Circuit timing analysis, linear maps, and semantic morphisms · Circuit timing analysis, linear...

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 morphismsIFIP WG 2.8, Nov. 2012 1 / 29

Outline

Timing analysis

Linear transformations

Semantics and implementation

Timing analysis

Simple timing analysis

Computation with a time delay:

Timing analysis

Trivial timings

Timing analysis

Sequential composition

Timing analysis

Parallel composition

Timing analysis

But ...

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

Timing analysis

Multi-path analysis

I Max delay for each input/output pair

I How do delays compose?

Timing analysis

How do delays compose?

Timing analysis

V � U = W , where

Wi ,k = Maxj

(Ui ,j + Vj ,k)

Look familiar? Matrix multiplication?

Timing analysis

V � U = W , where

Wi ,k = Maxj

(Ui ,j + Vj ,k)

Look familiar?

Matrix multiplication?

Timing analysis

V � U = W , where

Wi ,k = Maxj

(Ui ,j + Vj ,k)

Look familiar? Matrix multiplication?

Timing analysis

MaxPlus algebra

type Delay = MaxPlus Double

data MaxPlus a = MP a

instance Ord a⇒ AdditiveGroup (MaxPlus a) whereMP a +̂ MP b = MP (a ‘max ‘ b)

instance (Ord a,Num a)⇒ VectorSpace (MaxPlus a) wheretype Scalar (MaxPlus a) = aa ·MP b = MP (a + b)

Oops – We also need a zero.VectorSpace is overkill. Module over a semi-ring suffices.

Timing analysis

MaxPlus algebra

type Delay = MaxPlus Double

data MaxPlus a = −∞ | Fi a

instance Ord a⇒ AdditiveGroup (MaxPlus a) where0 = −∞MP a +̂ MP b = MP (a ‘max ‘ b)−∞ +̂ = −∞

+̂−∞ = −∞instance (Ord a,Num a)⇒ VectorSpace (MaxPlus a) where

type Scalar (MaxPlus a) = aa ·MP b = MP (a + b)· −∞ = −∞

Representation?

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

I Matrices

I Functions

I What else?

Representation?

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

I Matrices

I Functions

I What else?

Matrices

a11 · · · a1m...

. . ....

an1 · · · anm

Static typing?

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)

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)

Represent via type family (old)

class VectorSpace v ⇒ HasBasis v wheretype 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).

Represent as GADT

data a ( b whereDot :: InnerSpace b ⇒

b → (b ( Scalar b)(:4) :: VS3 a c d ⇒ -- vector spaces with same scalar field

(a ( c)→ (a ( d)→ (a ( c × d)

Semantics

[[·]] :: (a ( b)→ (a→ b)[[Dot b]] = dot b[[f :4 g ]] = [[f ]] 4[[g ]]

where, on functions,

(f 4 g) a = (f a, g a)

Recall:

data a ( b whereDot :: InnerSpace b ⇒ b → (b ( Scalar b)(:4) :: VS3 a c d ⇒ (a ( c)→ (a ( d)→ (a ( c × d)

Semantic type class morphisms

Category instance specification:

[[id ]] ≡ id[[g ◦ f ]] ≡ [[g ]] ◦ [[f ]]

Arrow instance specification:

[[f 4 g ]] ≡ [[f ]] 4[[g ]][[f × g ]] ≡ [[f ]]×[[g ]]

(4) :: Arrow (;)⇒ (a ; c)→ (a ; d)→ (a ; c × d)(×) :: Arrow (;)⇒ (a ; c)→ (b ; d)→ (a× b ; c × d)

The Category and Arrow laws then follow.

Deriving a Category instance

One case:

[[(f :4 g) ◦ h]]≡ ([[f ]] 4[[g ]]) ◦ [[h]]≡ [[f ]] ◦ [[h]] 4[[g ]] ◦ [[h]]≡ [[f ◦ h 4 g ◦ h]]

(where f ◦ h 4 g ◦ h ≡ (f ◦ h) 4(g ◦ h)). Uses:

(f 4 g) ◦ h ≡ f ◦ h 4 g ◦ h

Implementation:

(f :4 g) ◦ h = f ◦ h :4 g ◦ h

Deriving a Category instance

[[Dot s ◦ Dot b]]≡ dot s ◦ dot b≡ dot (s · b)≡ [[Dot (s · b)]]

[[Dot (a, b) ◦ (f :4 g)]]≡ dot (a, b) ◦ ([[f ]] 4[[g ]])≡ add ◦ (dot a ◦ [[f ]] 4 dot b ◦ [[g ]])≡ dot a ◦ [[f ]] +̂ dot b ◦ [[g ]]≡ [[Dot a ◦ f +̂ Dot b ◦ g ]]

dot (a, b) ≡ add ◦ (dot a× dot b)

(k × h) ◦ (f 4 g) ≡ k ◦ f 4 h ◦ g

[[f +̂ g ]] ≡ [[f ]] +̂[[g ]]

Deriving an Arrow instance

[[f 4 g ]]≡ [[f ]] 4[[g ]]≡ [[f :4 g ]]

[[f × g ]]≡ [[f ]]×[[g ]]≡ [[f ]] ◦ fst 4[[g ]] ◦ snd≡ [[compFst f ]] 4[[compSnd g ]]≡ [[compFst f :4 compSnd g ]]

assuming

[[compFst f ]] ≡ [[f ]] ◦ fst[[compSnd g ]] ≡ [[g ]] ◦ snd

Composing with fst and snd

compFst :: VS3 a b c ⇒ a ( c → a× b ( ccompSnd :: VS3 a b c ⇒ b ( c → a× b ( c

Derivation:

dot a ◦ fst ≡ dot (a, 0)

(f 4 g) ◦ fst ≡ f ◦ fst 4 g ◦ fst

Implementation:

compFst (Dot a) = Dot (a, 0)compFst (f :4 g) = compFst f 4 compFst g

compSnd (Dot b) = Dot (0, b)compSnd (f :4 g) = compSnd f 4 compSnd g

Adding linear maps

[[Dot b +̂ Dot c]]≡ dot b +̂ dot c≡ dot (b +̂ c)≡ [[Dot (b +̂ c)]]

[[(f :4 g) +̂(h :4 k)]]≡ ([[f ]] 4[[g ]]) +̂([[h]] 4[[k]])≡ ([[f ]] +̂[[h]]) 4([[g ]] +̂[[k]])≡ [[(f +̂ h) 4(g +̂ k)]]

Other cases don’t type-check.Uses (on functions):

(f 4 g) +̂(h 4 k) ≡ (f +̂ h) 4(g +̂ k)

What next?

I Fancier timing analysis

I What else is linear?

I More examples of semantic type class morphisms

Circuit timing analysis, linear maps, and semantic morphisms · Circuit timing analysis, linear...

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