minimal natural deduction.. what about classical logic? Γ, x:A ⊢ M : B (Ax) (→I) Γ, x:A ⊢ x : A Γ ⊢ λx.M : A→B Γ ⊢ M : A→B Γ ⊢ N : A (→E) Γ ⊢ (M N) : B

125 The λμ-calculus on one slide (almost)
Presented by Parigot (1992) to tackle classical logic Many variants since - we present slight refinement Syntax extends λ-calculus: new class of (Greek) variables α... M, N ::= x | λx.M | (M N)

126 The λμ-calculus on one slide (almost)
Presented by Parigot (1992) to tackle classical logic Many variants since - we present slight refinement Syntax extends λ-calculus: new class of (Greek) variables α... M, N ::= x | λx.M | (M N) | μα.M | [α]N

127 The λμ-calculus on one slide (almost)
Presented by Parigot (1992) to tackle classical logic Many variants since - we present slight refinement Syntax extends λ-calculus: new class of (Greek) variables α... M, N ::= x | λx.M | (M N) | μα.M | [α]N

128 The λμ-calculus on one slide (almost)
Presented by Parigot (1992) to tackle classical logic Many variants since - we present slight refinement Syntax extends λ-calculus: new class of (Greek) variables α... M, N ::= x | λx.M | (M N) | μα.M | [α]N Types for the calculus are expressed using judgements of the form: Γ ⊢ M : A | Δ Δ provides type assumptions for Greek variables The underlying logic doesn’t look much like classical natural deduction...

129 Extending λμ-calculus
Γ, x:A ⊢ x : A | Δ Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

130 Extending λμ-calculus
Γ, x:A ⊢ x : A | Δ Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

131 Extending λμ-calculus
Judgements have multiple conclusions Γ, x:A ⊢ x : A | Δ Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

132 Extending λμ-calculus
Judgements have multiple conclusions  Γ, x:A ⊢ x : A | Δ Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

133 Extending λμ-calculus
Judgements have multiple conclusions  Parigot: derivations can be transformed Γ, x:A ⊢ x : A | Δ Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

134 Extending λμ-calculus
Judgements have multiple conclusions  Parigot: derivations can be transformed Γ, x:A ⊢ x : A | Δ Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

135 Extending λμ-calculus
Judgements have single conclusions  Parigot: derivations can be transformed Γ, ¬Δ, x:A ⊢ x : A Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

136 Extending λμ-calculus
Judgements have single conclusions  Γ, ¬Δ, x:A ⊢ x : A Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

137 Extending λμ-calculus
Γ, ¬Δ, x:A ⊢ x : A Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

138 Extending λμ-calculus
Γ, ¬Δ, x:A ⊢ x : A Γ, x:A ⊢ M : B | Δ Γ ⊢ N : A | Δ Γ ⊢ λx.M : A→B | Δ Γ ⊢ [α]N :  | α:A, Δ Γ ⊢ M : A→B | Δ Γ ⊢ N : A | Δ Γ ⊢ M :  | α:A, Δ Γ ⊢ (M N) : B | Δ Γ ⊢ μα.M : A | Δ

139 Extending λμ-calculus
Γ, ¬Δ, x:A ⊢ x : A Γ, ¬Δ, x:A ⊢ M : B Γ, ¬Δ ⊢ N : A Γ, ¬Δ ⊢ λx.M : A→B Γ, ¬Δ, α: ¬A ⊢ [α]N :  Γ, ¬Δ ⊢ M : A→B Γ, ¬Δ ⊢ N : A Γ, ¬Δ, α: ¬A ⊢ M :  Γ, ¬Δ ⊢ (M N) : B Γ, ¬Δ ⊢ μα.M : A

140 Extending λμ-calculus
Γ, ¬Δ, x:A ⊢ x : A Γ, ¬Δ, x:A ⊢ M : B Γ, ¬Δ ⊢ N : A Γ, ¬Δ ⊢ λx.M : A→B Γ, ¬Δ, α: ¬A ⊢ [α]N :  Γ, ¬Δ ⊢ M : A→B Γ, ¬Δ ⊢ N : A Γ, ¬Δ, α: ¬A ⊢ M :  Γ, ¬Δ ⊢ (M N) : B Γ, ¬Δ ⊢ μα.M : A

141 Extending λμ-calculus
Greek ‘special’ assumptions are restricted  Γ, ¬Δ, x:A ⊢ x : A Γ, ¬Δ, x:A ⊢ M : B Γ, ¬Δ ⊢ N : A Γ, ¬Δ ⊢ λx.M : A→B Γ, ¬Δ, α: ¬A ⊢ [α]N :  Γ, ¬Δ ⊢ M : A→B Γ, ¬Δ ⊢ N : A Γ, ¬Δ, α: ¬A ⊢ M :  Γ, ¬Δ ⊢ (M N) : B Γ, ¬Δ ⊢ μα.M : A

142 Extending λμ-calculus
Greek ‘special’ assumptions are restricted  Idea: merge the two alphabets Γ, ¬Δ, x:A ⊢ x : A Γ, ¬Δ, x:A ⊢ M : B Γ, ¬Δ ⊢ N : A Γ, ¬Δ ⊢ λx.M : A→B Γ, ¬Δ, α: ¬A ⊢ [α]N :  Γ, ¬Δ ⊢ M : A→B Γ, ¬Δ ⊢ N : A Γ, ¬Δ, α: ¬A ⊢ M :  Γ, ¬Δ ⊢ (M N) : B Γ, ¬Δ ⊢ μα.M : A

143 Extending λμ-calculus
Greek ‘special’ assumptions are restricted  Idea: merge the two alphabets Γ, ¬Δ, x:A ⊢ x : A Γ, ¬Δ, x:A ⊢ M : B Γ, ¬Δ ⊢ N : A Γ, ¬Δ ⊢ λx.M : A→B Γ, ¬Δ, α: ¬A ⊢ [α]N :  Γ, ¬Δ ⊢ M : A→B Γ, ¬Δ ⊢ N : A Γ, ¬Δ, α: ¬A ⊢ M :  Γ, ¬Δ ⊢ (M N) : B Γ, ¬Δ ⊢ μα.M : A

144 Extending λμ-calculus
Greek ‘special’ assumptions are restricted  Idea: merge the two alphabets Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A Γ ⊢ λx.M : A→B Γ, α: ¬A ⊢ [α]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, α: ¬A ⊢ M :  Γ ⊢ (M N) : B Γ ⊢ μα.M : A

145 Extending λμ-calculus
Greek ‘special’ assumptions are restricted  Idea: merge the two alphabets Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A Γ ⊢ λx.M : A→B Γ, α: ¬A ⊢ [α]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, α: ¬A ⊢ M :  Γ ⊢ (M N) : B Γ ⊢ μα.M : A

146 Extending λμ-calculus
Greek ‘special’ assumptions are restricted  Idea: merge the two alphabets Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A Γ ⊢ λx.M : A→B Γ, α: ¬A ⊢ [α]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, α: ¬A ⊢ M :  Γ ⊢ (M N) : B Γ ⊢ μα.M : A

147 Extending λμ-calculus
Greek ‘special’ assumptions are restricted  Idea: merge the two alphabets Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A Γ ⊢ λx.M : A→B Γ, x: ¬A ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  Γ ⊢ (M N) : B Γ ⊢ μx.M : A

148 Extending λμ-calculus
All assumptions have equal status  Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A Γ ⊢ λx.M : A→B Γ, x: ¬A ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  Γ ⊢ (M N) : B Γ ⊢ μx.M : A

149 Extending λμ-calculus
Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A Γ ⊢ λx.M : A→B Γ, x: ¬A ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  Γ ⊢ (M N) : B Γ ⊢ μx.M : A

150 Extending λμ-calculus
(Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A (→I) Γ ⊢ λx.M : A→B Γ, x: ¬A ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

151 Extending λμ-calculus
(Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A (→I) ? Γ ⊢ λx.M : A→B Γ, x: ¬A ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

152 Extending λμ-calculus
Special case of (¬E) (Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ N : A (→I) Γ ⊢ λx.M : A→B Γ, x: ¬A ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

153 Extending λμ-calculus
Special case of (¬E) (Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ x : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

154 Extending λμ-calculus
Special case of (¬E) First premise must be an axiom  (Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ x : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

155 Extending λμ-calculus
Special case of (¬E) First premise must be an axiom  (Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ x : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [x]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

156 Extending λμ-calculus
Special case of (¬E) First premise must be an axiom  (Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ x : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [ x ]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

157 Extending λμ-calculus
Special case of (¬E) First premise must be an axiom  (Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ x : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [ x ]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

158 Extending λμ-calculus
Special case of (¬E) First premise can be any proof  (Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

159 Extending λμ-calculus
(Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

160 Extending λμ-calculus
(Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

161 Extending λμ-calculus
The rule (¬I) is entirely missing  (Ax) Γ, x:A ⊢ x : A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

162 Extending λμ-calculus
Γ, x:A ⊢ M :   (Ax) (¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

163 Extending λμ-calculus
Γ, x:A ⊢ M :  (Ax) (¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

164 Extending λμ-calculus
Γ, x:A ⊢ M :  (Ax) (¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

165 Extending λμ-calculus
Γ, x:A ⊢ M :  (Ax) (¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

166 Extending λμ-calculus
Γ, x:A ⊢ M :  (Ax) (¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

167 Extending λμ-calculus
Classical Natural Deduction Γ, x:A ⊢ M :  (Ax) (¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

168 Extending λμ-calculus
Classical Natural Deduction Γ, x:A ⊢ M :  (Ax) (¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

169 Extending λμ-calculus
Classical Natural Deduction Γ, x:A ⊢ M :  (Ax) (¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

170 The νλμ-calculus Γ, x:A ⊢ M :  Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A
(¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

171 The νλμ-calculus The νλμ-calculus Γ, x:A ⊢ M :  Γ, x:A ⊢ x : A
(¬I) Γ, x:A ⊢ x : A Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

172 Important features The νλμ-calculus Γ, x:A ⊢ M :  Γ, x:A ⊢ x : A
Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

173 Important features The νλμ-calculus Γ, x:A ⊢ M :  Γ, x:A ⊢ x : A
Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

174 Important features The νλμ-calculus Γ, x:A ⊢ M :  Γ, x:A ⊢ x : A
Γ ⊢ νx.M : ¬A Γ, x:A ⊢ M : B Γ ⊢ M : ¬A Γ ⊢ N : A (→I) (¬E) Γ ⊢ λx.M : A→B Γ ⊢ [M]N :  Γ ⊢ M : A→B Γ ⊢ N : A Γ, x: ¬A ⊢ M :  (→E) (PC) Γ ⊢ (M N) : B Γ ⊢ μx.M : A

175 Important features  νx.M ¬A [M]N μx.M

176 Important features  νx.M ¬A [M]N μx.M

177 Important features  νx.M ¬A [M]N μx.M

178 Important features  ¬A νx.M [M]N μx.M

179 Important features  ¬A νx.M [M]N μx.M

180 Important features  ¬A νx.M [M]N μx.M

181 Important features  ¬A νx.M [M]N μx.M

182 Important features  ¬A νx.M [M]N μx.M

183 Important features  ¬A νx.M [M]N μx.M

184 Important features  ¬A νx.M [M]N μx.M

185 Important features  ¬A νx.M [M]N μx.M

186 Important features  ¬A νx.M [M]N μx.M

187 Important features  ¬A νx.M [M]N μx.M

188 Important features  ¬A νx.M [M]N μx.M

189 Important features  ¬A νx.M [M]N μx.M

190 Important features  - type of “finished programs” – essentially no output value ¬A - negated type for continuation with input of type A νx.M - terms to explicitly represent continuations terms which consume an input and produce -type [M]N - continuation application Pass the argument N to the continuation M μx.M - control operator Capture the surrounding ‘context’ and bind it to x What about reductions? Many alternative ideas exist, particularly for the μ-bound terms Identify a common theme, and generalise the existing work  ¬A νx.M [M]N μx.M

191 Reductions… λx.M N → M< N \ x > [νx.M]N → M< N \ x >
Two “non-classical” reduction rules arise naturally: λx.M N → M< N \ x > [νx.M]N → M< N \ x > These are the same rules one would use if basing a calculus on minimal logic So far, nothing to distinguish functions and continuations - “might as well” define ¬A as A→ ? However, once we add reductions related to terms μx.M then we can see a difference..

192 μ-reductions As with any binder, consider how μ can be eliminated

193 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC)

194 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC)

195 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC)

196 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC)

197 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC) on μx.M : A

198 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

199 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

200 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

201 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

202 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

203 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

204 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

205 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

206 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

207 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

208 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

209 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A μx.M

210 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A μx.M : A

211 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A z : A

212 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC) νz. Γ ⊢ μx.M : A z : A : ¬A

213 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC) νz. Γ ⊢ μx.M : A z : A : ¬A

214 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC) νz. Γ ⊢ μx.M : A z : A : ¬A

215 μ-reductions As with any binder, consider how μ can be eliminated
Recall the typing rule: Find a continuation of type ¬A Substitute the term for x Term of type ¬A is a continuation with a ‘hole’ of type A Any context around μx.M must have a ‘hole’ of type A Γ, x: ¬A ⊢ M :  (PC) Γ ⊢ μx.M : A Γ, x: ¬A ⊢ M :  (PC) νz. Γ ⊢ μx.M : A z : A : ¬A

216 μ-reductions Γ, x: ¬A ⊢ M :  (PC) νz. Γ ⊢ μx.M : A z : A : ¬A

217 μ-reductions Γ, x: ¬A ⊢ M :  (PC) νz. E{ } Γ ⊢ μx.M : A z : A : ¬A

218 μ-reductions Γ ⊢ νz.E{z} : ¬A Consider typing the term
Γ, x: ¬A ⊢ M :  (PC) νz. E{ } z : A : ¬A Γ ⊢ μx.M : A

219 μ-reductions Γ ⊢ νz.E{z} : ¬A Consider typing the term
Γ, x: ¬A ⊢ M :  (PC) νz. E{ } z : A : ¬A Γ ⊢ μx.M : A

220 μ-reductions Γ ⊢ νz.E{z} : ¬A Consider typing the term
Γ, x: ¬A ⊢ M :  (PC) νz. E{ } z : A : ¬A Γ ⊢ μx.M : A

221 μ-reductions Γ ⊢ νz.E{z} : ¬A Consider typing the term
Γ, x: ¬A ⊢ M :  (PC) νz. E{ } : ¬A Γ ⊢ μx.M : A z : A

222 μ-reductions Γ, z:A ⊢ E{z} :  Γ ⊢ νz.E{z} : ¬A
Consider typing the term Γ, x: ¬A ⊢ M :  (PC) νz. E{ } z : A : ¬A Γ ⊢ μx.M : A

223 μ-reductions Γ, z:A ⊢ E{z} :  Γ ⊢ νz.E{z} : ¬A
Consider typing the term This only “works” if E{z} has type  What if we have a context with a different type? What if we want a compatible reduction rule? i.e., can enlarge the context, preserving reductions consuming the “whole” context at once is not wise! Γ, x: ¬A ⊢ M :  (PC) νz. E{ } z : A : ¬A Γ ⊢ μx.M : A

224 μ-reductions We tackle these problems by defining local rules for gradually moving μ-bindings outwards e.g., Γ ⊢ (μx.M N) : B

225 μ-reductions We tackle these problems by defining local rules for gradually moving μ-bindings outwards e.g., Γ ⊢ (μx.M N) : B

226 μ-reductions Γ ⊢ (μx.M N) : B

227 μ-reductions Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B

228 μ-reductions N Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B

229 μ-reductions N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B

230 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B

231 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B Γ ⊢ ? : B

232 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B Γ, y: ¬B ⊢ ? :  (PC) Γ ⊢ μy.? : B

233 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M has the right type.. Γ, y: ¬B ⊢ ? :  (PC) Γ ⊢ μy.? : B

234 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M has the right type.. Γ, y: ¬B ⊢ ? :  (PC) Γ ⊢ μy.? : B M requires x:¬(A→B)

235 M requires x:¬(A→B) .. we have y: ¬B and N : A available
μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  (PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M has the right type.. Γ, y: ¬B ⊢ ? :  (PC) Γ ⊢ μy.? : B M requires x:¬(A→B) .. we have y: ¬B and N : A available

236 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M has the right type.. Γ, y: ¬B ⊢ ? :  (PC) Γ ⊢ μy.? : B M requires x:¬(A→B) .. we have y: ¬B and N : A available From ¬B and A we can prove ¬(A→B)

237 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M has the right type.. Substitute the corresponding term for x in M Γ, y: ¬B ⊢ ? :  (PC) Γ ⊢ μy.? : B M requires x:¬(A→B) .. we have y: ¬B and N : A available From ¬B and A we can prove ¬(A→B)

238 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B Substitute the corresponding term for x in M M* Γ, y: ¬B ⊢ M< νz.[y](z N) / x > :  Γ, y: ¬B ⊢ ? :  (PC) Γ ⊢ μy.? : B

239 Γ, y: ¬B ⊢ M< νz.[y](z N) / x > : 
μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  (PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M* Γ, y: ¬B ⊢ M< νz.[y](z N) / x > :  (PC) Γ ⊢ μy.? : B

240 Γ, y: ¬B ⊢ M< νz.[y](z N) / x > : 
μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  (PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M* Γ, y: ¬B ⊢ M< νz.[y](z N) / x > :  (PC) Γ ⊢ μy.? : B

241 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M* Γ, y: ¬B ⊢ M< νz.[y](z N) / x > :  (PC) Γ ⊢ μy.M< νz.[y](z N) / x > : B

242 μ-reductions M N Γ, x: ¬(A→B) ⊢ M :  Γ ⊢ μx.M : A→B Γ ⊢ N : A
(PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M* Γ, y: ¬B ⊢ M< νz.[y](z N) / x > :  (PC) Γ ⊢ μy.M< νz.[y](z N) / x > : B

243 μ-reductions M Reduction rule: μx.M N → μy.M< νz.[y](z N) / x >
Γ, x: ¬(A→B) ⊢ M :  (PC) Γ ⊢ μx.M : A→B Γ ⊢ N : A (→E) Γ ⊢ (μx.M N) : B M* Γ, y: ¬B ⊢ M< νz.[y](z N) / x > :  (PC) Γ ⊢ μy.M< νz.[y](z N) / x > : B

244 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x >

245 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x >

246 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x >

247 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x >

248 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x >

249 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x >

250 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x > [N]μx.M → M< νz.[N]z / x >

251 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x > [N]μx.M → M< νz.[N]z / x >

252 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x > [N]μx.M → M< νz.[N]z / x > μ-binders work outwards through function applications

253 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x > [N]μx.M → M< νz.[N]z / x > μ-binders work outwards through function applications When a continuation application is reached, they stop

254 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x > [N]μx.M → M< νz.[N]z / x > μ-binders work outwards through function applications When a continuation application is reached, they stop Let Ec be a context consisting of any number of function applications, enclosed by a continuation application

255 μ-reduction rules By this process, we derive the following rules
μx.M N → μy.M< νz.[y](z N) / x > N μx.M → μy.M< νz.[y](N z) / x > [μx.M]N → M< νz.[z]N / x > [N]μx.M → M< νz.[N]z / x > μ-binders work outwards through function applications When a continuation application is reached, they stop Let Ec be a context consisting of any number of function applications, enclosed by a continuation application Then Ec{μx.M} → M< νz.Ec{z} / x >

256 μ-reduction rules The n: Ec{μx.M} → M< νz.Ec{z} / x >
Furthermore, if enclosed in further context E E{Ec{μx.M}} → E{M< νz.Ec{z} / x >} The μ-binding captures the surrounding context up to the nearest enclosing continuation application That is, it behaves like a delimited control operator! The special kind of application [M]N provides an implicit delimiter for the control behaviour Now, we can explain the advantage of explicit negation Replacing ¬A with A→ amounts to replacing [M]N with (M N), i.e., removing the delimiter in the term

257 Back to the control operators…
E{Ec{μx.M}} → E{M< νz.Ec{z} / x >} How can we represent existing control operators? Recall ‘abort’ A was typed as: i.e., -elimination Logically, we express -elimination by (PC) Equivalent if x doesn’t occur in M Now consider the computational behaviour: if x doesn’t occur in M, E{Ec{μx.M}} → E{M< νz.Ec{z} / x >} Γ ⊢ M :  (A) Γ ⊢ (A M) : C Γ, x: ¬C ⊢ M :  (PC) Γ ⊢ μx.M : C

258 Back to the control operators…
E{Ec{μx.M}} → E{M< νz.Ec{z} / x >} How can we represent existing control operators? Recall ‘abort’ A was typed as: i.e., -elimination Logically, we express -elimination by (PC) Equivalent if x doesn’t occur in M Now consider the computational behaviour: if x doesn’t occur in M, E{Ec{μx.M}} → E{M< νz.Ec{z} / x >} = E{M} Γ ⊢ M :  (A) Γ ⊢ (A M) : C Γ, x: ¬C ⊢ M :  (PC) Γ ⊢ μx.M : C

259 Back to the control operators…
E{Ec{μx.M}} → E{M< νz.Ec{z} / x >} How can we represent existing control operators? Recall ‘abort’ A was typed as: i.e., -elimination Logically, we express -elimination by (PC) Equivalent if x doesn’t occur in M Now consider the computational behaviour: if x doesn’t occur in M, E{Ec{μx.M}} → E{M< νz.Ec{z} / x >} = E{M} This is exactly the behaviour of (delimited) ‘abort’! Γ ⊢ M :  (A) Γ ⊢ (A M) : C Γ, x: ¬C ⊢ M :  (PC) Γ ⊢ μx.M : C

260 Back to the control operators…
E{Ec{μx.M}} → E{M< νz.Ec{z} / x >} ‘control’ C was typed with ((A→B)→)→A This is a classically-valid formula : write the simplest possible proof of it in classical natural deduction Extract the corresponding νλμ-term: λx.μy.(x λz.μw.[y]z) : ((A→B)→)→A Now consider the computational behaviour: E{Ec{λx.μy.(x λz.μw.[y]z) M}} → E{Ec{μy.(M λz.μw.[y]z)}} → E{(M λz.μw.Ec{z})} This is the behaviour of the delimited version of C

261 Summary - things we can do
νλμ-calculus has Curry-Howard-style correspondence with classical natural deduction Naturally expresses delimited control operators First Curry-Howard calculus to do so It can also simulate many similar calculi λ-calculus, λμ (and variants), symmetric λ-calculus, etc.. Reductions expressive from logical perspective Can encode sequent proofs as natural deduction proofs and simulate cut elimination with our proof reductions First encoding of classical cut elimination in natural deduction paradigm (?)

262 Future Work - things I can’t do (yet)
Strong Normalisation Property: Typeable terms only have terminating reductions Existing techniques seem not to apply Reducibility candidates? Generalised by Barbanera&Berardi Generalised further by Urban Do not seem to work with  as a proper type Encodings into other strongly normalising calculi? General reductions in νλμ make this hard No known encoding into another calculus Any ideas?

263 Future Work - things I can’t do (yet)
Strong Normalisation Property: Typeable terms only have terminating reductions Existing techniques seem not to apply Reducibility candidates? Generalised by Barbanera&Berardi Generalised further by Urban Do not seem to work with  as a proper type Encodings into other strongly normalising calculi? General reductions in νλμ make this hard No known encoding into another calculus Any ideas? 

264 In conclusion..  λx.μk.[k]((Fix f. λy.match y with [] --> 1
n:rest --> if n=0 then μj.[k]0 else n*(f rest)) x) 

265 In conclusion..  λx.μk.[k]((Fix f. λy.match y with [] --> 1
n:rest --> if n=0 then μj.[k]0 else n*(f rest)) x) 

266 Thank you for listening!
In conclusion.. λx.μk.[k]((Fix f. λy.match y with [] --> 1 n:rest --> if n=0 then μj.[k]0 else n*(f rest)) x)  Thank you for listening!

267 Thank you for listening!
In conclusion.. λx.μk.[k]((Fix f. λy.match y with [] --> 1 n:rest --> if n=0 then μj.[k]0 else n*(f rest)) x) For more details (and proper references), please see my PhD thesis (Ch. 5 & 6) Available from or Google for “Alex Summers”  Thank you for listening!