[PPT] - The Duality of Construction Paul Downen Zena M. Ariola University PowerPoint Presentation

SLIDE 1

The Duality of Construction

Paul Downen Zena M. Ariola

University of Oregon

April 9, 2014

SLIDE 2

The sequent calculus

SLIDE 3

Sequent calculus vs. Natural deduction

◮ Natural deduction tells us about pure functional

programming

◮ Sequent calculus tells us about programming with

duality

◮ Flow of Information: producers are dual to consumers ◮ Evaluation: Call-by-value is dual to call-by-name ◮ Construction: data structures are dual to co-data (abstract

bjects with procedural interface)

SLIDE 4

Previous Work

◮ Curien and Herbelin (2000) ◮ Wadler (2003, 2005) ◮ Zeilberger (2008, 2009) ◮ Munch-Maccagnoni (2009) and Curien (2010)

SLIDE 5

Sequent calculus: a symmetric language

Commands c v| |e Producers v Consumers e (contexts) Function abstraction λx.v Function call (call stack) v · e Input variable x Output variable (co-variable) α Output abstraction µα.c Input abstraction (let binding) ˜ µx.c . . . . . .

SLIDE 6

Flow of information v| |˜ µx.c c {v/x}

flow of information productive info

SLIDE 7

Flow of information c {e/α} µα.c| |e

flow of information consumptive info

SLIDE 8

Not so fast. . .

SLIDE 9

Fundamental dilemma of classical computation

µα.c1| |˜ µx.c2 c1 {(˜ µx.c2)/α} c2 {(µα.c1)/x}

SLIDE 10

Fundamental dilemma of classical computation

µ .c1| |˜ µ .c2 c1 c2

SLIDE 11

Impact of strategy on substitution

◮ Call-by-value: Refined notion of “value”

◮ Subset of producers (terms) ◮ Variables stand in for values

◮ Call-by-name: Refined notion of “strict context”

(“co-value”)

◮ Subset of consumers (co-terms) ◮ Co-variables stand in for co-values

SLIDE 12

Parameterizing by the strategy

◮ Single calculus uses unspecified “values” and

“co-values”

◮ Only substitute (co-)values for (co-)variables ◮ Strategy = definition of (co-)values ◮ Impact of strategy isolated to the two

parameterized rules for substitution

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Parameterizing by the strategy (µE) µα.c| |E = c {E/α} (˜ µV) V | |˜ µx.c = c {V /x} (ηµ) µα.v| |α = v (η˜

µ)

˜ µx.x| |e = e

SLIDE 14

Some strategies (and their dual)

Call-by-value:

◮ Variables are values ◮ Every consumer is a co-value

is dual to. . . Call-by-name:

◮ Every producer is a value ◮ Co-variables are co-values

SLIDE 15

Some strategies (and their dual)

Call-by-value: V ∈ Value ::= x E ∈ CoValue ::= e is dual to. . . Call-by-name: V ∈ Value ::= v E ∈ CoValue ::= α

SLIDE 16

Some strategies (and their dual)

Lazy call-by-value (aka call-by-need):

◮ Values same as call-by-value ◮ Co-values may contain delayed let-bindings

is dual to. . . Lazy call-by-name:

◮ Values may contain delayed co-let-bindings (callcc) ◮ Co-values same as call-by-name

SLIDE 17

Some strategies (and their dual)

Lazy call-by-value (aka call-by-need): V ∈ Value ::= x E ∈ CoValue ::= α | | ˜ µx.v| |˜ µy.x| |E is dual to. . . Lazy call-by-name: V ∈ Value ::= x | | µα.µβ.V | |α| |e E ∈ CoValue ::= α

SLIDE 18

Two dual approaches to

rganize information

SLIDE 19

Data types

◮ Defined by rules of creation (constructors) ◮ Producer: fixed shapes given by constructors ◮ Consumer: case analysis on constructions ◮ Like ADTs in ML and Haskell

SLIDE 20

Declaring sums as data (G)ADT: data Either a b where Left :: a → Either a b Right :: b → Either a b Sequent: data a ⊕ b where Left : a ⊢ a ⊕ b| Right : b ⊢ a ⊕ b|

SLIDE 21

Declaring sums as data

◮ Producer: two constructors (left or right)

Left(v1) Right(v2)

◮ Consumer: consider shape of input

◮ “If I’m given Left, do this” ◮ “If I’m given Right, do that”

˜ µ[Left(x).c1| Right(y).c2]

SLIDE 22

Co-data types

◮ Defined by rules of observation (messages) ◮ Consumer: fixed shapes given by observations ◮ Producer: case analysis on messages ◮ Like interfaces for abstract objects

SLIDE 23

Declaring products as co-data codata a & b where First : |a & b ⊢ a Second : |a & b ⊢ b

SLIDE 24

Declaring products as co-data

◮ Consumer: two observations (first or second)

First[e1] Second[e2]

◮ Producer: consider shape of output

◮ “If I’m asked for first, do this” ◮ “If I’m asked for second, do that”

µ(First[α].c1| Second[β].c2)

SLIDE 25

Declaring functions as co-data codata a → b where Call : a|a → b ⊢ b

SLIDE 26

Declaring functions as co-data

◮ Consumer: one observation (function call)

◮ Argument ◮ What to do with result

Call[v, e]

◮ Producer: consider shape of output

◮ Function pops argument off call-stack

µ(Call[x, α].c) = λx.µα.c

SLIDE 27

Evaluating data and co-data

◮ Two fundamental principles of data and co-data:

◮ β: Case analysis breaks apart structure ◮ η: Forwarding is unobservable

◮ Does not perform substitution

◮ And therefore does not reference strategy ◮ Hold in the presence of effects (control, non-termination)

SLIDE 28

SLIDE 29

Evaluating sums as data

(β)

Left(v)
˜

µ[ Left(x). c1 | Right(y). c2]

= v|

|˜ µx.c1 (β)

Right(v)
˜

µ[ Left(x). c1 | Right(y). c2]

= v|

|˜ µy.c2 (η) ˜ µ[ Left(x). Left(x)| |γ | Right(y). Right(y)| |γ] = γ

SLIDE 30

Evaluating products as co-data

(β)

µ( First[α].

c1 | Second[β]. c2)

First[e]
= µα.c1|

|e (β)

µ( First[α].

c1 | Second[β]. c2)

Second[e]
= µβ.c2|

SLIDE 31

General characterization of data and co-data

◮ Constructors dual to messages, case abstractions dual to

abstract objects

◮ All basic connectives of linear/polarized logic fit into same

general pattern

◮ The ordinary: →, ⊗, ⊕, &, . . . ◮ The exotic: `, ¬, . . .

◮ All other behavior derived from β, η, and substitution:

◮ Usual call-by-name and call-by-value λ-calculus β and η rules ◮ Wadler’s (2003) ς rules for lifting components out of structures

SLIDE 32

Summary

◮ Single theory of the sequent calculus

parameterized by various strategies

◮ User-defined data and co-data defined by β and η

independent of strategy

◮ Illustrate call-by-name, call-by-value, and lazy

versions of both

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Summary

◮ Generalize known dualities of computation

◮ General duality between various strategies ◮ General duality between data and co-data types

◮ Two or more strategies in the same program

◮ Use kinds to denote strategies ◮ Well-kindedness preserves consistency ◮ Extends the polarized view of evaluation strategy

SLIDE 34

Questions?

Answers!

SLIDE 35

Interleaving multiple strategies

SLIDE 36

Conflicts between strategies µα.c1| |˜ µx.c2 µα.c1 ˜ µx.c2 CBV non-value co-value CBN value non-co-value

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Conflicts between strategies µα.c1| |˜ µx.c2 µα.c1 ˜ µx.c2 CBV non-value co-value CBN value non-co-value OK

SLIDE 38

Conflicts between strategies µα.c1| |˜ µx.c2 µα.c1 ˜ µx.c2 CBV non-value co-value CBN value non-co-value OK

SLIDE 39

Conflicts between strategies µα.c1| |˜ µx.c2 µα.c1 ˜ µx.c2 CBV non-value co-value CBN value non-co-value non-deterministic

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Conflicts between strategies µα.c1| |˜ µx.c2 µα.c1 ˜ µx.c2 CBV non-value co-value CBN value non-co-value stuck

SLIDE 41

Well-kindedness preserves consistency

Γ ⊢ v :: S|∆ Γ|e :: S ⊢ ∆ v| |e : Γ ⊢ ∆ Cut

◮ CBV |

|CBV : well-kinded, call-by-value command

◮ CBN|

|CBN: well-kinded, call-by-name command

◮ CBV |

|CBN: ill-kinded, non-deterministic command

◮ CBN|

|CBV : ill-kinded, stuck command

SLIDE 42

The polarized regime . . . as an instance of the general theory:

◮ Only two kinds (therefore only two strategies)

◮ Positive: call-by-value ◮ Negative: call-by-name

◮ Pick strategy of (co-)data types to maximize η

◮ Positive: data ◮ Negative: co-data

SLIDE 43

Annotating variables

Γ, x :: S ⊢ xS :: S|∆ Var Γ|αS :: S ⊢ α :: S, ∆ CoVar c : (Γ ⊢ α :: S, ∆) Γ ⊢ µαS.c :: S|∆ Act c : (Γ, x :: S ⊢ ∆) Γ|˜ µxS.c :: S ⊢ ∆ CoAct

SLIDE 44

The problem with annotating commands

◮ Annotating commands (cuts) with a strategy:

◮ v|

|eV: call-by-value

◮ v|

|eN : call-by-name

◮ Loss of determinism

µ