The Virtue of Vicious Circles Baltasar Trancn y Widemann TU Ilmenau - - PowerPoint PPT Presentation

Introduction Technique Applications Conclusion

The Virtue of Vicious Circles

Baltasar Trancón y Widemann

TU Ilmenau

2016-10-25

Trancón y Widemann The Virtue of Vicious Circles 1 / 34

Introduction Technique Applications Conclusion

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

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Introduction Technique Applications Conclusion

History

Background: functional programming & compiler construction Ideas & experiments 2002–2004 PhD Thesis 2005–2006 Few spin-off papers –2008 In limbo since

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Introduction Technique Applications Conclusion

Data Objects

Data structures in memory

cells local primitive data references/pointers (to cells & other pointers)

Arise where terms are not expressive enough

semantic networks programming language intermediate representations self-referential stuff in general

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Introduction Technique Applications Conclusion

Data Objects

Data structures in memory

cells local primitive data references/pointers (to cells & other pointers)

Arise where terms are not expressive enough

semantic networks programming language intermediate representations self-referential stuff in general

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Introduction Technique Applications Conclusion

Deep Equality

Graph structure is not fully abstract Equate objects with equivalent structure

same observation paths not just isomorphism several degrees of freedom in recursive types

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Introduction Technique Applications Conclusion

Deep Equality Illustrated

Non-cyclic freedom: sharing 1 1 2 3 5 3 5 Cyclic freedom: unrolling

1

internal (period) unrolling

2

external (phase) unrolling

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Introduction Technique Applications Conclusion

Deep Equality Illustrated

Non-cyclic freedom: sharing 1 1 2 3 5 Cyclic freedom: unrolling

1

internal (period) unrolling

2

external (phase) unrolling

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Introduction Technique Applications Conclusion

Deep Equality Illustrated

Non-cyclic freedom: sharing 1 1 2 3 5 Cyclic freedom: unrolling

1

internal (period) unrolling

2

external (phase) unrolling

1 2 3 5

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Introduction Technique Applications Conclusion

Deep Equality Illustrated

Non-cyclic freedom: sharing 1 1 2 3 5 Cyclic freedom: unrolling

1

internal (period) unrolling

2

external (phase) unrolling

1 2 3 5 2 3 5

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Introduction Technique Applications Conclusion

Deep Equality Illustrated

Non-cyclic freedom: sharing 1 1 2 3 5 Cyclic freedom: unrolling

1

internal (period) unrolling

2

external (phase) unrolling

1 2 3 5 2

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Introduction Technique Applications Conclusion

Circles

. . . arise naturally

recursive let mutual destructive assignment unification without occurrence check

. . . are often embarassing for semantics & practice

circular references are not part–whole relationships exhaustive (inductive) traversal fails

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Introduction Technique Applications Conclusion

Circles

. . . arise naturally

recursive let mutual destructive assignment unification without occurrence check

. . . are often embarassing for semantics & practice

circular references are not part–whole relationships exhaustive (inductive) traversal fails

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Introduction Technique Applications Conclusion

Illustration

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Introduction Technique Applications Conclusion

Illustration

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Introduction Technique Applications Conclusion

Illustration

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Introduction Technique Applications Conclusion

Illustration

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Introduction Technique Applications Conclusion

Motivating Example #1

data List a = Cons a (List a) | Nil map f (Cons x xs) = Cons (f x) (map f xs) map f Nil = Nil let yoink = Cons 2 (Cons 3 (Cons 5 yoink)) in map (+1) (Cons 1 yoink)

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Introduction Technique Applications Conclusion

Motivating Example #2

1 2 3 5

YOINK!

1

Is there an even number?

YES

example easy

2

Is there a perfect number?

NO

no example hard

3

Are all numbers prime?

NO

counterex. easy

4

Are all numbers Fibonacci?

YES

no counterex. hard No good writing a Haskell program

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Introduction Technique Applications Conclusion

Motivating Example #2

1 2 3 5

YOINK!

1

Is there an even number?

YES

example easy

2

Is there a perfect number?

NO

no example hard

3

Are all numbers prime?

NO

counterex. easy

4

Are all numbers Fibonacci?

YES

no counterex. hard No good writing a Haskell program

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Introduction Technique Applications Conclusion

Motivating Example #2

1 2 3 5

YOINK!

1

Is there an even number?

YES

example easy

2

Is there a perfect number?

NO

no example hard

3

Are all numbers prime?

NO

counterex. easy

4

Are all numbers Fibonacci?

YES

no counterex. hard No good writing a Haskell program

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Introduction Technique Applications Conclusion

Summary

Graph-shaped data are relevant but their graph shape is often not Algorithms should be aware

use graph structure for termination respect deep equality

Trancón y Widemann The Virtue of Vicious Circles 10 / 34

Introduction Technique Applications Conclusion

Summary

Graph-shaped data are relevant but their graph shape is often not Algorithms should be aware

use graph structure for termination respect deep equality

Trancón y Widemann The Virtue of Vicious Circles 10 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

Trancón y Widemann The Virtue of Vicious Circles 10 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

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Introduction Technique Applications Conclusion Pointers Cycles Details

Pointers & Cells

A snapshot of memory is a map of addresses to cells

local bits pointers to (addresses of) cells

The semantics of a pointer (wrt a snapshot) is a pointed map

• bservation by (transitive) dereferencing

F(X) = 2∗ × X∗ α : A → F(A)

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Introduction Technique Applications Conclusion Pointers Cycles Details

Pointers & Cells

A snapshot of memory is a map of addresses to cells

local bits pointers to (addresses of) cells

The semantics of a pointer (wrt a snapshot) is a pointed map

• bservation by (transitive) dereferencing

F(X) = 2∗ × X∗ α : A → F(A)

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Introduction Technique Applications Conclusion Pointers Cycles Details

Pointers & Cells

A snapshot of memory is a map of addresses to cells

local bits pointers to (addresses of) cells

The semantics of a pointer (wrt a snapshot) is a pointed map

• bservation by (transitive) dereferencing

F(X) = 2∗ × X∗ α : A → F(A)

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Introduction Technique Applications Conclusion Pointers Cycles Details

Pointer Coalgebra

Memory states are coalgebras Pointers are pointed coalgebras Deep equality is bisimilarity Hierarchy of models initial finite noncircular data – too small rational finite circular data – about right final infinite data – too large

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Introduction Technique Applications Conclusion Pointers Cycles Details

Pointer Coalgebra

Memory states are coalgebras Pointers are pointed coalgebras Deep equality is bisimilarity Hierarchy of models initial finite noncircular data – too small rational finite circular data – about right final infinite data – too large

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Introduction Technique Applications Conclusion Pointers Cycles Details

Pointer Coalgebra

Memory states are coalgebras Pointers are pointed coalgebras Deep equality is bisimilarity Hierarchy of models initial finite noncircular data – too small rational finite circular data – about right final infinite data – too large

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Introduction Technique Applications Conclusion Pointers Cycles Details

Iteration & Coiteration in a Nutshell

Consider simpler functor P(X) = X + 1 Iteration uses initial model N, [succ, zero] Coiteration uses final model N ∪ {∞}, pred

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Introduction Technique Applications Conclusion Pointers Cycles Details

Iteration & Coiteration in a Nutshell

Consider simpler functor P(X) = X + 1 Iteration uses initial model N, [succ, zero] Coiteration uses final model N ∪ {∞}, pred

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Introduction Technique Applications Conclusion Pointers Cycles Details

Iteration & Coiteration in a Nutshell

Consider simpler functor P(X) = X + 1 Iteration uses initial model N, [succ, zero] Coiteration uses final model N ∪ {∞}, pred

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Introduction Technique Applications Conclusion Pointers Cycles Details

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

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Introduction Technique Applications Conclusion Pointers Cycles Details

Motto

Those who cannot remember the past are condemned to repeat it. (Ruiz de Santayana 1906)

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Introduction Technique Applications Conclusion Pointers Cycles Details

Call Stack

Standard implementation of recursive functions Incarnation = code + (top) stack frame Cycle = identical stacked frames Suffices for cycle detection Handling?

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Introduction Technique Applications Conclusion Pointers Cycles Details

Call Stack

Standard implementation of recursive functions Incarnation = code + (top) stack frame Cycle = identical stacked frames Suffices for cycle detection Handling?

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5 6

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5 6 4

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5 6 4 3

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations I

1 2 3 5 6 4 3 2

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Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations II

1 2 3 5 Destination-passing style (Minamide 1998) Tail recursion modulo cons(tructor) (Warren 1980)

can be used to eliminate tail calls, or alternatively to handle duckling loops!

Lazy evaluation similar; consumer- not producer-driven

Trancón y Widemann The Virtue of Vicious Circles 17 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations II

1 2 3 5 Destination-passing style (Minamide 1998) Tail recursion modulo cons(tructor) (Warren 1980)

can be used to eliminate tail calls, or alternatively to handle duckling loops!

Lazy evaluation similar; consumer- not producer-driven

Trancón y Widemann The Virtue of Vicious Circles 17 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations II

1 2 3 5 2 Destination-passing style (Minamide 1998) Tail recursion modulo cons(tructor) (Warren 1980)

can be used to eliminate tail calls, or alternatively to handle duckling loops!

Lazy evaluation similar; consumer- not producer-driven

Trancón y Widemann The Virtue of Vicious Circles 17 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations II

1 2 3 5 2 3 Destination-passing style (Minamide 1998) Tail recursion modulo cons(tructor) (Warren 1980)

can be used to eliminate tail calls, or alternatively to handle duckling loops!

Lazy evaluation similar; consumer- not producer-driven

Trancón y Widemann The Virtue of Vicious Circles 17 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations II

1 2 3 5 2 3 4 Destination-passing style (Minamide 1998) Tail recursion modulo cons(tructor) (Warren 1980)

can be used to eliminate tail calls, or alternatively to handle duckling loops!

Lazy evaluation similar; consumer- not producer-driven

Trancón y Widemann The Virtue of Vicious Circles 17 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations II

1 2 3 5 2 3 4 6 Destination-passing style (Minamide 1998) Tail recursion modulo cons(tructor) (Warren 1980)

can be used to eliminate tail calls, or alternatively to handle duckling loops!

Lazy evaluation similar; consumer- not producer-driven

Trancón y Widemann The Virtue of Vicious Circles 17 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations II

1 2 3 5 2 3 4 6 Destination-passing style (Minamide 1998) Tail recursion modulo cons(tructor) (Warren 1980)

can be used to eliminate tail calls, or alternatively to handle duckling loops!

Lazy evaluation similar; consumer- not producer-driven

Trancón y Widemann The Virtue of Vicious Circles 17 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Order of Operations II

1 2 3 5 2 3 4 6 Destination-passing style (Minamide 1998) Tail recursion modulo cons(tructor) (Warren 1980)

can be used to eliminate tail calls, or alternatively to handle duckling loops!

Lazy evaluation similar; consumer- not producer-driven

Trancón y Widemann The Virtue of Vicious Circles 17 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Handler #1

Retain stack frames Fix output before recursive call Cycle detected = ⇒ copy outputs Generic algorithm (calling convention)

applied to recursive function definition implements coiteration of body as coalgebra

Proof by bisimulation construction & graph coloring

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Introduction Technique Applications Conclusion Pointers Cycles Details

Handler #1

Retain stack frames Fix output before recursive call Cycle detected = ⇒ copy outputs Generic algorithm (calling convention)

applied to recursive function definition implements coiteration of body as coalgebra

Proof by bisimulation construction & graph coloring

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Introduction Technique Applications Conclusion Pointers Cycles Details

Illustration

1 2 3 5

YOINK!

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Introduction Technique Applications Conclusion Pointers Cycles Details

Illustration

1 2 3 5

YOINK!

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Introduction Technique Applications Conclusion Pointers Cycles Details

Illustration

1 2 3 5

YOINK!

2

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Introduction Technique Applications Conclusion Pointers Cycles Details

Illustration

1 2 3 5

YOINK!

2 3

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Introduction Technique Applications Conclusion Pointers Cycles Details

Illustration

1 2 3 5

YOINK!

2 3 4

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Introduction Technique Applications Conclusion Pointers Cycles Details

Illustration

1 2 3 5

YOINK!

2 3 4 6

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Introduction Technique Applications Conclusion Pointers Cycles Details

Illustration

1 2 3 5

YOINK!

2 3 4 6

YOINK!

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Introduction Technique Applications Conclusion Pointers Cycles Details

Handler #2

Boolean function encoded as positive Hilbert calculus Semantics depend on data no cycles unique fixpoint (induction) cycles complete lattice of fixpoints (Tarski) Cycle detected = ⇒ abort with constant value Generic algorithm

constant false/true select least/greatest fixpoint, resp. change of strategy safe at stratification boundaries

Possibly for other (finite) base lattices?

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Introduction Technique Applications Conclusion Pointers Cycles Details

Handler #2

Boolean function encoded as positive Hilbert calculus Semantics depend on data no cycles unique fixpoint (induction) cycles complete lattice of fixpoints (Tarski) Cycle detected = ⇒ abort with constant value Generic algorithm

constant false/true select least/greatest fixpoint, resp. change of strategy safe at stratification boundaries

Possibly for other (finite) base lattices?

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Introduction Technique Applications Conclusion Pointers Cycles Details

More Handlers?

Similar approach by Kozen’s group (Jeannin, Kozen, and Silva 2013) Pluggable cycle handlers

fixpoints numerical solvers . . .

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Introduction Technique Applications Conclusion Pointers Cycles Details

Cycle Handling vs. Laziness

Handler #1 for free in lazy evaluation but Final semantics conflate cycles & true infinity Handler #2 impossible in lazy evaluation Combination yields many interesting algorithms General(?) recipe:

1

take ordinary recursive algorithm

2

add cycle handler

3

• btain cycle-aware algorithm – or fail

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Introduction Technique Applications Conclusion Pointers Cycles Details

Cycle Handling vs. Laziness

Handler #1 for free in lazy evaluation but Final semantics conflate cycles & true infinity Handler #2 impossible in lazy evaluation Combination yields many interesting algorithms General(?) recipe:

1

take ordinary recursive algorithm

2

add cycle handler

3

• btain cycle-aware algorithm – or fail

Trancón y Widemann The Virtue of Vicious Circles 22 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

Cycle Handling vs. Laziness

Handler #1 for free in lazy evaluation but Final semantics conflate cycles & true infinity Handler #2 impossible in lazy evaluation Combination yields many interesting algorithms General(?) recipe:

1

take ordinary recursive algorithm

2

add cycle handler

3

• btain cycle-aware algorithm – or fail

Trancón y Widemann The Virtue of Vicious Circles 22 / 34

Introduction Technique Applications Conclusion Pointers Cycles Details

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

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Introduction Technique Applications Conclusion Pointers Cycles Details

Efficiency

Cycle detection by stack inspection is costly Save work by invariant every cycle contains a marked pointer Trivial in pure functional programming

mark the YOINK! pointers

For regular functions (homomorphisms) check only upon mark Also for reference-counting garbage collection

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Introduction Technique Applications Conclusion Pointers Cycles Details

Efficiency

Cycle detection by stack inspection is costly Save work by invariant every cycle contains a marked pointer Trivial in pure functional programming

mark the YOINK! pointers

For regular functions (homomorphisms) check only upon mark Also for reference-counting garbage collection

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Introduction Technique Applications Conclusion Pointers Cycles Details

Efficiency

Cycle detection by stack inspection is costly Save work by invariant every cycle contains a marked pointer Trivial in pure functional programming

mark the YOINK! pointers

For regular functions (homomorphisms) check only upon mark Also for reference-counting garbage collection

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Introduction Technique Applications Conclusion Pointers Cycles Details

Implementation

Requires read/write access to call stack

not supported by usual backends (C, JVM, .NET, . . . )

Possible solution: stackless transform

add code to load/unload stack frames to heap objects portable implementation of continuations

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Introduction Technique Applications Conclusion Pointers Cycles Details

Implementation

Requires read/write access to call stack

not supported by usual backends (C, JVM, .NET, . . . )

Possible solution: stackless transform

add code to load/unload stack frames to heap objects portable implementation of continuations

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Addition

Rational numbers as sequences of digits Addition is not naturally coiterative

depends on carry from the infinite right

Consider half-addition instead

sum/carry digitwise (cf. map) shift & repeat

Converges

each digit can overflow at most once

Trancón y Widemann The Virtue of Vicious Circles 25 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Addition

Rational numbers as sequences of digits Addition is not naturally coiterative

depends on carry from the infinite right

Consider half-addition instead

sum/carry digitwise (cf. map) shift & repeat

Converges

each digit can overflow at most once

Trancón y Widemann The Virtue of Vicious Circles 25 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Addition

Rational numbers as sequences of digits Addition is not naturally coiterative

depends on carry from the infinite right

Consider half-addition instead

sum/carry digitwise (cf. map) shift & repeat

Converges

each digit can overflow at most once

Trancón y Widemann The Virtue of Vicious Circles 25 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Division

Long division

compute most significant digit by trial subtraction repeat with remainder

Reverse loop structure

inner iteration

• uter coiteration

Problem: cycle modulo bisimilarity

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Division

Long division

compute most significant digit by trial subtraction repeat with remainder

Reverse loop structure

inner iteration

• uter coiteration

Problem: cycle modulo bisimilarity

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Division

Long division

compute most significant digit by trial subtraction repeat with remainder

Reverse loop structure

inner iteration

• uter coiteration

Problem: cycle modulo bisimilarity

Trancón y Widemann The Virtue of Vicious Circles 26 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Usual suspects on lists

Usual list functions generalize by adding handler #1

append insert/delete (periodically)

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Filter

Filter is not naturally coinductive

can delete infinitely many elements straight away

Two-phase approach: mark & sweep

1

cross out unwanted elements

2

eight all crossed out or iteratively filterable

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Filter

Filter is not naturally coinductive

can delete infinitely many elements straight away

Two-phase approach: mark & sweep

1

cross out unwanted elements

2

eight all crossed out or iteratively filterable

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Quicksort

I G O A L W A Y S (O N)

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Quicksort

I G O A L W A Y S (O N) I G O A L W A Y S (O N) G A A I G O A L W A Y S (O N) I I G O A L W A Y S (O N) O L W Y S (O N)

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Quicksort

I G O A L W A Y S (O N) I G O A L W A Y S (O N) G A A

G A A A A G A A G G A A

I G O A L W A Y S (O N) I I G O A L W A Y S (O N) O L W Y S (O N)

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Quicksort

I G O A L W A Y S (O N) I G O A L W A Y S (O N) G A A

G A A A A G A A G G A A

I G O A L W A Y S (O N) I I G O A L W A Y S (O N) O L W Y S (O N)

O L W Y S (O N) L (N) O L W Y S (O N) O (O) O L W Y S (O N) W Y S

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Quicksort

I G O A L W A Y S (O N) I G O A L W A Y S (O N) G A A

G A A A A G A A G G A A

I G O A L W A Y S (O N) I I G O A L W A Y S (O N) O L W Y S (O N)

O L W Y S (O N) L (N) O L W Y S (O N) O (O) O L W Y S (O N) W Y S

W Y S S W Y S W W Y S Y

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Quicksort

I G O A L W A Y S (O N) I G O A L W A Y S (O N) G A A

G A A A A G A A G G A A

I G O A L W A Y S (O N) I I G O A L W A Y S (O N) O L W Y S (O N)

O L W Y S (O N) L (N) O L W Y S (O N) O (O) O L W Y S (O N) W Y S

W Y S S W Y S W W Y S Y

A A G I L (N)

Trancón y Widemann The Virtue of Vicious Circles 29 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

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Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Polynomial Datatypes

Sums, Products & Recursion Structural subtyping

sum with less variants product with more components transitively closed lifted to functions

Dynamic conversion by access operation mapping

table format recursive types = ⇒ circular table references

Type conversion tables form category

cycle-aware algorithms for subtype check & composition constant-time conversion of structured values

Trancón y Widemann The Virtue of Vicious Circles 30 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Polynomial Datatypes

Sums, Products & Recursion Structural subtyping

sum with less variants product with more components transitively closed lifted to functions

Dynamic conversion by access operation mapping

table format recursive types = ⇒ circular table references

Type conversion tables form category

cycle-aware algorithms for subtype check & composition constant-time conversion of structured values

Trancón y Widemann The Virtue of Vicious Circles 30 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Polynomial Datatypes

Sums, Products & Recursion Structural subtyping

sum with less variants product with more components transitively closed lifted to functions

Dynamic conversion by access operation mapping

table format recursive types = ⇒ circular table references

Type conversion tables form category

cycle-aware algorithms for subtype check & composition constant-time conversion of structured values

Trancón y Widemann The Virtue of Vicious Circles 30 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Example

BinTree[α] ::= Branch2

• BinTree[α], BinTree[α]
• 1

| Leaf(α)

2

23Tree[β] ::= Branch3

• 23Tree[β], 23Tree[β], 23Tree[β]
• 1

| Branch2

• 23Tree[β], 23Tree[β]
• 2

| Leaf(β)

3

c : (α <: β) →

• BinTree[α] <: 23Tree[β]
• c(d) =
• 1 →

2

c(d), c(d)

• 2 →

3 (d)

• Trancón y Widemann

The Virtue of Vicious Circles 31 / 34

Introduction Technique Applications Conclusion Arithmetics Lists Subtyping

Example

BinTree[α] ::= Branch2

• BinTree[α], BinTree[α]
• 1

| Leaf(α)

2

23Tree[β] ::= Branch3

• 23Tree[β], 23Tree[β], 23Tree[β]
• 1

| Branch2

• 23Tree[β], 23Tree[β]
• 2

| Leaf(β)

3

c : (α <: β) →

• BinTree[α] <: 23Tree[β]
• c(d) =
• 1 →

2

c(d), c(d)

• 2 →

3 (d)

• Trancón y Widemann

The Virtue of Vicious Circles 31 / 34

Introduction Technique Applications Conclusion

1

Introduction

2

Technique Pointer Coalgebra Cycle Detection & Handling Technical Details

3

Applications Rational Arithmetics Circular Lists Structural Polynomial Subtyping

4

Conclusion

Trancón y Widemann The Virtue of Vicious Circles 31 / 34

Introduction Technique Applications Conclusion

Take-Home Message

Adequate semantics of immutable data with pointers

rational coalgebra between initial and final model graph modulo deep equality

Applications in knowledge representation & compilers Cycle detection & handling

coiteration search problems

Trancón y Widemann The Virtue of Vicious Circles 32 / 34

Introduction Technique Applications Conclusion

Take-Home Message

Adequate semantics of immutable data with pointers

rational coalgebra between initial and final model graph modulo deep equality

Applications in knowledge representation & compilers Cycle detection & handling

coiteration search problems

Trancón y Widemann The Virtue of Vicious Circles 32 / 34

Introduction Technique Applications Conclusion

Take-Home Message

Adequate semantics of immutable data with pointers

rational coalgebra between initial and final model graph modulo deep equality

Applications in knowledge representation & compilers Cycle detection & handling

coiteration search problems

Trancón y Widemann The Virtue of Vicious Circles 32 / 34

Introduction Technique Applications Conclusion

Outlook

Higher-order functions – from Set to CPO? Programming language syntax? Portable safe implementation Other lattices – abstract interpretation framework?

Trancón y Widemann The Virtue of Vicious Circles 33 / 34

References

Jeannin, Jean-Baptiste, Dexter Kozen, and Alexandra Silva (2013). “Language Constructs for Non-Well-Founded Computation”. In: Programming Languages and Systems (ESOP 2013). Ed. by Matthias Felleisen and Philippa Gardner. Vol. 7792. Lecture Notes in Computer Science. Springer-Verlag,

• pp. 61–80. DOI: ✶✵✳✶✵✵✼✴✾✼✽✲✸✲✻✹✷✲✸✼✵✸✻✲✻❴✹.

Minamide, Yasuhiko (1998). “A functional representation of data structures with a hole”. In: Proceedings of the 25th ACM SIGPLAN-SIGACT symposium on Principles of programming languages (POPL ’98). San Diego, California, United States: ACM, pp. 75–84. ISBN: 0-89791-979-3. DOI: ✶✵✳✶✶✹✺✴✷✻✽✾✹✻✳✷✻✽✾✺✸. Ruiz de Santayana, Jorge Augustín Nicolás (1906). The Life of Reason. Warren, David H. D. (1980). DAI Research Report 141. University of Edinburgh. xkcd (2009). Ducklings. URL: ❤tt♣✿✴✴①❦❝❞✳❝♦♠✴✺✸✼✴. Trancón y Widemann The Virtue of Vicious Circles 35 / 34