[PPT] - A Computational Understanding of Classical (Co)Recursion P a ul PowerPoint Presentation

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PPDP 2020, September 8–10

A Computational Understanding

f Classical (Co)Recursion

Paul Downen and Zena M. Ariola

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Topic

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Topic

Both programs and proofs with loops

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Topic

Both programs and proofs with loops
(Co)Recursion and (Co)Induction

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Topic

Both programs and proofs with loops
(Co)Recursion and (Co)Induction
“Terminating” or “Productive”

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Topic

Both programs and proofs with loops
(Co)Recursion and (Co)Induction
“Terminating” or “Productive”
Extend to non-termination, effects

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Methodology

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Methodology

Duality

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Methodology

Duality
Computational

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Methodology

Duality
Computational
Curry-Howard

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Methodology

Duality
Computational
Curry-Howard
sequent calculus as abstract machines

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Methodology

Duality
Computational
Curry-Howard
sequent calculus as abstract machines
Classical

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Recursive Programs

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Recursion on Natural Numbers

In System T

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Recursion on Natural Numbers

Simply-typed -calculus plus inductive

λ Nat = Zero ∣ Succ Nat

In System T

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Recursion on Natural Numbers

Simply-typed -calculus plus inductive

λ Nat = Zero ∣ Succ Nat recA

Nat : Nat → A → (Nat → A → A) → A

In System T

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Recursion on Natural Numbers

Simply-typed -calculus plus inductive

λ Nat = Zero ∣ Succ Nat recA

Nat : Nat → A → (Nat → A → A) → A

rec M as {Zero → N ∣ Succ x → y . P}

where M, x : Nat; N and P, y : A

In System T

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Recursion on Natural Numbers

Simply-typed -calculus plus inductive

λ Nat = Zero ∣ Succ Nat recA

Nat : Nat → A → (Nat → A → A) → A

rec M as {Zero → N ∣ Succ x → y . P}

where M, x : Nat; N and P, y : A

case M of Zero → N Succ x → P := rec M of Zero → M Succ x → _ . P

In System T

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Recursion on Natural Numbers

Simply-typed -calculus plus inductive

λ Nat = Zero ∣ Succ Nat recA

Nat : Nat → A → (Nat → A → A) → A

rec M as {Zero → N ∣ Succ x → y . P}

where M, x : Nat; N and P, y : A

case M of Zero → N Succ x → P := rec M of Zero → M Succ x → _ . P iter M as Zero → N Succ → y . P := rec M as Zero → N Succ _ → y . P

In System T

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Examples of Recursion

In System T

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Examples of Recursion

plus Zero y = y plus (Succ x′ ) y = Succ (plus x′ y)

In System T

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Examples of Recursion

plus Zero y = y plus (Succ x′ ) y = Succ (plus x′ y)

plus = λx . λy . iter x as Zero → y Succ → z . Succ z

In System T

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Examples of Recursion

plus Zero y = y plus (Succ x′ ) y = Succ (plus x′ y)

plus = λx . λy . iter x as Zero → y Succ → z . Succ z

In System T

pred Zero = Zero pred (Succ x′ ) = x′

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Examples of Recursion

plus Zero y = y plus (Succ x′ ) y = Succ (plus x′ y)

plus = λx . λy . iter x as Zero → y Succ → z . Succ z

In System T

pred Zero = Zero pred (Succ x′ ) = x′ pred = λx . case x of Zero → Zero Succ x′ → x′

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Examples of Recursion

plus Zero y = y plus (Succ x′ ) y = Succ (plus x′ y)

plus = λx . λy . iter x as Zero → y Succ → z . Succ z

In System T

pred Zero = Zero pred (Succ x′ ) = x′ pred = λx . case x of Zero → Zero Succ x′ → x′

minus x Zero = x minus x (Succ y′ ) = pred (minus x y′ )

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Examples of Recursion

plus Zero y = y plus (Succ x′ ) y = Succ (plus x′ y)

plus = λx . λy . iter x as Zero → y Succ → z . Succ z

In System T

pred Zero = Zero pred (Succ x′ ) = x′ pred = λx . case x of Zero → Zero Succ x′ → x′

minus x Zero = x minus x (Succ y′ ) = pred (minus x y′ )

minus = λx . λy . iter y as Zero → x Succ → z . pred z

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Recursion vs Iteration

Expressiveness vs Cost

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Recursion vs Iteration

iter M as Zero → N Succ → y . P := rec M as Zero → N Succ _ → y . P

Expressiveness vs Cost

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Recursion vs Iteration

iter M as Zero → N Succ → y . P := rec M as Zero → N Succ _ → y . P rec M as Zero → N Succ x → y . P := snd(iter M as Zero → (Zero, N) Succ → (x, y) . (Succ x, P))

Expressiveness vs Cost

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Recursion vs Iteration

iter M as Zero → N Succ → y . P := rec M as Zero → N Succ _ → y . P rec M as Zero → N Succ x → y . P := snd(iter M as Zero → (Zero, N) Succ → (x, y) . (Succ x, P))

goes from

to time

pred (Succn Zero) O(1) O(n)

Expressiveness vs Cost

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Recursion vs Iteration

iter M as Zero → N Succ → y . P := rec M as Zero → N Succ _ → y . P rec M as Zero → N Succ x → y . P := snd(iter M as Zero → (Zero, N) Succ → (x, y) . (Succ x, P))

goes from

to time

pred (Succn Zero) O(1) O(n)

goes from

to

minus (Succn Zero) (Succm Zero) O(n) O(n2 + nm)

Expressiveness vs Cost

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Recursion vs Iteration

iter M as Zero → N Succ → y . P := rec M as Zero → N Succ _ → y . P rec M as Zero → N Succ x → y . P := snd(iter M as Zero → (Zero, N) Succ → (x, y) . (Succ x, P))

goes from

to time

pred (Succn Zero) O(1) O(n)

goes from

to

minus (Succn Zero) (Succm Zero) O(n) O(n2 + nm)

Native

has the same performance penalty as encoding in CBV

rec

Expressiveness vs Cost

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Recursion vs Iteration

iter M as Zero → N Succ → y . P := rec M as Zero → N Succ _ → y . P rec M as Zero → N Succ x → y . P := snd(iter M as Zero → (Zero, N) Succ → (x, y) . (Succ x, P))

goes from

to time

pred (Succn Zero) O(1) O(n)

goes from

to

minus (Succn Zero) (Succm Zero) O(n) O(n2 + nm)

Native

has the same performance penalty as encoding in CBV

rec

Recursive result always computed; full traversal is mandatory

Expressiveness vs Cost

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Recursion in an Abstract Machine

Building the Recursive Continuation

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Recursion in an Abstract Machine

Building the Recursive Continuation

⟨M| |E⟩

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Recursion in an Abstract Machine

Building the Recursive Continuation

⟨M N| |E⟩ ↦ ⟨M| |N ⋅ E⟩ ⟨λx . M| |N ⋅ E⟩ ↦ ⟨M[N/x]| |E⟩ ⟨M| |E⟩

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Recursion in an Abstract Machine

Building the Recursive Continuation

⟨M N| |E⟩ ↦ ⟨M| |N ⋅ E⟩ ⟨λx . M| |N ⋅ E⟩ ↦ ⟨M[N/x]| |E⟩

ralt := { Zero → N ∣ Succ x → y . P }

⟨M| |E⟩

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Recursion in an Abstract Machine

Building the Recursive Continuation

⟨M N| |E⟩ ↦ ⟨M| |N ⋅ E⟩ ⟨λx . M| |N ⋅ E⟩ ↦ ⟨M[N/x]| |E⟩

⟨rec M as ralt| |E⟩ ↦ ⟨M| |rec ralt with E⟩ ralt := { Zero → N ∣ Succ x → y . P }

⟨M| |E⟩

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Recursion in an Abstract Machine

Building the Recursive Continuation

⟨M N| |E⟩ ↦ ⟨M| |N ⋅ E⟩ ⟨λx . M| |N ⋅ E⟩ ↦ ⟨M[N/x]| |E⟩

⟨rec M as ralt| |E⟩ ↦ ⟨M| |rec ralt with E⟩ ⟨Zero| |rec ralt with E⟩ ↦ ⟨N| |E⟩ ralt := { Zero → N ∣ Succ x → y . P }

⟨M| |E⟩

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Recursion in an Abstract Machine

Building the Recursive Continuation

⟨M N| |E⟩ ↦ ⟨M| |N ⋅ E⟩ ⟨λx . M| |N ⋅ E⟩ ↦ ⟨M[N/x]| |E⟩

⟨rec M as ralt| |E⟩ ↦ ⟨M| |rec ralt with E⟩ ⟨Zero| |rec ralt with E⟩ ↦ ⟨N| |E⟩ ⟨Succ M| |rec ralt with E⟩ ↦ ⟨P[M/x, rec M as ralt/y]| |E⟩ ralt := { Zero → N ∣ Succ x → y . P }

⟨M| |E⟩

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Corecursive Programs

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What’s the Dual of Natural Numbers?

Nat⊥

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What’s the Dual of Natural Numbers?

is dual to

Zero : 1 → Nat Run : Nat⊥ → ⊥

Nat⊥

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What’s the Dual of Natural Numbers?

is dual to

Zero : 1 → Nat Run : Nat⊥ → ⊥

is dual to

Succ : Nat → Nat Tail : Nat⊥ → Nat⊥

Nat⊥

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What’s the Dual of Natural Numbers?

is dual to

Zero : 1 → Nat Run : Nat⊥ → ⊥

is dual to

Succ : Nat → Nat Tail : Nat⊥ → Nat⊥

is an infinite stream of computations

Nat⊥

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What’s the Dual of Natural Numbers?

is dual to

Zero : 1 → Nat Run : Nat⊥ → ⊥

is dual to

Succ : Nat → Nat Tail : Nat⊥ → Nat⊥

is an infinite stream of computations

Nat⊥

You can run them, but they don’t return

Nat⊥

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What’s the Dual of Natural Numbers?

is dual to

Zero : 1 → Nat Run : Nat⊥ → ⊥

is dual to

Succ : Nat → Nat Tail : Nat⊥ → Nat⊥

is an infinite stream of computations

Nat⊥

You can run them, but they don’t return

Nat Values: Zero Succ V Nat Continuation: rec {Zero → N ∣ Succ x → y . P} with E

Nat⊥

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What’s the Dual of Natural Numbers?

is dual to

Zero : 1 → Nat Run : Nat⊥ → ⊥

is dual to

Succ : Nat → Nat Tail : Nat⊥ → Nat⊥

is an infinite stream of computations

Nat⊥

You can run them, but they don’t return

Nat Values: Zero Succ V Nat Continuation: rec {Zero → N ∣ Succ x → y . P} with E Nat⊥ Value: corec {Run → E ∣ Tail α → β . F} with V Nat⊥ Continuations: Run Tail E

Nat⊥

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Corecursion on Streams

In an Abstract Machine

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Corecursion on Streams

Generalize

to

Nat⊥ Stream A

In an Abstract Machine

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Corecursion on Streams

Generalize

to

Nat⊥ Stream A

Infinite stream of computations that return an A

In an Abstract Machine

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Corecursion on Streams

Generalize

to

Nat⊥ Stream A

Infinite stream of computations that return an A
and

Head : Stream A → A Tail : Stream A → Stream A

In an Abstract Machine

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Corecursion on Streams

Generalize

to

Nat⊥ Stream A

Infinite stream of computations that return an A
and

Head : Stream A → A Tail : Stream A → Stream A

Nat⊥ Value: corec {Run → E ∣ Tail β → γ . F} with V Nat⊥ Conts.: Run Tail E

In an Abstract Machine

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Corecursion on Streams

Generalize

to

Nat⊥ Stream A

Infinite stream of computations that return an A
and

Head : Stream A → A Tail : Stream A → Stream A

Nat⊥ Value: corec {Run → E ∣ Tail β → γ . F} with V Nat⊥ Conts.: Run Tail E

Stream A Value: corec {Head α → E ∣ Tail β → γ . F} with V Stream A Conts.: Head E Tail E

In an Abstract Machine

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Corecursion on Streams

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”
Contexts named by

; invoked by jumps

μα . J ⟨M| |α⟩

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”
Contexts named by

; invoked by jumps

μα . J ⟨M| |α⟩

Destructors: when

Head M : A Tail M : Stream A M : Stream A

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”
Contexts named by

; invoked by jumps

μα . J ⟨M| |α⟩

Destructors: when

Head M : A Tail M : Stream A M : Stream A

Generator: corec {Head → x . N ∣ Tail β → y . P} with M

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”
Contexts named by

; invoked by jumps

μα . J ⟨M| |α⟩

Destructors: when

Head M : A Tail M : Stream A M : Stream A

Generator: corec {Head → x . N ∣ Tail β → y . P} with M

Accumulator

, named and in the branches

M x y

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”
Contexts named by

; invoked by jumps

μα . J ⟨M| |α⟩

Destructors: when

Head M : A Tail M : Stream A M : Stream A

Generator: corec {Head → x . N ∣ Tail β → y . P} with M

Accumulator

, named and in the branches

M x y

Head branch: computes first element from current accumulator

N x

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”
Contexts named by

; invoked by jumps

μα . J ⟨M| |α⟩

Destructors: when

Head M : A Tail M : Stream A M : Stream A

Generator: corec {Head → x . N ∣ Tail β → y . P} with M

Accumulator

, named and in the branches

M x y

Head branch: computes first element from current accumulator

N x

Tail branch: computes one of two options

P

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”
Contexts named by

; invoked by jumps

μα . J ⟨M| |α⟩

Destructors: when

Head M : A Tail M : Stream A M : Stream A

Generator: corec {Head → x . N ∣ Tail β → y . P} with M

Accumulator

, named and in the branches

M x y

Head branch: computes first element from current accumulator

N x

Tail branch: computes one of two options

P

Continue: return a new accumulator value from current used for next corecursive loop

y

In the λμ-Calculus

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Corecursion on Streams

Functional, direct-style
Don’t mention continuations directly; implicit “evaluation contexts”
Contexts named by

; invoked by jumps

μα . J ⟨M| |α⟩

Destructors: when

Head M : A Tail M : Stream A M : Stream A

Generator: corec {Head → x . N ∣ Tail β → y . P} with M

Accumulator

, named and in the branches

M x y

Head branch: computes first element from current accumulator

N x

Tail branch: computes one of two options

P

Continue: return a new accumulator value from current used for next corecursive loop

y

End: send a fully-formed stream to context ; this corecursive loop is finished

β

In the λμ-Calculus

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Examples of Corecursion

In an Abstract Machine

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Examples of Corecursion

count x = x, x + 1, x + 2, x + 3…

In an Abstract Machine

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Examples of Corecursion

count x = x, x + 1, x + 2, x + 3… count = λx . corec {Head → y . y ∣ Tail _ → z . Succ z} with x

In an Abstract Machine

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Examples of Corecursion

count x = x, x + 1, x + 2, x + 3… count = λx . corec {Head → y . y ∣ Tail _ → z . Succ z} with x scons x (y0, y1, y2…) = x, y0, y1, y2…

In an Abstract Machine

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Examples of Corecursion

count x = x, x + 1, x + 2, x + 3… count = λx . corec {Head → y . y ∣ Tail _ → z . Succ z} with x scons x (y0, y1, y2…) = x, y0, y1, y2… scons = λx . λys . corec {Head → _ . x ∣ Tail α → _ . μδ . ⟨ys| |α⟩} with _

In an Abstract Machine

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Examples of Corecursion

count x = x, x + 1, x + 2, x + 3… count = λx . corec {Head → y . y ∣ Tail _ → z . Succ z} with x scons x (y0, y1, y2…) = x, y0, y1, y2… scons = λx . λys . corec {Head → _ . x ∣ Tail α → _ . μδ . ⟨ys| |α⟩} with _ app [x0, x1, …, xn] (y0, y1, y2…) = x0, x1, …, xn, y0, y1, y2…

In an Abstract Machine

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Examples of Corecursion

count x = x, x + 1, x + 2, x + 3… count = λx . corec {Head → y . y ∣ Tail _ → z . Succ z} with x scons x (y0, y1, y2…) = x, y0, y1, y2… scons = λx . λys . corec {Head → _ . x ∣ Tail α → _ . μδ . ⟨ys| |α⟩} with _ app [x0, x1, …, xn] (y0, y1, y2…) = x0, x1, …, xn, y0, y1, y2… app = λxs . λys . corec Head → Cons x xs . x Tail _ → Cons x xs . xs Head → Nil . Head ys Tail α → Nil . μδ . ⟨Tail ys| |α⟩ with xs

In an Abstract Machine

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Corecursion vs Coiteration

Expressiveness vs Cost; CBV vs CBN

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Corecursion vs Coiteration

coiter { Head α → E Tail → γ . F} with V := corec { Head α → E Tail _ → γ . F} with V

Expressiveness vs Cost; CBV vs CBN

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Corecursion vs Coiteration

coiter { Head α → E Tail → γ . F} with V := corec { Head α → E Tail _ → γ . F} with V ⟨Left V| |[E, F]⟩ ↦ ⟨V| |E⟩ ⟨Right V| |[E, F]⟩ ↦ ⟨V| |F⟩

Expressiveness vs Cost; CBV vs CBN

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Corecursion vs Coiteration

coiter { Head α → E Tail → γ . F} with V := corec { Head α → E Tail _ → γ . F} with V ⟨Left V| |[E, F]⟩ ↦ ⟨V| |E⟩ ⟨Right V| |[E, F]⟩ ↦ ⟨V| |F⟩ corec { Head α → E Tail β → γ . F} with V := coiter { Head α → [Head α, E] Tail → [β, γ] . [Tail β, F]} with Right V

Expressiveness vs Cost; CBV vs CBN

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Corecursion vs Coiteration

coiter { Head α → E Tail → γ . F} with V := corec { Head α → E Tail _ → γ . F} with V ⟨Left V| |[E, F]⟩ ↦ ⟨V| |E⟩ ⟨Right V| |[E, F]⟩ ↦ ⟨V| |F⟩ corec { Head α → E Tail β → γ . F} with V := coiter { Head α → [Head α, E] Tail → [β, γ] . [Tail β, F]} with Right V

(Amortized) overhead cost; consider

:

scons x ys

Expressiveness vs Cost; CBV vs CBN

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Corecursion vs Coiteration

coiter { Head α → E Tail → γ . F} with V := corec { Head α → E Tail _ → γ . F} with V ⟨Left V| |[E, F]⟩ ↦ ⟨V| |E⟩ ⟨Right V| |[E, F]⟩ ↦ ⟨V| |F⟩ corec { Head α → E Tail β → γ . F} with V := coiter { Head α → [Head α, E] Tail → [β, γ] . [Tail β, F]} with Right V

(Amortized) overhead cost; consider

:

scons x ys

Native

: adds

verhead to cost of

corec Head(Tailn+1(scons x ys)) O(1) Head(Tailn ys)

Expressiveness vs Cost; CBV vs CBN

SLIDE 79

Corecursion vs Coiteration

coiter { Head α → E Tail → γ . F} with V := corec { Head α → E Tail _ → γ . F} with V ⟨Left V| |[E, F]⟩ ↦ ⟨V| |E⟩ ⟨Right V| |[E, F]⟩ ↦ ⟨V| |F⟩ corec { Head α → E Tail β → γ . F} with V := coiter { Head α → [Head α, E] Tail → [β, γ] . [Tail β, F]} with Right V

(Amortized) overhead cost; consider

:

scons x ys

Native

: adds

verhead to cost of

corec Head(Tailn+1(scons x ys)) O(1) Head(Tailn ys)

Encoded

: adds

verhead to cost of

corec Head(Tailn+1(scons x ys)) O(n) Head(Tailn ys)

Expressiveness vs Cost; CBV vs CBN

SLIDE 80

Corecursion vs Coiteration

coiter { Head α → E Tail → γ . F} with V := corec { Head α → E Tail _ → γ . F} with V ⟨Left V| |[E, F]⟩ ↦ ⟨V| |E⟩ ⟨Right V| |[E, F]⟩ ↦ ⟨V| |F⟩ corec { Head α → E Tail β → γ . F} with V := coiter { Head α → [Head α, E] Tail → [β, γ] . [Tail β, F]} with Right V

(Amortized) overhead cost; consider

:

scons x ys

Native

: adds

verhead to cost of

corec Head(Tailn+1(scons x ys)) O(1) Head(Tailn ys)

Encoded

: adds

verhead to cost of

corec Head(Tailn+1(scons x ys)) O(n) Head(Tailn ys)

Native CBN

has same overhead as encoding; Native CBV more efficient

corec corec

Expressiveness vs Cost; CBV vs CBN

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Corecursion vs Coiteration

coiter { Head α → E Tail → γ . F} with V := corec { Head α → E Tail _ → γ . F} with V ⟨Left V| |[E, F]⟩ ↦ ⟨V| |E⟩ ⟨Right V| |[E, F]⟩ ↦ ⟨V| |F⟩ corec { Head α → E Tail β → γ . F} with V := coiter { Head α → [Head α, E] Tail → [β, γ] . [Tail β, F]} with Right V

(Amortized) overhead cost; consider

:

scons x ys

Native

: adds

verhead to cost of

corec Head(Tailn+1(scons x ys)) O(1) Head(Tailn ys)

Encoded

: adds

verhead to cost of

corec Head(Tailn+1(scons x ys)) O(n) Head(Tailn ys)

Native CBN

has same overhead as encoding; Native CBV more efficient

corec corec

Corollary by duality of

and

rec iter

Expressiveness vs Cost; CBV vs CBN

SLIDE 82

(Co)Inductive Reasoning

SLIDE 83

Finite Induction

By Inversion on the Input

Infinite Induction

By Inversion on the Input

Γ, x : Nat ⊢ Φ(x)

SLIDE 91

SLIDE 111

Infinite Coinduction

By Inversion on the Output

SLIDE 112

Infinite Coinduction

By Inversion on the Output

Γ, α ÷ Stream A ⊢ Φ(α)

SLIDE 113

SLIDE 125

A Coinduction Principle

Based on Control Flow

Γ, α ÷ Stream A ⊢ Φ(α) Γ, β ÷ A ⊢ Φ(Head β) Γ, α ÷ Stream A, Φ(α) ⊢ Φ(Tail α)

Bisimulation = (∀s, s′ : Stream A . Φ(s, s′ ) ⇒ Head s = Head s′ : A) ⇒ (∀s, s′ : Stream A . Φ(s, s′ ) ⇒ Φ(Tail s, Tail s′ )) ⇒ (∀s, s′ : Stream A . Φ(s, s′ ) ⇒ s = s′ : Stream A)

⇓

SLIDE 126

Proof by Coinduction

SLIDE 127

Proof by Coinduction

repeat x = x, x, x… alt = 0,1,0,1… evens (x0, x1, x2…) = x0, x2, x4…

SLIDE 128

Proof by Coinduction

Theorem: evens alt = repeat 0 : Stream A repeat x = x, x, x… alt = 0,1,0,1… evens (x0, x1, x2…) = x0, x2, x4…

SLIDE 129

Proof by Coinduction

Theorem: evens alt = repeat 0 : Stream A

S.T.S: α ÷ Stream A ⊢ ⟨evens alt|

|α⟩ = ⟨repeat 0| |α⟩

repeat x = x, x, x… alt = 0,1,0,1… evens (x0, x1, x2…) = x0, x2, x4…

SLIDE 130

Proof by Coinduction

Proof: By coinduction on …

α ÷ Stream A

Theorem: evens alt = repeat 0 : Stream A

S.T.S: α ÷ Stream A ⊢ ⟨evens alt|

|α⟩ = ⟨repeat 0| |α⟩

repeat x = x, x, x… alt = 0,1,0,1… evens (x0, x1, x2…) = x0, x2, x4…

SLIDE 131

Proof by Coinduction

Proof: By coinduction on …

α ÷ Stream A

:

α = Head β ⟨evens alt| |Head β⟩ = ⟨0| |β⟩ = ⟨repeat 0| |Head β⟩

Theorem: evens alt = repeat 0 : Stream A

S.T.S: α ÷ Stream A ⊢ ⟨evens alt|

|α⟩ = ⟨repeat 0| |α⟩

repeat x = x, x, x… alt = 0,1,0,1… evens (x0, x1, x2…) = x0, x2, x4…

SLIDE 132

Proof by Coinduction

Proof: By coinduction on …

α ÷ Stream A

:

α = Head β ⟨evens alt| |Head β⟩ = ⟨0| |β⟩ = ⟨repeat 0| |Head β⟩

: Assume CoIH

and show …

α = Tail β ⟨evens alt| |β⟩ = ⟨repeat 0| |β⟩ ⟨evens alt| |Tail β⟩ = ⟨repeat 0| |Tail β⟩

Theorem: evens alt = repeat 0 : Stream A

S.T.S: α ÷ Stream A ⊢ ⟨evens alt|

|α⟩ = ⟨repeat 0| |α⟩

repeat x = x, x, x… alt = 0,1,0,1… evens (x0, x1, x2…) = x0, x2, x4…

SLIDE 133

Proof by Coinduction

Proof: By coinduction on …

α ÷ Stream A

:

α = Head β ⟨evens alt| |Head β⟩ = ⟨0| |β⟩ = ⟨repeat 0| |Head β⟩

: Assume CoIH

and show …

α = Tail β ⟨evens alt| |β⟩ = ⟨repeat 0| |β⟩ ⟨evens alt| |Tail β⟩ = ⟨repeat 0| |Tail β⟩ ⟨evens alt| |Tail β⟩ = ⟨evens (Tail(Tail alt))| |β⟩ (def . evens) = ⟨evens alt| |β⟩ (def . alt) = ⟨repeat 0| |β⟩ (CoIH) = ⟨repeat 0| |Tail β⟩ (def . repeat)

Theorem: evens alt = repeat 0 : Stream A

S.T.S: α ÷ Stream A ⊢ ⟨evens alt|

|α⟩ = ⟨repeat 0| |α⟩

repeat x = x, x, x… alt = 0,1,0,1… evens (x0, x1, x2…) = x0, x2, x4…

SLIDE 134

Weak vs Strong (Co)Induction

And Effectful Computation

SLIDE 135

Weak vs Strong (Co)Induction

Strong (co)induction proves any property Φ

And Effectful Computation

SLIDE 136

Weak vs Strong (Co)Induction

Strong (co)induction proves any property Φ
Strong induction is unsound in CBN

And Effectful Computation

SLIDE 137

Weak vs Strong (Co)Induction

Strong (co)induction proves any property Φ
Strong induction is unsound in CBN
Strong coinduction is unsound in CBV

And Effectful Computation

SLIDE 138

Weak vs Strong (Co)Induction

Strong (co)induction proves any property Φ
Strong induction is unsound in CBN
Strong coinduction is unsound in CBV
Weak (co)induction restricts Φ

And Effectful Computation

SLIDE 139

Weak vs Strong (Co)Induction

Strong (co)induction proves any property Φ
Strong induction is unsound in CBN
Strong coinduction is unsound in CBV
Weak (co)induction restricts Φ
Weak induction on : must be strict on like

x x ⟨x| |E⟩ = ⟨x| |E′ ⟩

And Effectful Computation

SLIDE 140

Weak vs Strong (Co)Induction

Strong (co)induction proves any property Φ
Strong induction is unsound in CBN
Strong coinduction is unsound in CBV
Weak (co)induction restricts Φ
Weak induction on : must be strict on like

x x ⟨x| |E⟩ = ⟨x| |E′ ⟩

Weak induction on : must be productive on like

α α ⟨V| |α⟩ = ⟨V′ | |α⟩

And Effectful Computation

SLIDE 141

Weak vs Strong (Co)Induction

Strong (co)induction proves any property Φ
Strong induction is unsound in CBN
Strong coinduction is unsound in CBV
Weak (co)induction restricts Φ
Weak induction on : must be strict on like

x x ⟨x| |E⟩ = ⟨x| |E′ ⟩

Weak induction on : must be productive on like

α α ⟨V| |α⟩ = ⟨V′ | |α⟩

Weak (co)induction is always sound

And Effectful Computation

SLIDE 142

Lessons Learned

SLIDE 143

Lessons Learned

Duality — Ideas for free!

SLIDE 144

Lessons Learned

Duality — Ideas for free!
Impact of evaluation, computation, effects, divergence
CBV: strong induction and efficient corecursion
CBN: strong coinduction and efficient recursion
Future work: Call-by-push-value or polarities could get best of both worlds

SLIDE 145

Lessons Learned

Duality — Ideas for free!
Impact of evaluation, computation, effects, divergence
CBV: strong induction and efficient corecursion
CBN: strong coinduction and efficient recursion
Future work: Call-by-push-value or polarities could get best of both worlds
(Co)Induction are both inversion principles
Induction: inversion on input, guided by information flow
Coinduction: inversion on output, guided by control flow