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String-diagram semantics for functional languages with data-flow - - PowerPoint PPT Presentation

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String-diagram semantics for functional languages with data-flow Steven Cheung & Dan Ghica & Koko Moruya University of Birmingham 5th Jan 2018 GoI machine [Mackie 95] [Danos & Regnier 99] Geometry of interaction [Girard

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String-diagram semantics for functional languages with data-flow

Steven Cheung & Dan Ghica & Koko Moruya University of Birmingham 5th Jan 2018

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GoI machine

[Mackie ’95] [Danos & Regnier ’99]

  • Geometry of interaction [Girard ’89]
  • Token passing machine
  • a λ-term → a graph (proof-net)
  • evaluation = specific path travelled by the token
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GoI machine

(λx.x + x) (1 + 1)

@ D + λ + C ! ! ! 1 1

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GoI machine

(λx.x + x) (1 + 1)

@ D + λ + C ! ! ! 1 1

Contraction Abstraction Constants box structure Dereliction Binary addition Application

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GoI machine

@ D + λ + C ! ! ! 1 1

? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

R : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

R : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

R : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

1 : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : 1 : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : 1 : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

1 : 1 : ? : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

1 : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : 1 : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

? : 1 : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

1 : 1 : ? : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

2 : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : 2 : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : 2 : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

L : 2 : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

2 : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

2 : 2 : ? : □

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GoI machine

@ D + λ + C ! ! ! 1 1

4 : □

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GoI machine

@ D + λ + C ! ! ! 1 1

R : 4 : □

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GoI machine

@ D + λ + C ! ! ! 1 1

R : 4 : □

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GoI machine

@ D + λ + C ! ! ! 1 1

R : 4 : □

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GoI machine

@ D + λ + C ! ! ! 1 1

4 : □

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GoI machine

@ D + λ + C ! ! ! 1 1

4 : □

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GoI machine

[Mackie ’95] [Danos & Regnier ’99]

  • Fixed graph = constants space
  • Compilation to circuit [Ghica ’07]
  • ✔ call-by-name, ? call-by-value
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Dynamic GoI machine

[Muroya & Ghica ’17]

  • Framework for defining FPL semantics
  • Combine token passing & graph rewriting
  • different interleaving = different strategy
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Dynamic GoI machine

[Muroya & Ghica ’17]

(λx.x + x) (1 + 1)

@ D + λ + C ! ! ! 1 1

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1

(λx.x + x) (1 + 1)

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Dynamic GoI machine

@ D + λ + C ! ! ! 1 1 @ D ! λ + C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2

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Dynamic GoI machine

@ D ! λ + C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2

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Dynamic GoI machine

@ D ! λ + C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2

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Dynamic GoI machine

@ D ! λ + C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2

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Dynamic GoI machine

@ D ! λ + C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2

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Dynamic GoI machine

@ D ! λ + C ! 2

@ ! λ 2 + C

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2

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Dynamic GoI machine

@ ! λ 2 + C

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2

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Dynamic GoI machine

@ ! λ 2 + C

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2]

@ ! λ 2 + C + 2 ! C

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2]

+ 2 ! C

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2]

+ 2 ! C

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2]

+ 2 ! C

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2]

+ 2 ! C

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2]

+ 2 ! C

+ 2 ! C ! 2

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2]

+ 2 ! C ! 2

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2]

+ 2 ! C ! 2

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Dynamic GoI machine

+ 2 ! C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2]

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Dynamic GoI machine

+ 2 ! C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2]

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Dynamic GoI machine

+ 2 ! C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2]

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Dynamic GoI machine

+ 2 ! C ! 2

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2] ↦ 2 + 2 [x ↦ 2]

+ 2 ! ! 2

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2] ↦ 2 + 2 [x ↦ 2]

+ 2 ! ! 2

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2] ↦ 2 + 2 [x ↦ 2]

+ 2 ! ! 2

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SLIDE 86

Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2] ↦ 2 + 2 [x ↦ 2]

+ 2 ! ! 2

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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2] ↦ 2 + 2 [x ↦ 2] ↦ 4 [x ↦ 2]

+ 2 ! ! 2

! 4

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DGoIM & data-flow model

  • GoI graph ~ data-flow graph
  • preserve data-flow by suppressing certain rewrites
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Dynamic GoI machine

(λx.x + x) (1 + 1) ↦ (λx.x + x) 2 ↦ x + x [x ↦ 2] ↦ x + 2 [x ↦ 2]

+ 2 ! C

+ 2 ! C ! 2

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SLIDE 90

DGoIM & data-flow model

+ 2 ¡ Ɔ

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SLIDE 91

DGoIM & data-flow model

+ 2 ¡ Ɔ

? : □

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SLIDE 92

DGoIM & data-flow model

+ 2 ¡ Ɔ

? : ? : □

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SLIDE 93

DGoIM & data-flow model

+ 2 ¡ Ɔ

? : ? : □

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SLIDE 94

DGoIM & data-flow model

+ 2 ¡ Ɔ

? : ? : □

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SLIDE 95

DGoIM & data-flow model

+ 2 ¡ Ɔ

2 : ? : □

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SLIDE 96

DGoIM & data-flow model

+ 2 ¡ Ɔ

2 : ? : □

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SLIDE 97

DGoIM & data-flow model

+ 2 ¡ Ɔ

? : 2 : ? : □

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SLIDE 98

DGoIM & data-flow model

+ 2 ¡ Ɔ

? : 2 : ? : □

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SLIDE 99

DGoIM & data-flow model

+ 2 ¡ Ɔ

? : 2 : ? : □

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SLIDE 100

DGoIM & data-flow model

+ 2 ¡ Ɔ

2 : 2 : ? : □

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SLIDE 101

DGoIM & data-flow model

+ 2 ¡ Ɔ

2 : 2 : ? : □

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SLIDE 102

DGoIM & data-flow model

+ 2 ¡ Ɔ

4 : □

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SLIDE 103

DGoIM & data-flow model

+ 2 ¡ Ɔ

4 : □

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SLIDE 104

DGoIM & data-flow model

  • Natural and intuitive
  • Transitions are local
  • makes implementation more tractable
  • Examples:
  • PL for Machine learning
  • PL for Self-adjusting computation
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Machine learning

Direct mode Training mode

data = {(xi, yi)}

x0

m(x0) m(x) = W × x + b trained parameters W, b

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SLIDE 106

TensorFlow

import tensorflow as tf # Model parameters W = tf.Variable([1], dtype=tf.float32) b = tf.Variable([0], dtype=tf.float32) # Model input and output x = tf.placeholder(tf.float32) linear_model = W * x + b y = tf.placeholder(tf.float32)

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SLIDE 107

TensorFlow

import tensorflow as tf # Model parameters W = tf.Variable([1], dtype=tf.float32) b = tf.Variable([0], dtype=tf.float32) # Model input and output x = tf.placeholder(tf.float32) linear_model = W * x + b y = tf.placeholder(tf.float32) sess = tf.Session() print(sess.run(y, {x:7})) # output 7

Direct mode

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TensorFlow

import tensorflow as tf # Model parameters W = tf.Variable([1], dtype=tf.float32) b = tf.Variable([0], dtype=tf.float32) # Model input and output x = tf.placeholder(tf.float32) linear_model = W * x + b y = tf.placeholder(tf.float32) # loss loss = tf.reduce_sum(tf.square(linear_model - y)) # optimizer

  • ptimizer = tf.train.GradientDescentOptimizer(0.01)

train = optimizer.minimize(loss) # training data x_train = [1, 2, 3, 4] y_train = [0, -1, -2, -3] # training loop init = tf.global_variables_initializer() sess = tf.Session() sess.run(init) # reset values to wrong for i in range(1000): sess.run(train, {x: x_train, y: y_train}) # evaluate training accuracy curr_W, curr_b, curr_loss = sess.run([W, b, loss], {x: x_train, y: y_train}) print("W: %s b: %s loss: %s"%(curr_W, curr_b, curr_loss))

Training mode

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SLIDE 109

DSL for machine learning

  • Goals:
  • Proper functional language for machine learning
  • Support both direct and training mode seamlessly
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SLIDE 110

DSL for machine learning

let loss f p = … in let grad_desc f p loss rate = … in let linear_model x = {1} * x + {0} in let output = linear_model 7 in let f@p = linear_model in let trained_model = f (grad_desc f p loss rate) in …

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SLIDE 111

DSL for machine learning

let loss f p = … in let grad_desc f p loss rate = … in let linear_model x = {1} * x + {0} in let output = linear_model 7 in let f@p = linear_model in let trained_model = f (grad_desc f p loss rate) in …

λ ? D + * ᗡ ᗡ D ¿ ¿ ! ¡ ¡ 1

Parameters

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SLIDE 112

DSL for machine learning

let loss f p = … in let grad_desc f p loss rate = … in let linear_model x = {1} * x + {0} in let output = linear_model 7 in let f@p = linear_model in let trained_model = f (grad_desc f p loss rate) in …

Direct mode

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SLIDE 113

DSL for machine learning

let loss f p = … in let grad_desc f p loss rate = … in let linear_model x = {1} * x + {0} in let output = linear_model 7 in let f@p = linear_model in let trained_model = f (grad_desc f p loss rate) in … let f@p = linear_model in … f ≜ λp. λx. p[0] * x + p[1] p ≜ [1, 0] decoupling

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SLIDE 114

DSL for machine learning

let f@p = linear_model in … f ≜ λp. λx. p[0] * x + p[1] p ≜ [1, 0]

A ! λ ? D + * ᗡ ᗡ D ¿ ¿ ! ¿ ¿ ¡ ¡ 1

! ! D ! [1,0] λ D P2 ! λ ? D + * D D D ? ? ! ? ?

Decoupling

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SLIDE 115

DSL for machine learning

let loss f p = … in let grad_desc f p loss rate = … in let linear_model x = {1} * x + {0} in let output = linear_model 7 in let f@p = linear_model in let trained_model = f (grad_desc f p loss rate) in …

Training mode

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SLIDE 116

Parameters management

Confidence interval Confidence interval

let linear W b x = W * x + b in let lower p = linear (nth 0 p) (nth 1 p) in let upper p = linear (nth 0 p) (nth 2 p) in let ci p = (lower [(nth 0 p);(nth 1 p)], upper [(nth 0 p);(nth 2 p)]) in let newP = optimise_ci ci [1;1;2] … in let newCi = ci newP in …

Without decoupling

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SLIDE 117

Parameters management

let linear W b x = W * x + b in let lower p = linear (nth 0 p) (nth 1 p) in let upper p = linear (nth 0 p) (nth 2 p) in let ci p = (lower [(nth 0 p);(nth 1 p)], upper [(nth 0 p);(nth 2 p)]) in let newP = optimise_ci ci [1;1;2] … in let newCi = ci newP in …

With decoupling Without decoupling Confidence interval Confidence interval

let linear W b x = W * x + b in let a = {1} in let lower = linear a {1} in let upper = linear a {2} in let ci = (lower , upper) in let f@p = ci in let newP = optimise_ci f p … in let newCi = f newP in …

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SLIDE 118

DSL for machine learning

  • Proved soundness (including garbage collection)
  • Looking into equational theory (beta, eta, etc)
  • Implementation:
  • https://cwtsteven.github.io/GoI-TF-Visualiser/
  • as an extension to OCaml (work in progress)
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SLIDE 119

Self-adjusting Computation

[Acar ’05]

  • Concern the input-output relationship in a program
  • Efficient updates
  • using dependency graph and memoization
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SLIDE 120

Self-adjusting Computation

[Acar ’05]

  • Think of spreadsheets

1

x

2

m

+ 1 2

y

3

n

+ 1 5

z

+

let x = 1 in let y = 2 in let m = x + 1 in let n = y + 1 in let z = m + n in set x to 3; stabilise();

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SLIDE 121

Self-adjusting Computation

[Acar ’05]

  • Think of spreadsheets

3

x

2

m

+ 1 2

y

3

n

+ 1 5

z

+

let x = 1 in let y = 2 in let m = x + 1 in let n = y + 1 in let z = m + n in set x to 3; stabilise();

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SLIDE 122

Self-adjusting Computation

[Acar ’05]

  • Think of spreadsheets

3

x

4

m

+ 1 2

y

3

n

+ 1 7

z

+

let x = 1 in let y = 2 in let m = x + 1 in let n = y + 1 in let z = m + n in set x to 3; stabilise();

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SLIDE 123

Self-adjusting Computation

[Acar ’05]

  • Think of spreadsheets

3

x

4

m

+ 1 2

y

3

n

+ 1 7

z

+

let x = 1 in let y = 2 in let m = x + 1 in let n = y + 1 in let z = m + n in set x to 3; stabilise(); Circular dependence

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SLIDE 124

DSL for SAC

  • Allow circular dependence
  • Break down stabilisation into step by step propagation
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SLIDE 125

DSL for SAC

M 1 ! M 1 + ! 2 3

Cell x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y Value Dependency

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SLIDE 126

DSL for SAC

let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y

C M 1 + + ! 2 M 3 ! 3

x y

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SLIDE 127

DSL for SAC

let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y

C M 1 + + ! 2 M 3 ! 3

x y ? : □ ? : □

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SLIDE 128

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y ? : ? : □ ? : ? : □

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SLIDE 129

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y ? : ? : □ ? : ? : □

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SLIDE 130

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y 3 : ? : □ 2 : ? : □

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SLIDE 131

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y ? : 3 : ? : □ ? : 2 : ? : □

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SLIDE 132

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y ? : 3 : ? : □ ? : 2 : ? : □

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SLIDE 133

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y ? : 3 : ? : □ 1 : 2 : ? : □

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SLIDE 134

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y 3 : 3 : ? : □ 3 : □

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SLIDE 135

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y 3 : 3 : ? : □ 3 : □

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SLIDE 136

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y 6 : □ 3 : □

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SLIDE 137

DSL for SAC

C M 1 + + ! 2 M 3 ! 3

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y 6 : □ 3 : □

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SLIDE 138

DSL for SAC

x y

C M 6 + + ! 2 M 3 ! 3

let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y b = true

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SLIDE 139

DSL for SAC

x y let x = {1} in let y = {x + 2} in set x to y + 3; let b = prop in let b = prop in y b = true

C M 6 + + ! 2 M 8 ! 3

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SLIDE 140

Comparison

  • Simulate stabilise()
  • let rec stabilise = if prop then stabilise else ()
  • Synchronous languages
  • Create data-flow equations
  • X(n) = Y(n-1) + Z(n)
  • Lustre: X = pre(Y) + Z
  • Ours: let x = {y} + z
  • prop = explicit clock tick
  • set x to … = alter data-flow equations
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SLIDE 141

Comparison

  • FRP vs Self-adjusting computation
  • Synchronous languages vs Ours
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SLIDE 142

DSL for SAC

  • Proving soundness and other equational laws
  • Implementation:
  • https://cwtsteven.github.io/GoI-SAC-Visualiser/
  • as an extension to OCaml (work in progress)
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SLIDE 143

Conclusion

  • Example PLs:
  • PL for machine learning
  • PL for self-adjusting computation
  • GoI-style semantics
  • natural and intuitive for data flow models