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Circuit timing analysis, linear maps, and semantic morphisms Conal - - PowerPoint PPT Presentation

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Circuit timing analysis, linear maps, and semantic morphisms Conal Elliott Tabula IFIP WG 2.8, Nov. 2012 Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 1 / 29 Outline Timing

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Circuit timing analysis, linear maps, and semantic morphisms

Conal Elliott

Tabula

IFIP WG 2.8, Nov. 2012

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 1 / 29

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Outline

Timing analysis Linear transformations Semantics and implementation

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 2 / 29

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Timing analysis

Timing analysis

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 3 / 29

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Timing analysis

Simple timing analysis

Computation with a time delay:

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 4 / 29

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Timing analysis

Trivial timings

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 5 / 29

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Timing analysis

Sequential composition

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 6 / 29

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Timing analysis

Parallel composition

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 7 / 29

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Timing analysis

But ...

Oops: Same circuit (f × g), different timings.

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 8 / 29

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Timing analysis

Multi-path analysis

◮ Max delay for each input/output pair ◮ How do delays compose?

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 9 / 29

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Timing analysis

How do delays compose?

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 10 / 29

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Timing analysis

How do delays compose?

V ⊙ U = W , where Wi,k = Max

j

(Ui,j + Vj,k)

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 11 / 29

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Timing analysis

How do delays compose?

V ⊙ U = W , where Wi,k = Max

j

(Ui,j + Vj,k) Look familiar?

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 11 / 29

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Timing analysis

How do delays compose?

V ⊙ U = W , where Wi,k = Max

j

(Ui,j + Vj,k) Look familiar? Matrix multiplication?

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 11 / 29

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Timing analysis

MaxPlus algebra

type Delay = MaxPlus Double data MaxPlus a = MP a instance Ord a ⇒ AdditiveGroup (MaxPlus a) where MP a ˆ + MP b = MP (a ‘max‘ b) instance (Ord a, Num a) ⇒ VectorSpace (MaxPlus a) where type Scalar (MaxPlus a) = a a · MP b = MP (a + b) Oops – We also need a zero. VectorSpace is overkill. Module over a semi-ring suffices.

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 12 / 29

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Timing analysis

MaxPlus algebra

type Delay = MaxPlus Double data MaxPlus a = −∞ | Fi a instance Ord a ⇒ AdditiveGroup (MaxPlus a) where 0 = −∞ MP a ˆ + MP b = MP (a ‘max‘ b) − ∞ ˆ + = −∞ ˆ + −∞ = −∞ instance (Ord a, Num a) ⇒ VectorSpace (MaxPlus a) where type Scalar (MaxPlus a) = a a · MP b = MP (a + b) · −∞ = −∞

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 13 / 29

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Linear transformations

Linear transformations

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 14 / 29

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Linear transformations

Representation?

How might we represent linear maps/transformations a ⊸ b?

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 15 / 29

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Linear transformations

Representation?

How might we represent linear maps/transformations a ⊸ b?

◮ Matrices ◮ Functions ◮ What else?

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 15 / 29

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Linear transformations

Matrices

   a11 · · · a1m . . . ... . . . an1 · · · anm    Static typing?

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 16 / 29

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Linear transformations

Statically sized matrices

type Mat m n a = Vec m (Vec n a) (◦) :: (IsNat m, IsNat o) ⇒ Mat n o D → Mat m n D → Mat o m D no ◦ mn = crossF dot (transpose no) mn crossF :: (IsNat m, IsNat o) ⇒ (a → b → c) → Vec o a → Vec m b → Mat o m c crossF f as bs = (λa → f a < $ > bs) < $ > as dot :: (Ord a, Num a) ⇒ Vec n a → Vec n a → a u ‘dot‘ v = sum (zipWithV (∗) u v)

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 17 / 29

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Linear transformations

Generalizing

type Mat m n a = m (n a) (◦) :: (Functor m, Applicative n, Traversable n, Applicative o) ⇒ Mat n o D → Mat m n D → Mat o m D no ◦ mn = crossF dot (sequenceA no) mn crossF :: (Functor m, Functor o) ⇒ (a → b → c) → o a → m b → Mat o m c crossF f as bs = (λa → f a < $ > bs) < $ > as dot :: (Foldable n, Applicative n, Ord a, Num a) ⇒ n a → n a → a u ‘dot‘ v = sum (liftA2 (∗) u v)

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 18 / 29

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Linear transformations

Represent via type family (old)

class VectorSpace v ⇒ HasBasis v where type Basis v :: ∗ coord :: v → (Basis v → Scalar v) Linear map as memoized function from basis: newtype a ⊸ b = L (Basis a →M b) See Beautiful differentiation (ICFP 2009).

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 19 / 29

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Linear transformations

Represent as GADT

data a ⊸ b where Dot :: InnerSpace b ⇒ b → (b ⊸ Scalar b) (:△) :: VS3 a c d ⇒

  • - vector spaces with same scalar field

(a ⊸ c) → (a ⊸ d) → (a ⊸ c × d)

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 20 / 29

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Semantics and implementation

Semantics and implementation

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 21 / 29

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Semantics and implementation

Semantics

[ [·] ] :: (a ⊸ b) → (a → b) [ [Dot b] ] = dot b [ [f :△ g] ] = [ [f ] ] △[ [g] ] where, on functions, (f △ g) a = (f a, g a) Recall: data a ⊸ b where Dot :: InnerSpace b ⇒ b → (b ⊸ Scalar b) (:△) :: VS3 a c d ⇒ (a ⊸ c) → (a ⊸ d) → (a ⊸ c × d)

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 22 / 29

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Semantics and implementation

Semantic type class morphisms

Category instance specification: [ [id] ] ≡ id [ [g ◦ f ] ] ≡ [ [g] ] ◦ [ [f ] ] Arrow instance specification: [ [f △ g] ] ≡ [ [f ] ] △[ [g] ] [ [f × g] ] ≡ [ [f ] ] ×[ [g] ] where (△) :: Arrow (❀) ⇒ (a ❀ c) → (a ❀ d) → (a ❀ c × d) (×) :: Arrow (❀) ⇒ (a ❀ c) → (b ❀ d) → (a × b ❀ c × d) The Category and Arrow laws then follow.

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 23 / 29

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Semantics and implementation

Deriving a Category instance

One case: [ [(f :△ g) ◦ h] ] ≡ ([ [f ] ] △[ [g] ]) ◦ [ [h] ] ≡ [ [f ] ] ◦ [ [h] ] △[ [g] ] ◦ [ [h] ] ≡ [ [f ◦ h △ g ◦ h] ] (where f ◦ h △ g ◦ h ≡ (f ◦ h) △(g ◦ h)). Uses: (f △ g) ◦ h ≡ f ◦ h △ g ◦ h Implementation: (f :△ g) ◦ h = f ◦ h :△ g ◦ h

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 24 / 29

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Semantics and implementation

Deriving a Category instance

[ [Dot s ◦ Dot b] ] ≡ dot s ◦ dot b ≡ dot (s · b) ≡ [ [Dot (s · b)] ] [ [Dot (a, b) ◦ (f :△ g)] ] ≡ dot (a, b) ◦ ([ [f ] ] △[ [g] ]) ≡ add ◦ (dot a ◦ [ [f ] ] △ dot b ◦ [ [g] ]) ≡ dot a ◦ [ [f ] ] ˆ + dot b ◦ [ [g] ] ≡ [ [Dot a ◦ f ˆ + Dot b ◦ g] ] Uses: dot (a, b) ≡ add ◦ (dot a × dot b) (k × h) ◦ (f △ g) ≡ k ◦ f △ h ◦ g [ [f ˆ + g] ] ≡ [ [f ] ] ˆ +[ [g] ]

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 25 / 29

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Semantics and implementation

Deriving an Arrow instance

[ [f △ g] ] ≡ [ [f ] ] △[ [g] ] ≡ [ [f :△ g] ] [ [f × g] ] ≡ [ [f ] ] ×[ [g] ] ≡ [ [f ] ] ◦ fst △[ [g] ] ◦ snd ≡ [ [compFst f ] ] △[ [compSnd g] ] ≡ [ [compFst f :△ compSnd g] ] assuming [ [compFst f ] ] ≡ [ [f ] ] ◦ fst [ [compSnd g] ] ≡ [ [g] ] ◦ snd

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 26 / 29

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Semantics and implementation

Composing with fst and snd

compFst :: VS3 a b c ⇒ a ⊸ c → a × b ⊸ c compSnd :: VS3 a b c ⇒ b ⊸ c → a × b ⊸ c Derivation: dot a

  • fst ≡ dot (a, 0)

(f △ g) ◦ fst ≡ f ◦ fst △ g ◦ fst Implementation: compFst (Dot a) = Dot (a, 0) compFst (f :△ g) = compFst f △ compFst g compSnd (Dot b) = Dot (0, b) compSnd (f :△ g) = compSnd f △ compSnd g

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 27 / 29

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Semantics and implementation

Adding linear maps

[ [Dot b ˆ + Dot c] ] ≡ dot b ˆ + dot c ≡ dot (b ˆ + c) ≡ [ [Dot (b ˆ + c)] ] [ [(f :△ g) ˆ +(h :△ k)] ] ≡ ([ [f ] ] △[ [g] ]) ˆ +([ [h] ] △[ [k] ]) ≡ ([ [f ] ] ˆ +[ [h] ]) △([ [g] ] ˆ +[ [k] ]) ≡ [ [(f ˆ + h) △(g ˆ + k)] ] Other cases don’t type-check. Uses (on functions): (f △ g) ˆ +(h △ k) ≡ (f ˆ + h) △(g ˆ + k)

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 28 / 29

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Semantics and implementation

What next?

◮ Fancier timing analysis ◮ What else is linear? ◮ More examples of semantic type class morphisms

Conal Elliott (Tabula) Circuit timing analysis, linear maps, and semantic morphisms IFIP WG 2.8, Nov. 2012 29 / 29