Type-checking Polymorphic Units for Astrophysics Research in Haskell

ExperienceReport:TypecheckingPolymorphicUnitsforAstrophysics

ResearchinHaskell

TakayukiMuranushiAdvancedInstituteforComputationalScience,

[email protected]

RichardA.EisenbergUniversityofPennsylvania

[email protected]

July23,

AirCanadaFlight143,aBoeing767233wasdepartingMontrealtoEdmonton.

Thefuelgaugeisnotworking. Thecrewsuseabackupsystemandmanuallycalculatedtheamountoffueltoberefueled.

1983 .

Manualcalculationmassoffuelrequired densityoffuel volumetoberefueled

[kg] [pound/L] [*L]

[kg] [kg/L] [L]

massoffuelrequired densityoffuel volumetoberefueled

Thecorrectcalculation

Flight143tookoffwith22,300pounds offueltoEdmonton,where22,300kg wasactuallyneeded.

Flight143madeanemergencylandingonrunway32LofGimliabandonedairport.

ItwasFamilyDayfestival;gocarts,campers,familiesandbarbecueswereon32L.

Noonewasseriouslyhurtnorkilled.

WadeH.Nelson(1997)

[kg] [kg/L] [L]

Mass Density Volume

[lb.] [lb./L] [L]

Asamelawofphysicscanberepresentedinmanydifferentunits

DimensionsLevel:

UnitsLevel:

Massoffuelrequired Densityoffuel

Volumetoberefueled

QuantityLevel:

quantityvalue =numericalvalue[unit ]

[kg] [lb./L] [L]

ToavoidmistakeslikeGimliGlider,wewouldliketousetypesystemtoenforcethecorrectnessofthedimensionsandunitsinourcalculations.

Suchcorrectnessoflawsofphysicsismorethanjustaboutspecificsetofunits;wecanrepresentonequantityinmanydifferentunits,buttheymeanthesamequantity.

unitsofmeasure aretosciencewhattypesaretoprogramming A.J.Kennedy

unitsofmeasure aretosciencewhattypesaretoprogramming A.J.Kennedy

Lawsofphysicsaredimensionmonomorphicandunitpolymorphic

T.Muranushi

[kg] [kg/L] [L] [lb.] [lb./L] [L]

Mass Density Volume DimensionsLevel:

UnitsLevel:

WealreadyhavetypesystemofunitsformanylanguagesincludingC,F#,simulink andofcourseinHaskell;wealreadyhavepolymorphism.Willtheyblend?

unitsofmeasurearetosciencewhattypesaretoprogramming A.J.Kennedy

Lawsofphysicsaredimensionmonomorphicandunitpolymorphic

T.Muranushi

Using`unittyped` byThijsAlkemade,IstartedanattempttoencodeknowledgeofphysicsinHaskell.

Aunitmonomorphicquantityfunctionrefuel::Fractionalf=>

ValueMassKiloGram f>ValueDensityKiloGramPerLiter f>ValueVolumeLiterf

refuelgasMassgasDen=gasMass|/|gasDen

refuel::Fractionalf=>ValueMassKiloGram f

>ValueDensityKiloGramPerLiter f>ValueVolumeLiterf


Aunitpolymorphicversion?

Thiscodedoesntcompile.

refuel::Fractionalf=>ValueMassuniMass f Here we replace the unit types

>ValueDensityuniDen f with type variables>ValueVolumeuniVol f

refuelgasMass gasDen =gasMass |/|gasDen

refuel::(Fractionalf,Convertible'MassuniMass,Convertible'DensityuniDen,Convertible'Volume uniVol,MapNeg negUniDen uniDen, negUniDen = 1 / uniDenMapMerge uniMass negUniDen uniVol uniMass * negUniDen = uniVol

)=>ValueMassuniMass f

>ValueDensityuniDen f>ValueVolumeuniVol f


refuel::(Fractionalf,Convertible'MassuniMass,Convertible'DensityuniDen,Convertible'Volume uniVol,MapNeg negUniDen uniDen, negUniDen = 1 / uniDenMapMerge uniMass negUniDen uniVol uniMass * negUniDen = uniVol

)=>ValueMassuniMass f



needstheseannotationstocompile:

refuel::Fractionalf=>ValueMassuniMass f


refuelgasMass gasDen =gasMass |/|gasDen

Thiscode

refuel::(Fractionalf,Convertible'Massumass,Convertible'Densityuden,Convertible'Volume uvol,MapNeg negUden uden,MapMerge umass negUden uvol)=>ValueMassumass f

>ValueDensityuden f>ValueVolumeuvol frefuelgasMassgasDen=gasMass|/|gasDen

Problemwithunitpolymorphismin`unittyped`toomuchtypeconstraint!!

gravityPoisson::(Fractionalx,dimLen~LengthDimension,dimPot~'['(Time,NTwo),'(Length,PTwo)],dimDen~Density,dimZhz ~'['(Time,NTwo)],Convertible'dimLenuniLen,Convertible'dimPotuniPot,Convertible'dimDenuniDen,Convertible'dimZhz uniZhz,Convertible'dimZhz uniZhz',MapMergedimLendimLen dimLen2,MapNegdimLen2dimLenNeg2,MapMergedimPotdimLenNeg2dimZhz,MapMergedimDen'['(Time,NTwo),'(Length,PThree),

'(Mass,NOne)]dimZhz,MapMergeuniLen uniLen uniLen2,MapNeguniLen2uniLenNeg2,MapMergeuniPot uniLenNeg2uniZhz,MapMergeuniDen '['(Second,NTwo),'(Meter,

PThree),'((KiloGram),NOne)]uniZhz')=>

(forall s.AD.Mode s=>Vec3(ValuedimLenuniLen (ADsx))>ValuedimPot

uniPot (ADsx))>(Vec3(ValuedimLenuniLen x)>(ValuedimDen

uniDen x))>(Vec3(ValuedimLenuniLen x)>(ValuedimZhz

uniZhz x))

gravityPoissongravitationalPotential densityr=laplacian gravitationalPotential r||(4*|pi|*|

densityr|*|g)

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)gravitationalPotentialToDensity ::forall xdimLen dimDen dimDen' dimLen2 dimNegLen2 dimZhz dimNegGC dimPotuniLen uniDen uniDen' uniLen2 uniNegLen2 uniZhz uniNegGC uniPot .(Fractional x, dimLen ~ LengthDimension, dimPot ~ '[ '(Time, NTwo), '(Length, PTwo)], dimDen ~ Density, MapEq dimDen' dimDen, Convertible' dimLen uniLen, Convertible' dimPot uniPot, Convertible' dimZhz uniZhz, Convertible' dimDen' uniDen', Convertible' dimDen uniDen, MapMerge dimLen dimLen dimLen2, 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', MapMerge uniLen uniLen uniLen2, 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') =>(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))gravitationalPotentialToDensity gravitationalPotential r= to (undefined :: Value dimDen uniDen x) $(laplacian gravitationalPotential r |/| (4 *| pi |*| g) :: Value dimDen' uniDen' x)

hydrostatic ::forall xdimLen dimPre dimDen dimAcc dimGpr dimNegLen dimNegDen dimAcc'uniLen uniPre uniDen uniAcc uniGpr uniNegLen uniNegDen uniAcc' .( Fractional x, dimLen ~ LengthDimension, dimPre ~ Pressure, dimDen ~ Density, dimAcc ~ Acceleration, dimNegDen ~ '[ '(Length, PThree), '(Mass, NOne) ], dimGpr ~ '[ '(Length, NTwo), '(Mass, POne) , '(Time, NTwo) ], Convertible' dimLen uniLen, Convertible' dimPre uniPre, Convertible' dimGpr uniGpr, Convertible' dimDen uniDen, 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') =>(forall s. AD.Mode s =>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)wheregradP :: Vec3 (Value dimLen uniLen x) > Vec3 (Value dimGpr uniGpr x)gradP = grad pressure

pressureToAcc ::forall xdimLen dimPre dimDen dimAcc dimGpr dimNegLen dimNegDen dimAcc'uniLen uniPre uniDen uniGpr uniNegLen uniNegDen uniAcc' .( Fractional x, dimLen ~ LengthDimension, dimPre ~ Pressure, dimDen ~ Density, dimAcc ~ Acceleration, dimNegDen ~ '[ '(Length, PThree), '(Mass, NOne) ], dimGpr ~ '[ '(Length, NTwo), '(Mass, POne) , '(Time, NTwo) ], Convertible' dimLen uniLen, Convertible' dimPre uniPre, Convertible' dimGpr uniGpr, Convertible' dimDen uniDen, 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') =>(forall s. AD.Mode s =>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)wheregradP :: Vec3 (Value dimLen uniLen x) > Vec3 (Value dimGpr uniGpr x)gradP = grad pressure

gravitationalPotentialToAcc ::forall xdimLen dimPot dimAcc' dimNegLen dimAccuniLen uniPot uniAcc' uniNegLen( Fractional x, dimLen ~ LengthDimension, dimPot ~ '[ '(Time, NTwo), '(Length, PTwo)], dimAcc' ~ '[ '(Length, POne), '(Time, NTwo)], dimAcc ~ Acceleration, Convertible' dimLen uniLen, Convertible' dimPot uniPot, Convertible' dimAcc' uniAcc', Convertible' dimAcc uniAcc', MapNeg dimLen dimNegLen, MapMerge dimPot dimNegLen dimAcc', MapEq dimAcc' dimAcc, MapNeg uniLen uniNegLen, MapMerge uniPot uniNegLen uniAcc') =>(forall s. AD.Mode s =>Vec3 (Value dimLen uniLen (AD s x)) > Value dimPot uniPot (AD s x))> (Vec3 (Value dimLen uniLen x) > Vec3 (Value dimAcc uniAcc' x))gravitationalPotentialToAcc pot r =(fmap $ to (undefined :: Value dimAcc uniAcc x)) $grad pot r

Weneedatleasttwotypeconstraintsperonearithmeticoperator,inordertoencodetypelevel unitcalculations.

Colorsindicate: Typeconstraints, Types,Values

Problem Solution

Oursolution:

Quantityrepresentationin`unittyped`

>:t88*|mile|/|hour ::Fractionalf=>

Value'['(Length,POne),'(Time,NOne)]'['(Mile,POne),'(Hour,NOne)]f

>:t88*|mile|/|hour ::Fractionalf=>

Value'['(Length,POne),'(Time,NOne)]'['(Mile,POne),'(Hour,NOne)]f

Typeconstructortakes(dimensions)(units)(numericalvalue)

>:t88 % mile:/hour::Fractionalf=>QuVelocitySIf ::Fractionalf=>

Qu'['(Length,One),'(Time,MOne)]'['(Length,Meter),'(Time,Second),]f

>:t88 % mile:/hour::Fractionalf=>QuVelocitySIf ::Fractionalf=>

Qu'['(Length,One),'(Time,MOne)]'['(Length,Meter),'(Time,Second),]f

Typeconstructortakes(dimensions)(mapfromdimensionstounits) (numericalvalue)

Quantityrepresentationin`units`

SystemofUnitsastypeargument Themapfromdimensionstounitsrepresentsasystemofunits;e.g.SIsystem,CGS(centimetergramsecond)system,etc.>88%Miles:/Hour::QuVelocitySIFloat39.33952m/s>:infoSItypeSI=MkLCSU

'[(Length,Meter),(Mass,Kilo:@Gram),(Time,Second),(Current,Ampere),(Temperature,Kelvin),(AmountOfSubstance,Mole),(LuminousIntensity,Lumen)]

>:infoCGStypeCGS=MkLCSU

'[(Length,Centi:@Meter),(Mass,Gram),(Time,Second)]

>88%Miles:/Hour::QuVelocitySIFloat39.33952m/s>:infoSItypeSI=MkLCSU

'[(Length,Meter),(Mass,Kilo:@Gram),(Time,Second),(Current,Ampere),(Temperature,Kelvin),(AmountOfSubstance,Mole),(LuminousIntensity,Lumen)]

>:infoCGStypeCGS=MkLCSU

'[(Length,Centi:@Meter),(Mass,Gram),(Time,Second)]

[Def]Acoherentsystemofunit

AJoule(1 [kg/m2s2])isthecoherentderivedunitofenergyinSI

Anerg(1 [g/cm2s2] = 10-7J)isthecoherentderivedunitofenergyincentimetergramsecondsystem

[kg] [kg/m3] [m3]

[SI mass] [SI density] [SI volume]

= {u1, u2, , un} {u1p u2q unr | p,q, , r }baseunits unitsderivedbyproductsofbaseunits

1:1mappingbetweendimensionsandunits

localcoherentsystemofunit istheonlyone,unconstrained,typevariable

Thecomputationisnondimensionalized;canbecarriedoutwithoutdetailsunits.

refuel::Fractionalf=>QuMass f>QuDensity f>QuVolume frefuelgasMassgasDen=gasMass|/|gasDen


Unitpolymorphiccalculationsin`units`

[kg] [kg/m3] [m3]

[SI mass] [SI density] [SI volume]

Unitpolymorphismwithfundeps gaverisetooverwhelmingcomplexesofconstrainedtypevariables

Takelocalcoherentsystemofunit asonlyonefreevariable







Anapplication:Avoidover/underflow

ljForce::Energy Float>Length Float>Length Float>Force Float

ljForceepssigmar=(24*|eps|*|sigma|pSix) |/|(r|pSeven)||(48*|eps|*|sigma|pTwelve)|/|(r|pThirteen)



Solution #1:

>letsigmaAr=3.4e8%MeterepsAr=1.68e21%Jouler=4.0e8%Meter

>(ljForceepsArsigmaArr::ForceSIFloat)#NewtonNaN

>letsigmaAr=3.4e8%MeterepsAr=1.68e21%Jouler=4.0e8%Meter

>(ljForceepsArsigmaArr::ForceSIFloat)#NewtonNaN

Exercise: How many Newtons is the Lennard-Jones force F between two argon atoms at distance 4, where



typeCU=MkLCSU'['(Length,Angstrom),'(Mass,ProtonMass),'(Time,Pico:@Second)]



typeCU=MkLCSU'['(Length,Angstrom),'(Mass,ProtonMass),'(Time,Pico:@Second)]

Solution #2:

>(ljForceepsArsigmaArr::ForceCUFloat)#Newton9.3407324e14>(ljForceepsArsigmaArr::ForceCUFloat)#Newton9.3407324e14

Exercise: How many Newtons is the Lennard-Jones force F between two argon atoms at distance 4, where

AstrophysicsresearchinHaskell

A27pageastrophysicspaperhasbeenwritteninHaskell;itsquantitativereasoningispoweredbytheunits library.

Experience Conclusion

Whenyoudesigntypesystemofunitsconsiderunitpolymorphism

because,withunitpolymorphism Wecanfaithfullyexpressunitindependentnatureoflawsofphysics.

Wecanwritequantityexpressions,whichuserscanlaterinterpretinunitsystemsoftheirchoice.

Wecanavoidoverflows/underflowsbyappropriatelychoosingsystemofunits.

Aneasywaytoimplementunitpolymorphismistotakelocalcoherentsystemofunit asonlyonefreevariable



Inthepaper

Extensibility Quantitycombinators Valuelevelunits Protectnumericalvaluesfrommanipulation

Thingscameafterpaper defaultLCSU TemplateHaskell

Comments Haskellissuchacoollanguagethatallowssomethinglikeùnits` atall.Itstypesystemissoprogrammablethatthesefeaturescanbebuiltontopof,insteadofbeingintegratedin(likeF#)orexternallyanalyzed(likeC)

Asfarasweknow,ùnits` istheonlypracticalsystemthatsupportsunitpolymorphism.

HeartfeltthankstoallpeoplesworkthatenabledGHC7.8,andtoRichardwhoimplementedùnits`.

Thanks! Questions?cabalinstallunits

andenjoyunitpolymorphism!

