Rank Polymorphism Viewed as a Constraint Problem Justin Slepak - - PowerPoint PPT Presentation

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Rank Polymorphism Viewed as a Constraint Problem Justin Slepak - - PowerPoint PPT Presentation

Mar 23, 2023 120 likes •1.61k views

Rank Polymorphism Viewed as a Constraint Problem Justin Slepak Panagiotis Manolios Olin Shivers jrslepak@ccs.neu.edu pete@ccs.neu.edu shivers@ccs.neu.edu Northeastern University 1 100 175 0.6 ( [(lo 0) (hi 0) ( 0)] (+ (* hi )

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Rank Polymorphism Viewed as a Constraint Problem

Justin Slepak Panagiotis Manolios Olin Shivers

jrslepak@ccs.neu.edu pete@ccs.neu.edu shivers@ccs.neu.edu

Northeastern University

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100 175 0.6 (λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α))))

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100 175 0.6 ((λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α)))) 100 175 0.6)

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RGB RGB

(λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α))))

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RGB RGB

((λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α)))) rgb1 ; 3 channels rgb2 ; 3 channels 0.6) ; scalar

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Credit: Wikimedia user Sadalsuud

(λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α))))

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Credit: Wikimedia user Sadalsuud

((λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α)))) sky ; row × col × chan labels ; row × col × chan img-mask) ; row × col × chan

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(λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α))))

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((λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α)))) film ; time × row × col × chan audience ; time × row × col × chan vid-mask) ; time × row × col × chan

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Credit: Wikimedia user Thetawave

(λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α))))

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Credit: Wikimedia user Thetawave

((λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α)))) scene1 ; time × row × col × chan scene2 ; time × row × col × chan [0.0 ... 1.0]) ; time

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(λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α))))

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(λ [(lo 0) (hi 0) (α 0)] (+ (* hi α) (* lo (- 1 α)))) Polymorphic in dimensionality

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Rank Polymorphic Programming Model

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1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1

3,7

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1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1

3,7

0 1 2 3 4 5 6 7

2,2,2

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1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1

3,7

0 1 2 3 4 5 6 7

2,2,2

  • 17
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1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1

3,7

0 1 2 3 4 5 6 7

2,2,2

  • atoms: non-aggregate elements

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1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1

3,7

0 1 2 3 4 5 6 7

2,2,2

  • atoms: non-aggregate elements

shape: size in each dimension

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1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1

3,7

2 0 1 2 3 4 5 6 7

2,2,2

3

atoms: non-aggregate elements shape: size in each dimension rank: length of shape i.e., number of dimensions

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dot 1 4 2 3

2,2

3 5

2

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dot 1 4 2 3

2,2

3 5

2

cell: basic unit function operates on

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dot 1 4 2 3

2,2

3 5

2

cell: basic unit function operates on

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dot 1 4 2 3

2,2

3 5

2

cell: basic unit function operates on frame: structure around cells

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dot 1 4 2 3

2,2

3 5

2

cell: basic unit function operates on frame: structure around cells function applied to each cell

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dot 1 4 2 3

2,2

3 5

2

cell: basic unit function operates on frame: structure around cells function applied to each cell results reassembled in frame

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Rank n array can split n+1 ways

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Rank n array can split n+1 ways 0 1 2 3 1 2 3 4 2 3 4 5

3,4

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Rank n array can split n+1 ways 0 1 2 3 1 2 3 4 2 3 4 5

3,4

0 1 2 3 1 2 3 4 2 3 4 5

3,4

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Rank n array can split n+1 ways 0 1 2 3 1 2 3 4 2 3 4 5

3,4

0 1 2 3 1 2 3 4 2 3 4 5

3,4

0 1 2 3 1 2 3 4 2 3 4 5

3,4

3

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dot 1 4 2 3

2,2

3 5

2

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dot 1 4 2 3

2,2

3 5

2

+ 1 4 2 3

2,2

3 5

2

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dot 1 4 2 3

2,2

3 5

2

+ 1 4 2 3

2,2

3 5

2

poly 1 4 2 3

2,2

3 5

2

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dot 1 4 2 3

2,2

3 5

2

+ 1 4 2 3

2,2

3 5

2

poly 1 4 2 3

2,2

3 5

2

Principal frame: maximum under prefix ordering

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dot 1 4 2 3

2,2

3 5

2

= 23 21

2

+ 1 4 2 3

2,2

3 5

2

= 4 7 7 8

2,2

poly 1 4 2 3

2,2

3 5

2

= 13 17

2

Principal frame: maximum under prefix ordering

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Trying different shapes

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5

2

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5

2

+ 1 4 2 3 7 8

3,2

3 5

2

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5

2

+ 1 4 2 3 7 8

3,2

3 5

2

poly 1 4 2 3 7 8

3,2

3 5

2

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5

2

= 23 21 61

3

+ 1 4 2 3 7 8

3,2

3 5

2

poly 1 4 2 3 7 8

3,2

3 5

2

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5

2

= 23 21 61

3

+ 1 4 2 3 7 8

3,2

3 5

2

→ Shape error: no principal frame poly 1 4 2 3 7 8

3,2

3 5

2

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5

2

= 23 21 61

3

+ 1 4 2 3 7 8

3,2

3 5

2

→ Shape error: no principal frame poly 1 4 2 3 7 8

3,2

3 5

2

→ Shape error: no principal frame

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Trying different shapes

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5 1

3

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5 1

3

+ 1 4 2 3 7 8

3,2

3 5 1

3

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5 1

3

+ 1 4 2 3 7 8

3,2

3 5 1

3

poly 1 4 2 3 7 8

3,2

3 5 1

3

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5 1

3

→ Domain error: dot requires equal length args + 1 4 2 3 7 8

3,2

3 5 1

3

poly 1 4 2 3 7 8

3,2

3 5 1

3

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5 1

3

→ Domain error: dot requires equal length args + 1 4 2 3 7 8

3,2

3 5 1

3

= 4 7 7 8 8 9

3,2

poly 1 4 2 3 7 8

3,2

3 5 1

3

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Trying different shapes dot 1 4 2 3 7 8

3,2

3 5 1

3

→ Domain error: dot requires equal length args + 1 4 2 3 7 8

3,2

3 5 1

3

= 4 7 7 8 8 9

3,2

poly 1 4 2 3 7 8

3,2

3 5 1

3

→ 13 17 15

3

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Typing Rank Polymorphic Code

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1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1

3,7

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[[1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 0 1 1 1]]

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[[1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 0 1 1 1]] : (Arr (Shp 3 7) Int)

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[[1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 0 1 1 1]] : (Arr (Shp 3 7) Int) 0 1 2 3 4 5 6 7

2,2,2

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[[1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 0 1 1 1]] : (Arr (Shp 3 7) Int) [[[0 1] [2 3]] [[4 5] [6 7]]]

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[[1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 0 1 1 1]] : (Arr (Shp 3 7) Int) [[[0 1] [2 3]] [[4 5] [6 7]]] : (Arr (Shp 2 2 2) Int)

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[[1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 0 1 1 1]] : (Arr (Shp 3 7) Int) [[[0 1] [2 3]] [[4 5] [6 7]]] : (Arr (Shp 2 2 2) Int)

  • 57
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[[1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 0 1 1 1]] : (Arr (Shp 3 7) Int) [[[0 1] [2 3]] [[4 5] [6 7]]] : (Arr (Shp 2 2 2) Int)

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[[1 0 0 1 1 0 1] [0 1 0 1 0 1 1] [0 0 1 0 1 1 1]] : (Arr (Shp 3 7) Int) [[[0 1] [2 3]] [[4 5] [6 7]]] : (Arr (Shp 2 2 2) Int) 0 : (Arr (Shp) Int)

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+ dot poly

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+ : (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int)) dot poly

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+ : (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int)) dot : (Π ((l Dim)) (-> ((Arr (Shp l) Int) (Arr (Shp l) Int)) (Arr (Shp) Int))) poly

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+ : (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int)) dot : (Π ((l Dim)) (-> ((Arr (Shp l) Int) (Arr (Shp l) Int)) (Arr (Shp) Int))) poly : (Π ((l Dim)) (-> ((Arr (Shp l) Int) (Arr (Shp) Int)) (Arr (Shp) Int)))

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+ : (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int)) dot : (Π ((l Dim)) (-> ((Arr (Shp l) Int) (Arr (Shp l) Int)) (Arr (Shp) Int))) poly : (Π ((l Dim)) (-> ((Arr (Shp l) Int) (Arr (Shp) Int)) (Arr (Shp) Int))) append

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+ : (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int)) dot : (Π ((l Dim)) (-> ((Arr (Shp l) Int) (Arr (Shp l) Int)) (Arr (Shp) Int))) poly : (Π ((l Dim)) (-> ((Arr (Shp l) Int) (Arr (Shp) Int)) (Arr (Shp) Int))) append : (∀ ((T Atom)) (Π ((l1 Dim) (l2 Dim) (c Shp)) (-> ((Arr (++ (Shp l1) c) T) (Arr (++ (Shp l2) c) T)) (Arr (++ (Shp (+ l1 l2)) c) T))))

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(+ [[1 4] [2 3] [7 8]] [3 5 1])

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(+ [[1 4] [2 3] [7 8]] [3 5 1]) + : (Arr (Shp) (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int)))

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(+ [[1 4] [2 3] [7 8]] [3 5 1]) + : (Arr (Shp) (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int))) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int)

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(+ [[1 4] [2 3] [7 8]] [3 5 1]) + : (Arr (Shp) (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int))) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int) [3 5 1] : (Arr (Shp 3) Int)

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(+ [[1 4] [2 3] [7 8]] [3 5 1]) + : (Arr (Shp) (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int))) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int) [3 5 1] : (Arr (Shp 3) Int) (Shp) ⊑ (Shp 3) ⊑ (Shp 3 2)

7

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(+ [[1 4] [2 3] [7 8]] [3 5 1]) + : (Arr (Shp) (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int))) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int) [3 5 1] : (Arr (Shp 3) Int) (Shp) ⊑ (Shp 3) ⊑ (Shp 3 2) (Arr (Shp 3 2) Int)

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(+ [[1 4] [2 3] [7 8]] [3 5 1]) [[+ +] [+ +] [+ +]] : (Arr (Shp 2 3) (-> ((Arr (Shp) Int) (Arr (Shp) Int)) (Arr (Shp) Int))) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int) [[3 3] [5 5] [1 1]] : (Arr (Shp 3 2) Int) (Shp) ⊑ (Shp 3) ⊑ (Shp 3 2) (Arr (Shp 3 2) Int)

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(+ [[1 4] [2 3] [7 8]] [3 5])

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(+ [[1 4] [2 3] [7 8]] [3 5]) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int) [3 5] : (Arr (Shp 2) Int)

74

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(+ [[1 4] [2 3] [7 8]] [3 5]) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int) [3 5] : (Arr (Shp 2) Int) (Shp 2) ⊑ / (Shp 3 2)

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(+ [[1 4] [2 3] [7 8]] [3 5]) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int) [3 5] : (Arr (Shp 2) Int) (Shp 2) ⊑ / (Shp 3 2) (Shp 3 2) ⊑ / (Shp 2)

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(+ [[1 4] [2 3] [7 8]] [3 5]) [[1 4] [2 3] [7 8]] : (Arr (Shp 3 2) Int) [3 5] : (Arr (Shp 2) Int) (Shp 2) ⊑ / (Shp 3 2) (Shp 3 2) ⊑ / (Shp 2)

Ill-typed

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Constraints for Rank Polymorphism

78

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First-order logic

79

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First-order logic

8

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First-order logic Free monoid over

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First-order logic Free monoid over

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First-order logic Free monoid over Free monoid over

associativity identity

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First-order logic Free monoid over Free monoid over

associativity identity nothing equal unless required

84

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First-order logic Free monoid over Free monoid over

associativity identity nothing equal unless required

85

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SLIDE 86

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SLIDE 87

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SLIDE 88

88

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SLIDE 89

Redundant but will be useful later

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Type checking:

Validate index arguments supplied by programmer

9

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Type checking:

Validate index arguments supplied by programmer Cell types?

91

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Type checking:

Validate index arguments supplied by programmer Cell types?

92

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Type checking:

Validate index arguments supplied by programmer Cell types? Principal frame?

93

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Type checking:

Validate index arguments supplied by programmer Cell types? Principal frame?

94

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SLIDE 95

Naïve solution

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Naïve solution

  • quantifiers for bound index vars

96

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Naïve solution

  • quantifiers for bound index vars
  • quantifier for principal frame

97

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SLIDE 98

Naïve solution

  • quantifiers for bound index vars
  • quantifier for principal frame
  • quantifiers for prefix/suffix completions

98

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SLIDE 99

Naïve solution

  • quantifiers for bound index vars
  • quantifier for principal frame
  • quantifiers for prefix/suffix completions

99

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SLIDE 100

Naïve solution

  • quantifiers for bound index vars
  • quantifier for principal frame
  • quantifiers for prefix/suffix completions

Write in terms of

1

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Naïve solution

  • quantifiers for bound index vars
  • quantifier for principal frame
  • quantifiers for prefix/suffix completions

Write in terms of Decidability issues

11

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Better solution

12

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Better solution

Canonicalize symbolic shapes

13

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SLIDE 104

Better solution

Canonicalize symbolic shapes

14

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Better solution

Canonicalize symbolic shapes

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Better solution

Canonicalize symbolic shapes

16

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SLIDE 107

In universal fragment, easy equality/prefix/suffix — including witnesses

17

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In universal fragment, easy equality/prefix/suffix — including witnesses

18

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In universal fragment, easy equality/prefix/suffix — including witnesses

19

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In universal fragment, easy equality/prefix/suffix — including witnesses Fold argument frame shapes (suffix witnesses) together

11

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In universal fragment, easy equality/prefix/suffix — including witnesses Fold argument frame shapes (suffix witnesses) together

111

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SLIDE 112

In universal fragment, easy equality/prefix/suffix — including witnesses Fold argument frame shapes (suffix witnesses) together

112

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SLIDE 113

In universal fragment, easy equality/prefix/suffix — including witnesses Fold argument frame shapes (suffix witnesses) together Type checking works entirely within universal fragment

113

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Type inference:

Identify index arguments elided by programmer

114

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Type inference:

Identify index arguments elided by programmer Function type (Pi (e ...) (-> ((Arr ι σ) ...) τ))

115

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Type inference:

Identify index arguments elided by programmer Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

116

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SLIDE 117

Type inference:

Identify index arguments elided by programmer Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

  • quantifiers for bound index vars

117

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SLIDE 118

Type inference:

Identify index arguments elided by programmer Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

  • quantifiers for bound index vars
  • quantifiers for frames

118

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SLIDE 119

Type inference:

Identify index arguments elided by programmer Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

  • quantifiers for bound index vars
  • quantifiers for frames
  • quantifiers for index arguments

119

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SLIDE 120

Type inference:

Identify index arguments elided by programmer Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

  • quantifiers for bound index vars
  • quantifiers for frames (eliminated before)
  • quantifiers for index arguments

12

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SLIDE 121

Type inference:

Identify index arguments elided by programmer Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

  • quantifiers for bound index vars
  • quantifiers for frames (eliminated before)
  • quantifiers for index arguments (solving for these)

121

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SLIDE 122

Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

122

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SLIDE 123

Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

123

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Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

124

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SLIDE 125

Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ...

125

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SLIDE 126

Function type Argument types (Pi (e ...) (-> ((Arr ι σ) ...) τ)) (Arr κ σ) ... Pretend are additional generators

126

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SLIDE 127

Why can we treat like additional generators?

127

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SLIDE 128

Why can we treat like additional generators? Algorithm for existential fragment (Makanin, 1977)

128

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SLIDE 129

Why can we treat like additional generators? Algorithm for existential fragment (Makanin, 1977) "How might subsequences align with each other?"

129

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SLIDE 130

Why can we treat like additional generators? Algorithm for existential fragment (Makanin, 1977) "How might subsequences align with each other?" Equations about smaller sequences

13

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Why can we treat like additional generators? Algorithm for existential fragment (Makanin, 1977) "How might subsequences align with each other?" Equations about smaller sequences No boundaries inside a generator

131

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SLIDE 132

Why can we treat like additional generators? Algorithm for existential fragment (Makanin, 1977) "How might subsequences align with each other?" Equations about smaller sequences No boundaries inside a generator ... or a universal variable

132

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SLIDE 133

Why can we treat like additional generators? Algorithm for existential fragment (Makanin, 1977) "How might subsequences align with each other?" Equations about smaller sequences No boundaries inside a generator ... or a universal variable Caveat: only works in conjunctive fragment

133

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SLIDE 134

Multiple solutions?

134

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SLIDE 135

Multiple solutions?

(append [[1 2] [3 4]] [[5 6] [7 8]])

135

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SLIDE 136

Multiple solutions?

(append [[1 2] [3 4]] [[5 6] [7 8]])

(Pi ((l1 Dim) (l2 Dim) (c Shape)) (Arr (Shp) (-> ((Arr (++ (Shp l1) c) Int) (Arr (++ (Shp l2) c) Int)) (Arr (++ (Shp (+ l1 l2)) c) Int))))

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SLIDE 137

Multiple solutions?

(append [[1 2] [3 4]] [[5 6] [7 8]])

(Pi ((l1 Dim) (l2 Dim) (c Shape)) (Arr (Shp) (-> ((Arr (++ (Shp l1) c) Int) (Arr (++ (Shp l2) c) Int)) (Arr (++ (Shp (+ l1 l2)) c) Int))))

c = (Shp 2) c = (Shp)

137

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SLIDE 138

Multiple solutions?

(append [[1 2] [3 4]] [[5 6] [7 8]])

(Pi ((l1 Dim) (l2 Dim) (c Shape)) (Arr (Shp) (-> ((Arr (++ (Shp l1) c) Int) (Arr (++ (Shp l2) c) Int)) (Arr (++ (Shp (+ l1 l2)) c) Int))))

c = (Shp 2) Major axis append c = (Shp)

138

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SLIDE 139

Multiple solutions?

(append [[1 2] [3 4]] [[5 6] [7 8]])

(Pi ((l1 Dim) (l2 Dim) (c Shape)) (Arr (Shp) (-> ((Arr (++ (Shp l1) c) Int) (Arr (++ (Shp l2) c) Int)) (Arr (++ (Shp (+ l1 l2)) c) Int))))

c = (Shp 2) Major axis append c = (Shp) Lifted row-wise append

139

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SLIDE 140

Multiple solutions?

(append [[1 2] [3 4]] [[5 6] [7 8]])

(Pi ((l1 Dim) (l2 Dim) (c Shape)) (Arr (Shp) (-> ((Arr (++ (Shp l1) c) Int) (Arr (++ (Shp l2) c) Int)) (Arr (++ (Shp (+ l1 l2)) c) Int))))

c = (Shp 2) Major axis append c = (Shp) Lifted row-wise append Established convention: Operate along major axis

14

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SLIDE 141

Multiple solutions?

(append [[1 2] [3 4]] [[5 6] [7 8]])

(Pi ((l1 Dim) (l2 Dim) (c Shape)) (Arr (Shp) (-> ((Arr (++ (Shp l1) c) Int) (Arr (++ (Shp l2) c) Int)) (Arr (++ (Shp (+ l1 l2)) c) Int))))

c = (Shp 2) Major axis append c = (Shp) Lifted row-wise append Established convention: Operate along major axis Require scalar frame shape

141

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SLIDE 142

Remaining work

142

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SLIDE 143

Remaining work

143

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SLIDE 144

Remaining work

Generating

144

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SLIDE 145

Remaining work

Generating Solving

145

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SLIDE 146

Remaining work

Solving Largely done (not by me — Makanin, 1977; Karhumäki, et al, 2000) ILP modulo theories (Manolios & Papavasileiou, 2013)

146

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SLIDE 147

Remaining work

Generating In progress

147

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SLIDE 148

Rank Polymorphism Viewed as a Constraint Problem

Justin Slepak Panagiotis Manolios Olin Shivers

jrslepak@ccs.neu.edu pete@ccs.neu.edu shivers@ccs.neu.edu

Northeastern University

Questions?

148