Type-checking Polymorphic Units for Astrophysics Research in Haskell

Experience Report: Type‐checking
Polymorphic Units for Astrophysics
Research in Haskell
Takayuki Muranushi
Advanced Institute for
Computational Science,
RIKEN
takayuki.muranushi@riken.jp
Richard A. Eisenberg
University of Pennsylvania
eir@cis.upenn.edu

July 23,
1983 .
• Air Canada Flight 143, a Boeing 767‐233 was
departing Montreal to Edmonton.
• The fuel gauge is not working.
• The crews use a backup system and manually
calculated the amount of fuel to be refueled.

Manual calculation
mass of
fuel required density of fuel volume to be refueled
÷ ＝
[kg] [pound/L] [*L]
The correct calculation
mass of
fuel required density of fuel volume to be refueled
÷ ＝
[kg] [kg/L] [L]
Flight 143 took off with 22,300 pounds of fuel to
Edmonton, where 22,300 kg was actually needed.

• Flight 143 made an emergency landing on
runway 32L of Gimli abandoned airport.
• It was “Family Day” festival; go‐carts, campers,
families and barbecues were on 32L.
• No one was seriously hurt nor killed.
Wade H. Nelson (1997)

A same law of physics can be
represented in many different units
Mass of
Density of fuel Volume to be
fuel required Dimensions Level:
Mass ÷ Density ＝
Volume
[kg] [kg/L] [L]
÷ ＝
[lb.] ÷ [lb./L] ＝ [L]
Units Level:
÷ ＝ refueled
Quantity Level:
quantity value = numerical value [ unit ]
[kg] ÷ [lb./L] ＝ [L]

“units‐of‐measure are to science what
types are to programming” ‐‐‐ A. J. Kennedy
• To avoid mistakes like Gimli Glider, we would
like to use type system to enforce the
correctness of the dimensions and units in our
calculations.
• Such correctness of “laws of physics” is more
than just about specific set of units; we can
represent one quantity in many different units,
but they mean the same quantity.

“Laws of physics are dimension‐monomorphic
and unit‐polymorphic”
‐‐‐ T. Muranushi
Dimensions Level:
Mass ÷ Density ＝ Volume
[kg] ÷ [kg/L] ＝ [L]
[lb.] ÷ [lb./L] ＝ [L]
Units Level:

“Laws of physics are dimension‐monomorphic
and unit‐polymorphic”
‐‐‐ T. Muranushi
• We already have type system of units for many
languages including C, F#, simulink and of
course in Haskell; we already have
polymorphism. Will they blend?

Using `unittyped` by Thijs Alkemade,
I started an attempt to encode knowledge
of physics in Haskell.

A unit‐monomorphic quantity function
refuel :: Fractional f =>
Value Mass KiloGram f
‐> Value Density KiloGramPerLiter f
‐> Value Volume Liter f
refuel gasMass gasDen = gasMass |/| gasDen
A unit‐polymorphic version?
Value Mass uniMass f ‐‐ Here we replace the unit types
‐> Value Density uniDen f ‐‐ with type variables
‐> Value Volume uniVol f ‐‐
This code doesn’t compile.

This code
Value Mass uniMass f
‐> Value Density uniDen f
‐> Value Volume uniVol f
needs these annotations to compile:
refuel :: (Fractional f,
Convertible' Mass uniMass,
Convertible' Density uniDen,
Convertible' Volume uniVol,
MapNeg negUniDen uniDen, ‐‐ negUniDen = 1 / uniDen
MapMerge uniMass negUniDen uniVol ‐‐ uniMass * negUniDen = uniVol
) =>
Value Mass uniMass f
‐> Value Density uniDen f
‐> Value Volume uniVol f

Problem with unit polymorphism in `unittyped`
too much type constraint!!
Colors indicate: Type constraints, Types, Values
Convertible' Mass umass,
Convertible' Density uden,
Convertible' Volume uvol,
MapNeg negUden uden,
MapMerge umass negUden uvol) =>
Value Mass umass f
gravityPoisson ::
(Fractional x
, dimLen ~ LengthDimension
, dimPot ~ '[ '(Time, NTwo), '(Length, PTwo)]
, dimDen ~ Density
, dimZhz ~ '[ '(Time, NTwo)]
, Convertible' dimLen uniLen
, Convertible' dimPot uniPot
, Convertible' dimDen uniDen
, Convertible' dimZhz uniZhz
, Convertible' dimZhz uniZhz'
, MapMerge dimLen dimLen dimLen2
, MapNeg dimLen2 dimLenNeg2
, MapMerge dimPot dimLenNeg2 dimZhz
, MapMerge dimDen '[ '(Time, NTwo), '(Length, PThree),
'(Mass, NOne) ] dimZhz
, MapMerge uniLen uniLen uniLen2
, MapNeg uniLen2 uniLenNeg2
, MapMerge uniPot uniLenNeg2 uniZhz
, MapMerge uniDen '[ '(Second, NTwo), '(Meter,
PThree), '((Kilo Gram), NOne) ] uniZhz'
) =>
(forall s. AD.Mode s =>
Vec3 (Value dimLen uniLen (AD s x)) ‐> Value dimPot
uniPot (AD s x))
‐> (Vec3 (Value dimLen uniLen x) ‐> (Value dimDen
uniDen x))
‐> (Vec3 (Value dimLen uniLen x) ‐> (Value dimZhz
uniZhz x))
gravityPoisson gravitationalPotential density r
= laplacian gravitationalPotential r |‐| (4 *| pi |*|
density r |*| g)
‐> Value Density uden f
‐> Value Volume uvol f
gravityPoisson ::
(Fractional x
, dimDen ~ Density
, dimZhz ~ '[ '(Time, NTwo)]
, Convertible' dimZhz uniZhz'
, MapNeg dimLen2 dimLenNeg2
, MapMerge dimPot dimLenNeg2 dimZhz
, MapMerge dimDen '[ '(Time, NTwo), '(Length, PThree), '(Mass, NOne) ] dimZhz
, MapNeg uniLen2 uniLenNeg2
, MapMerge uniPot uniLenNeg2 uniZhz
, MapMerge uniDen '[ '(Second, NTwo), '(Meter, PThree), '((Kilo Gram), NOne) ] uniZhz'
) =>
Vec3 (Value dimLen uniLen (AD s x)) ‐> Value dimPot uniPot (AD s x))
‐> (Vec3 (Value dimLen uniLen x) ‐> (Value dimDen uniDen x))
‐> (Vec3 (Value dimLen uniLen x) ‐> (Value dimZhz uniZhz x))
gravityPoisson gravitationalPotential density r
= laplacian gravitationalPotential r |‐| (4 *| pi |*| density r |*| g)
gravitationalPotentialToDensity ::
forall x
dimLen dimDen dimDen' dimLen2 dimNegLen2 dimZhz dimNegGC dimPot
uniLen uniDen uniDen' uniLen2 uniNegLen2 uniZhz uniNegGC uniPot .
(Fractional x
, dimDen ~ Density
, MapEq dimDen' dimDen
, Convertible' dimDen' uniDen'
, MapNeg dimLen2 dimNegLen2
, MapMerge dimPot dimNegLen2 dimZhz
, MapNeg '[ '(Time, NTwo), '(Length, PThree), '(Mass, NOne) ] dimNegGC
, dimNegGC ~ '[ '(Time, PTwo), '(Length, NThree), '(Mass, POne) ]
, MapMerge dimZhz dimNegGC dimDen'
, MapNeg uniLen2 uniNegLen2
, MapMerge uniPot uniNegLen2 uniZhz
, MapNeg '[ '(Second, NTwo), '(Meter, PThree), '((Kilo Gram), NOne) ] uniNegGC
, uniNegGC ~ '[ '(Second, PTwo), '(Meter, NThree), '((Kilo Gram), POne) ]
, MapMerge uniZhz uniNegGC uniDen'
) =>
‐> (Vec3 (Value dimLen uniLen x) ‐> (Value dimDen uniDen x))
gravitationalPotentialToDensity gravitationalPotential r
= to (undefined :: Value dimDen uniDen x) $
(laplacian gravitationalPotential r |/| (4 *| pi |*| g) :: Value dimDen' uniDen' x)
hydrostatic ::
forall x
dimLen dimPre dimDen dimAcc dimGpr dimNegLen dimNegDen dimAcc'
uniLen uniPre uniDen uniAcc uniGpr uniNegLen uniNegDen uniAcc' .
( Fractional x
, dimPre ~ Pressure
, dimDen ~ Density
, dimAcc ~ Acceleration
, dimNegDen ~ '[ '(Length, PThree), '(Mass, NOne) ]
, dimGpr ~ '[ '(Length, NTwo), '(Mass, POne) , '(Time, NTwo) ]
, Convertible' dimPre uniPre
, Convertible' dimGpr uniGpr
, Convertible' dimAcc uniAcc
, Convertible' dimAcc' uniAcc'
, MapNeg dimLen dimNegLen
, MapMerge dimPre dimNegLen dimGpr
, MapNeg dimDen dimNegDen
, MapMerge dimGpr dimNegDen dimAcc'
, MapEq dimAcc' dimAcc
, MapNeg uniLen uniNegLen
, MapMerge uniPre uniNegLen uniGpr
, MapNeg uniDen uniNegDen
, MapMerge uniGpr uniNegDen uniAcc'
) =>
Vec3 (Value dimLen uniLen (AD s x)) ‐> Value dimPre uniPre (AD s x))
‐> (Vec3 (Value dimLen uniLen x) ‐> Value dimDen uniDen x)
‐> (Vec3 (Value dimLen uniLen x) ‐> Vec3 (Value dimAcc uniAcc x))
‐> (Vec3 (Value dimLen uniLen x) ‐> Vec3 (Value dimAcc uniAcc x))
hydrostatic pressure density externalAcc r
= compose $ ¥i ‐> to (undefined :: Value dimAcc uniAcc x) $
(externalAcc r ! i) |+| (gradP r ! i) |/| (density r)
where
gradP :: Vec3 (Value dimLen uniLen x) ‐> Vec3 (Value dimGpr uniGpr x)
gradP = grad pressure
pressureToAcc ::
forall x
dimLen dimPre dimDen dimAcc dimGpr dimNegLen dimNegDen dimAcc'
uniLen uniPre uniDen uniGpr uniNegLen uniNegDen uniAcc' .
( Fractional x
, dimPre ~ Pressure
, dimDen ~ Density
, dimNegDen ~ '[ '(Length, PThree), '(Mass, NOne) ]
, dimGpr ~ '[ '(Length, NTwo), '(Mass, POne) , '(Time, NTwo) ]
, Convertible' dimPre uniPre
, Convertible' dimGpr uniGpr
, Convertible' dimAcc uniAcc'
, MapMerge dimPre dimNegLen dimGpr
, MapNeg dimDen dimNegDen
, MapMerge dimGpr dimNegDen dimAcc'
, MapMerge uniPre uniNegLen uniGpr
, MapNeg uniDen uniNegDen
, MapMerge uniGpr uniNegDen uniAcc'
) =>
Vec3 (Value dimLen uniLen (AD s x)) ‐> Value dimPre uniPre (AD s x))
‐> (Vec3 (Value dimLen uniLen x) ‐> Value dimDen uniDen x)
‐> (Vec3 (Value dimLen uniLen x) ‐> Vec3 (Value dimAcc uniAcc' x))
pressureToAcc pressure density r
= compose $ ¥i ‐> to (undefined :: Value dimAcc uniAcc' x) $
(gradP r ! i) |/| (density r)
where
gradP :: Vec3 (Value dimLen uniLen x) ‐> Vec3 (Value dimGpr uniGpr x)
gradP = grad pressure
gravitationalPotentialToAcc ::
forall x
dimLen dimPot dimAcc' dimNegLen dimAcc
uniLen uniPot uniAcc' uniNegLen
( Fractional x
, dimAcc' ~ '[ '(Length, POne), '(Time, NTwo)]
, Convertible' dimAcc uniAcc'
, MapMerge dimPot dimNegLen dimAcc'
, MapMerge uniPot uniNegLen uniAcc'
) =>
‐> (Vec3 (Value dimLen uniLen x) ‐> Vec3 (Value dimAcc uniAcc' x))
gravitationalPotentialToAcc pot r =
(fmap $ to (undefined :: Value dimAcc uniAcc x)) $
grad pot r
We need at least two
type constraints per one
arithmetic operator, in
order to encode type‐level
unit calculations.

↑ ✈ Problem
↓ ✈ Solution

Quantity representation in `unittyped`
Type constructor takes (dimensions) (units) (numerical value)
λ> :t 88 *| mile |/| hour
… :: Fractional f =>
Value '[ '(Length, POne), '(Time, NOne)]
'[ '(Mile, POne), '(Hour, NOne)] f
Quantity representation in `units`
Type constructor takes
(dimensions) (map from dimensions to units) (numerical value )
λ> :t 88 % mile :/ hour :: Fractional f => Qu Velocity SI f
… :: Fractional f =>
Qu '[ '(Length, One), '(Time, MOne)]
'[ '(Length, Meter), '(Time, Second), …] f

System of Units as type argument
• The map from dimensions to units represents
a system of units; e.g. SI system, CGS
(centimeter–gram–second) system, etc.
λ> 88 % Miles :/ Hour :: Qu Velocity SI Float
39.33952 m/s
λ> :info SI
type SI = MkLCSU
'[(Length, Meter), (Mass, Kilo :@ Gram), (Time, Second),
(Current, Ampere), (Temperature, Kelvin),
(AmountOfSubstance, Mole),
(LuminousIntensity, Lumen)]
λ> :info CGS
type CGS = MkLCSU
'[(Length, Centi :@ Meter), (Mass, Gram), (Time, Second)]

[Def] A coherent system of unit ℓ
ℓ = {u1, u2, … , un}∪ {u1
p u2
q …un
r | p,q, … , r ∈ }
base units units derived by products of base units
• A Joule (1 [kg/m2s2]) is the coherent derived unit of
energy in SI
• An erg (1 [g/cm2s2] = 10-7J) is the coherent derived
unit of energy in centimeter‐gram‐second system
1:1 mapping between dimensions and units
[kg] ÷ [kg/m3] ＝ [m3]
[SI mass]÷ [SI density] ＝ [SI volume]

Unit‐polymorphic calculations in `units`
Qu Mass ℓ f ‐> Qu Density ℓ f ‐> Qu Volume ℓ f
• local coherent system of unit ℓ is the only one,
unconstrained, type variable
• The computation is nondimensionalized; can be
carried out without details units.
[kg] ÷ [kg/m3] ＝ [m3]
[SI mass]÷ [SI density] ＝ [SI volume]

• Unit polymorphism with fundeps gave rise to
overwhelming complexes of constrained type
variables
Convertible' Mass umass,
Convertible' Density uden,
Convertible' Volume uvol,
MapNeg negUden uden,
MapMerge umass negUden uvol) =>
Value Mass umass f
‐> Value Density uden f
‐> Value Volume uvol f
• Take local coherent system of unit ℓ as only one
free variable

An application:
Avoid over/underflow

Exercise: How many Newtons is the Lennard-Jones
force F between two argon atoms at distance 4Å,
where
Solution #1:
ljForce :: Energy ℓ Float ‐> Length ℓ Float
‐> Length ℓ Float ‐> Force ℓ Float
ljForce eps sigma r
= (24 *| eps |*| sigma |ˆ pSix) |/| (r |ˆ pSeven)
|‐|(48 *| eps |*| sigma |ˆ pTwelve) |/| (r |ˆ pThirteen)
λ> let sigmaAr = 3.4e‐8 % Meter
epsAr = 1.68e‐21 % Joule
r = 4.0e‐8 % Meter
λ> (ljForce epsAr sigmaAr r :: Force SI Float) # Newton
NaN

Exercise: How many Newtons is the Lennard-Jones
force F between two argon atoms at distance 4Å,
where
Solution #2:
ljForce :: Energy ℓ Float ‐> Length ℓ Float
‐> Length ℓ Float ‐> Force ℓ Float
ljForce eps sigma r
= (24 *| eps |*| sigma |ˆ pSix) |/| (r |ˆ pSeven)
|‐|(48 *| eps |*| sigma |ˆ pTwelve) |/| (r |ˆ pThirteen)
type CU = MkLCSU '[ '(Length, Angstrom),
'(Mass, ProtonMass), '(Time, Pico :@ Second)]
λ> (ljForce epsAr sigmaAr r :: Force CU Float) # Newton
9.3407324e‐14

Astrophysics research in Haskell
• A 27‐page astrophysics paper has been
written in Haskell; its quantitative reasoning is
powered by the units library.

↑ ✈ Experience
↓ ✈ Conclusion

When you design type system of units
consider unit polymorphism
because, with unit polymorphism
• We can faithfully express unit‐independent
nature of laws of physics.
• We can write quantity expressions, which
users can later interpret in unit systems of
their choice.
• We can avoid overflows/ underflows by
appropriately choosing system of units.

• An easy way to implement unit
polymorphism is to take local coherent
system of unit ℓ as only one free variable

In the paper
• Extensibility
• Quantity combinators
• Value‐level units
• Protect numerical values from manipulation
Things came after paper
• defaultLCSU
• Template Haskell

Comments
• Haskell is such a cool language that allows
something like ùnits` at all. Its type system is
so programmable that these features can be built
on top of, instead of being integrated in (like F#)
or externally analyzed (like C)
• As far as we know, ùnits` is the only practical
system that supports unit polymorphism.
• Heartfelt thanks to all people’s work that enabled
GHC 7.8, and to Richard who implemented
ùnits`.

↑ ✈ Thanks!
↓ ✈ Questions?
cabal install units
and enjoy unit polymorphism!

Type-checking Polymorphic Units for Astrophysics Research in Haskell

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