generic programming for domain reasoners · generic programming for domain ... linear equations, or...
TRANSCRIPT
Chapte
r1
Gen
eric
prog
ram
min
gfo
rdo
mai
nre
asoners
Joh
an
Jeuri
ng,1,
2Jo
se ´
Ped
roMa
galh˜a
es,1
Ba
stia
an
Heere
n2C
ate
go
ry:
E v
alu
atio
n
Abs
trac
t:A
nex
erci
seas
sist
ant
isa
tool
that
supp
orts
stud
ents
lear
ning
p ro
ce-
dura
lsk
ills,
such
asso
lvin
gsy
stem
sof
linea
req
uatio
ns,
orre
writ
ing
alo
gic
ex-
pres
sion
todi
sjun
ctiv
eno
rmal
form
.Th
edo
mai
nre
ason
eris
the
com
pone
ntof
anex
erci
seas
sist
ant
that
deal
sw
ithta
sks
that
depe
ndon
the
sem
antic
sof
ap
artic
ular
dom
ain.
An
exer
cise
assi
stan
tty
pica
llyha
sm
ultip
ledo
mai
nr e
ason
ers,
and
the
beha
vior
of
each
of
thes
eis
,to
ala
rge
exte
nt,
dete
rmin
edby
the
dom
ain.
Gen
eric
prog
ram
min
gte
chni
ques
clai
mto
redu
ceco
dedu
plic
atio
n,to
mak
e
itea
sier
toch
ange
the
stru
ctur
eof
data
and
topr
ovid
eim
plem
enta
tions
of
man
yus
eful
func
tions
onm
ost
data
type
s.Th
ere
are
man
ylib
rarie
sfo
rge
neric
prog
ram
-m
ing,
all
with
diffe
rent
data
type
supp
ort,
expr
essi
vene
ssan
dea
seof
use.
W h
ilea
lot
of
effo
rtha
sbe
enpu
tin
tode
velo
ping
new
libra
ries
inth
ep
ast
few
year
s,th
ere
are
few
exam
ples
of
real
life
appl
icat
ions
of
gene
ricp
rogr
amm
ing.
Inth
ispa
per,
we
desc
ribe
our
expe
rienc
eof
appl
ying
gene
ricp
rogr
amm
ing
tech
niqu
esto
our
dom
ain
reas
oner
sim
plem
ente
din
Has
kell.
We
asse
ssan
dco
mpa
reth
ere
-su
lting
gene
ricdo
mai
nre
ason
ers
with
each
othe
ran
dw
ithth
ein
itial
,no
n-ge
neric
vers
ion,
u si
ngth
eso
ftwar
equ
ality
fact
ors
of
the
ISO
9126
inte
rnat
iona
lst
anda
rd.
1.1
INT
RO
DU
CT
ION
An
exer
cise
assi
stan
the
lps
ast
uden
tw
ithso
lvin
gex
erci
ses,
for
exam
ple
toso
lve
equa
tions
,or
dete
rmin
eth
ede
rivat
ive
of
afu
nctio
n.Th
ere
exis
thu
ndre
dsofe
xer-
cise
assi
stan
ts,
for
man
ym
athe
mat
ical
dom
ains
:lo
gic,
phys
ics,
etc.
An
impo
rtant
1Uni
vers
iteit
Utre
cht,
Dep
artm
ent
ofIn
form
atio
nan
dC
ompu
ting
Scie
nces
,PO
Box
80.0
89,
3508
TBU
trech
t,Th
eNe
ther
land
s2O
pen
Uni
vers
iteit
Ned
erla
nd,
Scho
olof
Com
pute
rSc
ienc
e,PO
Box
2960
,64
01D
LH
eerle
n,Th
eN
ethe
rland
s;{j
ohan
j,jp
m,
ba
stia
an}@
cs.u
u.n
l
1
2C
HA
PTE
R1.
GE
NE
RIC
PR
OG
RA
MM
ING
FO
RD
OM
AIN
RE
AS
ON
ER
S
part
of
anex
erci
seas
sist
ant
isth
edo
mai
nr e
ason
er:
aco
mpo
nent
whi
chtra
cks
the
step
sth
est
uden
tta
kes,
give
shi
nts,
diag
nose
ser
rors
,re
cord
sp
rogr
ess,
show
sw
orke
d-ou
tso
lutio
ns,
etc.
The
func
tiona
lity
of
the
dom
ain
reas
oner
fund
amen
-ta
llyde
pend
son
the
dom
ain,
the
rule
sth
ath
old
for
the
dom
ain
and
the
stra
tegi
esfo
rso
lvin
gex
erci
ses
with
inth
edo
mai
n.Th
eco
reco
mpo
nent
sof
ado
mai
nre
ason
erar
e:
•A
desc
riptio
nof
the
dom
ain,
cons
istin
gof
anab
stra
ctsy
ntax
toge
ther
with
apa
rser
arnipd
ta
pret
ty-p
rinte
r.
•R
ules
with
whi
chex
pres
sion
sin
the
dom
ain
are
man
ipul
ated
.Fo
rex
ampl
e,fRo
urleth
sewdo
itmhwa
ihnof
logi
cex
pres
sion
s,ethd
eoodm
isatirin
ba utri
eonm
r aunlie
pux
∨lat(ey
d∧.zF
)re
(xxa∨m
yp)l
e∧,(fox
rr∨th
z)e,d
Dome
Mor
gan’
silca
wex
p¬r(ex
ss∨i
oy)n
s , t
¬x
∧is¬
y,uet
tico.
•St
rate
gies
for
solv
ing
exer
cise
sin
the
dom
ain.
For
exam
ple,
for
r ew
ritin
ga
tSetrrm
ateign
iteos
disj
unct
ive
n eoxr
emrca
ilsfeo
sr min:
hfiers d
tr e
mov
eFBo
rooe lx
eaanm
cons
tant
s,im
plic
a-tio
nsan
deq
uiva
lenc
es,
then
push
nega
tions
insi
de,
and
final
lym
ove
occu
r-
renc
esof
∨up
war
ds.
Add
ition
ally
,fo
rea
chdo
mai
nw
ene
edfu
nctio
nalit
yfo
run
ifyin
gan
dre
writ
ing
term
s,ge
nera
ting
exer
cise
s,tra
vers
ing
term
s,de
term
inin
gth
eto
p-le
vel
equa
lity
of
two
term
s,et
c.In
the
past
few
year
s,w
eha
vede
velo
ped
dom
ain
r eas
oner
sw
hich
help
with
diag
nosi
ngst
uden
tbe
havi
our
inus
ing
exer
cise
assi
stan
tsfo
rca
lcul
atin
gw
ithfra
c-tio
ns,
perfo
rmin
gG
auss
ian
elim
inat
ion,
solv
ing
syst
ems
of
linea
req
uatio
nsan
dot
her
linea
ral
gebr
aex
erci
ses,
fact
orin
gp
olyn
omia
ls,
rew
ritin
ga
logi
cal
term
todi
sjun
ctiv
eno
rmal
form
,r e
writ
ing
rela
tion
alge
bra
term
s,de
velo
ping
(ext
ende
d)la
mbd
aca
lcul
usp
rogr
ams,
and
dete
rmin
ing
the
deriv
ativ
eor
inte
gral
of
afu
nc-
tion
[8,9
,15,
19].
Our
dom
ain
reas
oner
sof
fer
the
func
tiona
lity
desc
ribed
abov
eas
Web
ser-
vice
s[6
].S
ever
alex
erci
seas
sist
ants
,su
chas
Mat
hDox
[5],
Act
iveM
ath
[7]
and
the
Freu
dent
hal
Inst
itute
digi
tal
mat
hem
atic
sen
viro
nmen
t(D
WO
)[3
]us
eou
rdo
-m
ain
reas
onin
gW
ebse
rvic
es.
At
the
mom
ent
we
deve
lop
and
mai
ntai
nou
row
ndo
mai
nr e
ason
ers,
butin
the
futu
rew
eex
pect
user
sof
our
serv
ices
toad
dor
adap
tdo
mai
ns.
Furth
erm
ore,
we
are
invo
lved
intw
ore
cent
lyfu
nded
rese
arch
proj
ects
abou
tbr
idgi
ngth
ekn
owle
dge
gap
inm
athe
mat
ics
betw
een
seco
ndar
ysc
hool
san
dun
iver
sitie
s:M
ath
-brid
ge(f
rom
the
EU
eCon
tent
Plu
sp
rogr
amm
e)an
dN
KB
W2
(fro
mth
eS
UR
FFo
unda
tion,
the
Dut
chhi
gher
educ
atio
nan
dre
sear
chp
artn
ersh
ip
orga
niza
tion
for
ICT
).Fo
rth
ese
proj
ects
we
have
tode
liver
anu
mbe
rof
dom
ain
reas
oner
sea
chye
ar.
Toav
oid
reim
plem
entin
gth
esa
me
func
tiona
lity
for
ever
ydo
mai
nan
dto
sim
-pl
ifyth
epr
oces
sof
crea
ting
ane
wdo
mai
nre
ason
er,
we
have
u se
dge
neric
p ro
-gr
amm
ing
tech
niqu
es[2
]to
impl
emen
tr e
writ
ing,
trave
rsal
s,et
c.on
cean
dfo
ral
ldo
mai
ns.
The
appl
icat
ion
of
gene
ricp
rogr
amm
ing
tech
niqu
esto
our
dom
ain
r ea-
sone
rsis
the
proj
ect
that
we
eval
uate
i nth
isp
aper
:ho
wca
nge
neric
prog
ram
min
gbe
used
ina
larg
e,re
alis
ticap
plic
atio
n?W
hat
are
the
bene
fits
and
the
shor
t-co
min
gsof
the
solu
tion
t hat
u se
sge
neric
prog
ram
min
gte
chni
ques
?W
eex
pect
1.2.
US
ING
GE
NE
RIC
PR
OG
RA
MM
ING
3
gene
ricpr
ogra
mm
ing
tech
niqu
esto
sim
plify
the
addi
tion
of
new
dom
ain
reas
on-
ers
orto
chan
geex
istin
gon
es,
and
tor e
duce
the
code
b as
ean
dco
dedu
plic
atio
n.H
owev
er,
appl
icat
ion
of
gene
ricp
rogr
amm
ing
t ech
niqu
esal
sop
oses
som
ech
al-
leng
es.
We
have
toch
oose
asu
itabl
eap
proa
ch,
with
the
right
expr
essi
vene
ssan
dda
taty
pesu
ppor
t.A
dditi
onal
ly,
we
mus
ten
sure
that
we
can
k ee
pal
lof
the
orig
i-na
lfun
ctio
nalit
y,an
dth
atth
ege
neric
solu
tion
isin
deed
anim
prov
emen
tov
erth
eno
n-ge
neric
vers
ion.
We
have
impl
emen
ted
our
dom
ain
reas
oner
sin
Has
kell.
Initi
ally
,w
ech
ose
Has
kell
beca
use
itis
ala
ngua
gew
efe
elco
mfo
rtabl
ep
rogr
amm
ing
in,
and
be-
caus
edo
mai
nre
ason
ers
man
ipul
ate
expr
essi
ontre
es,
atw
hich
Has
kell
isgo
od.
Late
r,w
hen
we
deci
ded
tom
ake
our
dom
ain
reas
oner
sge
neric
,th
istu
rned
out
tobe
the
right
choi
ce,
sinc
eth
ere
are
man
yap
proa
ches
toge
neric
prog
ram
min
gin
Has
kell
[11,
23].
Type
s(d
omai
ns)
play
anim
porta
ntro
lein
our
softw
are,
and
we
also
wan
tto
use
type
dap
proa
ches
toge
neric
p ro
gram
min
g,w
hich
rule
sou
tan
appr
oach
like
DrI
FT
[24]
and
othe
rap
proa
ches
b as
edon
met
apro
gram
min
gor
code
gene
ratio
n.Th
ees
sent
ial
tech
nolo
gyus
edin
our
dom
ain
reas
oner
sis
rew
ritin
g(to
geth
erw
ithm
atch
ing)
.W
hen
r ew
ritin
g,w
em
ayap
ply
any
rew
ritin
g
rule
atan
ypo
sitio
ni n
ate
rm.
Of
cour
se,
ar e
writ
ing
rule
can
only
b eap
plie
dto
ate
rmw
ithin
the
dom
ain.
Itfo
llow
sth
atw
ene
edto
k no
ww
hich
subt
erm
sof
ate
rmbe
long
toth
edo
mai
nag
ain,
whi
chim
plie
st h
atw
ene
edto
know
the
recu
rsiv
est
ruct
ure
of
the
dom
ain.
Ther
ear
eon
lya
few
gene
ricpr
ogra
mm
ing
libra
ries
inH
aske
llth
atgi
veth
eus
erac
cess
toth
er e
curs
ive
stru
ctur
eof
valu
es:
Uni
plat
e[1
7],
Reg
ular
[18]
and
Mul
tirec
[22]
,in
ast
raig
htfo
rwar
dw
ay,
and
Scra
pY
our
Boi
lerp
late
(SY
B)
[14]
,in
ara
ther
cum
bers
ome
way
(see
the
Uni
plat
em
od-
ule
Dat
a.G
ener
ics.
Pla
teD
ata,
whi
chim
plem
ents
Uni
plat
eu
sing
SY
B).
We
have
thre
eim
plem
enta
tions
of
age
neric
dom
ain
reas
oner
:on
eu
sing
U n
ipla
te,
one
us-
ing
Mul
tirec
and
one
usin
gR
egul
ar.
This
eval
uatio
npa
per
disc
usse
sw
here
and
how
gene
ricpr
ogra
mm
ing
tech
-ni
ques
impa
ctth
equ
ality
of
our
softw
are.
Tost
ruct
ure
the
disc
ussi
on,
we
use
the
qual
itych
arac
teris
tics
defin
edin
the
ISO
9126
[13]
inte
rnat
iona
lst
anda
rdfo
rth
eev
alua
tion
of
softw
are
qual
ity.
We
use
thes
ech
arac
teris
tics
beca
use
they
help
togu
aran
tee
com
plet
enes
san
din
depe
nden
cein
our
com
paris
on.
Whe
nw
edi
scus
sth
ech
arac
teris
tics,
we
will
give
exam
ples
of
how
they
play
aro
lein
our
dom
ain
reas
oner
s.W
ew
illco
mpa
refo
urve
rsio
nsof
our
dom
ain
reas
oner
s:a
vers
ion
with
out
gene
ricp
rogr
amm
ing
tech
niqu
esan
dth
ree
vers
ions
usin
gge
neric
p ro
-gr
amm
ing,
one
for
each
of
the
Uni
plat
e,M
ultir
ecan
dR
egul
arlib
rarie
s.W
eho
peth
atou
rex
perie
nce
can
b eus
edas
are
fere
nce
for
othe
rpr
ojec
tsth
atco
nsid
erus
ing
gene
ricpr
ogra
mm
ing.
1.2
US
ING
GE
NE
RIC
PR
OG
RA
MM
ING
We
have
refa
ctor
edou
rex
istin
gdo
mai
nr e
ason
ers
tosp
litoff
com
pone
nts
that
can
b eim
plem
ente
db
ym
eans
of
gene
ricco
mpo
nent
s.Fo
rex
ampl
e,th
ese
cond
vers
ion
of
our
first
dom
ain
reas
oner
(200
5-20
06)
impl
emen
ted
linea
req
uatio
nsus
ing
the
follo
win
gda
taty
pes:
4C
HA
PTE
R1.
GE
NE
RIC
PR
OG
RA
MM
ING
FOR
DO
MA
IN R
EA
SO
NE
RS
type
Equa
tions
=[E
quat
ion]
data
Equ
atio
n=
Expr
:=:
Expr
data
Expr
=C
onR
atio
nal
|E
Var
Stri
ng||
EExVap
rr: S+
t:Ein
xgpr
||EEx
xpprr
:−:
EExxpp
rr||
EExxpp
rr::−×
::EEx
xpprr
||EEx
xpprr
::×/::
EExxpp
rr
The
dom
ain
reas
oner
cont
aine
dfu
nctio
nalit
yfo
rr e
writ
ing
linea
req
uatio
nsan
dm
athe
mat
ical
expr
essi
ons
that
are
used
ineq
uatio
ns
mod
ule
E q
uatio
nsR
ewrit
eAna
lysi
sw
here
equa
tions
Rew
rite
=..
.ex
prR
ewrit
e=
...
equa
tions
Subs
t=
...
expr
Subs
t=
...
ina
449-
line
mod
ule.
Inou
rcu
rren
tdo
mai
nre
ason
er,
the
dom
ain
of
linea
rex
-pr
essi
ons
does
not
cont
ain
spec
ific
func
tiona
lity
for
r ew
ritin
gan
dsu
bstit
utio
nan
ym
ore.
Inst
ead,
gene
ricfu
nctio
nsfo
rr e
writ
ing
and
subs
titut
ion
are
calle
dw
hen
the
gene
ricdo
mai
nre
ason
eris
u se
don
linea
rex
pres
sion
s.Th
ege
neric
r ew
ritin
gan
dsu
bstit
utio
nfu
nctio
nsth
emse
lves
are
impl
emen
ted
inte
rms
ofb
asic
gene
ricfu
nc-
tions
,su
chas
f old
,zi
p,cr
ush
and
map
[18]
.A
llin
all,
the
gene
ricm
odul
esfo
rre
writ
ing
cons
ist
of
370
lines
.Th
ege
neric
r ew
ritin
gfu
nctio
nalit
yis
used
inal
lot
her
dom
ains
asw
ell.
Ther
ear
ese
vera
lw
ays
inw
hich
gene
ricpr
ogra
mm
ing
can
b eus
edin
our
dom
ain
r eas
oner
s:
•To
obta
inin
stan
ces
of fu
nctio
nsw
hich
natu
rally
depe
ndon
the
stru
ctur
eof
the
data
type
,isnu
schta
astsh
oere
writ
ing
sfw unh
citcihon
n ,abtuu
rtaaall
lysod
eb paes
incddfu
o nnct
htioens
tslik
ef o
lda th
ndem
ap.
•To
use
gene
rictra
vers
alfu
nctio
nsto
appl
yso
me
func
tion
ata
parti
cula
rpo
si-
Ttioon
usine
apo
ssib
lyla
rge
lvf auln
ucetoio
fna
fam
ilyloy
fdat
atyp
es.
•To
not
have
toad
just
man
yfu
nctio
nsw
hen
exte
ndin
gor
r ede
finin
ga
dom
ain.
We
h av
eus
edge
neric
func
tions
for
allo
fth
ese
p ur
pose
s.
1.3
GE
NE
RIC
PR
OG
RA
MM
ING
LIB
RA
RIE
S
Ther
ear
ese
vera
llib
rarie
sfo
rge
neric
p ro
gram
min
gin
Has
kell.
T he
sedi
ffer
sub-
stan
tially
inex
pres
sive
ness
,da
taty
pes
supp
orte
dan
dus
abili
ty.
Rec
ently
,a
de-
taile
dco
mpa
rison
of
gene
ricpr
ogra
mm
ing
libra
ries
has
b ee
npe
rform
ed[2
1,23
].W
eh
ave
used
this
com
paris
onw
hen
choo
sing
age
neric
prog
ram
min
glib
rary
.
1.3.
GE
NE
RIC
PR
OG
RA
MM
ING
LIB
RA
RIE
S5
1.3.
1D
atat
ypes
inth
edo
mai
nre
ason
ers
Our
dom
ain
reas
oner
scu
rren
tlyr e
pres
ent
seve
ral
dom
ains
,su
chas
logi
cex
pres
-si
ons,
rela
tion
alge
bra
and
prog
ram
min
gin
asi
mpl
efu
nctio
nal
lang
uage
.E
ach
dom
ain
isre
pres
ente
dby
one
or
mor
eda
taty
pes.
Topr
ovid
ege
neric
func
tiona
lity
ona
dom
ain,
we
h av
eto
guar
ante
eth
atth
eda
taty
pes
r epr
esen
ting
the
dom
ain
can
betre
ated
gene
rical
ly.
D if
fere
ntlib
rarie
sfo
rge
neric
prog
ram
min
gha
vedi
ffere
ntle
vels
of
supp
ort
for
data
type
s.O
urda
taty
pes
h av
edi
ffere
ntle
vels
of c
ompl
exity
.Th
edo
mai
nofl
ogic
expr
es-
sion
s,fo
rin
stan
ce,
isre
pres
ente
db
ya
sing
le,
r ecu
rsiv
e,no
n-pa
ram
etric
data
type
:
data
Logi
c=
Logi
c:→
:Lo
gic
--im
plic
atio
n=|
Logi
c:↔
:Lo
gic
--eq
uiva
lenc
e||
LLoogg
iicc::∧↔
:LLo
oggiicc
--co
njun
ctio
n(a
nd)
||Lo
gic
::∧∨::
Logi
c--
disj
unct
ion
(or)
||LN
oogtL
ogic
--ne
gatio
n(n
ot)
||VNa
ortLS
otrginic
g--
varia
bles
||VTa
--tru
e
||T F
--fa
lse
The
dom
ains
of
rela
tion
alge
bra,
p ro
gram
min
gan
dca
lcul
atin
gw
ithfra
ctio
nsar
ere
pres
ente
dby
data
type
sof
sim
ilar
com
plex
ity.
Our
repr
esen
tatio
nof
mat
hem
atic
alex
pres
sion
sus
eslis
ts:
data
Expr
=E
xpr
:+:
Exp
r--
addi
tion
|E
xpr
:×:E
xpr
--m
ultip
licat
ion
||EEx
xpprr
:−:
EExxpp
rr--
subt
ract
ion
||N
egat
e:E E
xpxpr
--ne
gatio
n||
ENxegp
rat:/e
:EExx
pprr
--di
visi
on||
Sqr
tE :/
x:pE r
--sq
uare
root
||N S
qarttI
nteg
er--
natu
ral
num
ber
||N E
VaatrI n
nStetg
rinerg
--va
riabl
e||
Sym
rSS trti
rningg
[Exp
r]--
sym
bol
Due
toth
eir
p er
vasi
vene
ssin
Has
kell,
lists
shou
ldbe
give
na
p ar
ticul
artre
atm
ent
byth
ege
neric
prog
ram
min
glib
rary
.F
orin
stan
ce,
we
expe
ctth
eal
gebr
afo
rth
eSy
mco
nstru
ctor
of
the
Exp
rda
taty
peto
have
type
algS
ym::
Str
ing
→[r
]→
r.
This
will
not
h ap
pen
if,f
or
inst
ance
,w
etre
atth
e[E
xpr]
occu
rren
ceas
aco
nsta
ntty
pe. Th
edo
mai
no
fsy
stem
so
flin
ear
equa
tions
uses
afa
mily
of
p ar
amet
ricda
ta-
type
s: type
Line
arS
yste
m=
[Equ
atio
nE
xpr]
data
Equ
atio
na
=a
:=:
a
6C
HA
PTE
R1.
GE
NE
RIC
PR
OG
RA
MM
ING
FO
RD
OM
AIN
RE
AS
ON
ER
S
We
need
the
para
met
ricity
beca
use
we
can
r epr
esen
teq
uatio
nsof
diffe
rent
k in
dsof
expr
essi
ons.
The
dom
ain
for
p ol
ynom
ials
uses
anab
stra
ctda
taty
pe:
new
type
Pol
ynom
ial
a=
P(I
ntM
apa)
Usi
ngab
stra
ctda
taty
pes
incr
ease
sth
eef
ficie
ncy
and
conc
isen
ess
of
the
code
,bu
tpr
eclu
des
the
appl
icat
ion
of
gene
ricpr
ogra
mm
ing
tech
niqu
es.
The
expr
essi
vene
ssof
age
neric
p ro
gram
min
glib
rary
isor
thog
onal
toits
sup-
port
for
data
type
s.So
me
libra
ries
can
expr
ess
mor
ege
neric
func
tions
than
othe
rs.
For
our
dom
ain
reas
oner
s,w
ene
edfu
nctio
nalit
ysu
chas
rew
ritin
g,da
tage
nera
-tio
nan
dfo
ldin
g.Th
iscl
early
r est
ricts
our
choi
ceto
libra
ries
whi
chha
vea
notio
nof
the
recu
rsiv
est
ruct
ure
of
the
data
type
s:U
nipl
ate,
Reg
ular
and
M u
ltire
c.W
eha
veus
edea
chof
thes
eth
ree
libra
ries
atdi
ffere
ntst
ages
inou
rdo
mai
nr e
ason
ers.
We
will
give
abr
ief
over
view
of
each
of
the
libra
ries
and
our
reas
ons
for
u si
ngth
em.
1.3.
2U
nipl
ate
The
Uni
plat
elib
rary
[17]
prov
ides
com
bina
tors
for
gene
rictra
vers
als.
Itis
sim
ilar
inst
yle
toS
YB
,bu
tsi
mpl
ersi
nce
itdo
esno
tus
era
nk-2
poly
mor
phis
m.
Inth
esi
mpl
ified
vers
ion
we
use
for
our
dom
ain
r eas
oner
s,it
cons
ists
ofa
sing
lefu
nctio
nin
aty
pecl
ass:
clas
sU
nipl
ate
aw
here
unip
late
::a→
([a],
[a]→
a)
The
unip
late
func
tion
prov
ides
aw
ayto
acce
ssth
eim
med
iate
child
ren
of
ate
rm(th
efir
stco
mpo
nent
of th
er e
sult
p ai
r)an
dto
rebu
ilda
term
usin
ga
list
of re
plac
e-m
ent
child
ren
(the
seco
ndco
mpo
nent
of th
ep
air).
Eve
ryty
peto
beus
edw
ithth
islib
rary
has
tob e
give
nan
inst
ance
of
the
clas
sU
nipl
ate.
W e
show
part
of
the
inst
antia
tion
for
the
Logi
cda
taty
pe:
inst
ance
Uni
plat
eLo
gic
whe
reun
ipla
te(p
:→
:q)
=([
p,q]
,λ[p
,q]→
p:→
:q)
unip
late
((pp:↔
:q)
==..
...
.
The
othe
rco
nstru
ctor
sha
vea
sim
ilar
defin
ition
.G
ener
icfu
nctio
nsin
Uni
plat
ear
ew
ritte
nus
ing
the
unip
late
com
bina
tor.
The
func
tion
trans
form
b el
ow,
for
i nst
ance
,ap
plie
sa
trans
form
atio
nfu
nctio
non
ever
ysu
bele
men
to
fa
term
ina
botto
m-u
pfa
shio
n:
trans
form
::(U
nipl
ate
a)⇒
(a→
a)→
a→
atr
ansf
orm
fa
=n
ilpelta
(tceh a
i)ld⇒ re
n( ,a
ab→u il
da e)r
→)→=
aun
ipla
tea
in f
$bu
ilder
$m
ap(t
rans
form
f )ch
ildre
n
1.3.
GE
NE
RIC
PR
OG
RA
MM
ING
LIB
RA
RIE
S7
Inth
isfu
nctio
n,ch
ildre
nst
ands
for
the
r ecu
rsiv
ech
ildre
nof
the
inpu
tte
rm,
and
build
eris
the
func
tion
that
,giv
ena
list
ofr
epla
cem
ent
child
ren,
rebu
ilds
the
inpu
tte
rmb
yre
plac
ing
itsch
ildre
nw
ithel
emen
tsfr
omth
elis
t.B
oth
are
com
bine
dto
appl
yth
etra
nsfo
rmat
ion
funct
ionf
toev
ery
r ecu
rsiv
eel
emen
toft
hein
put
term
.W
eus
edU
nipl
ate
asth
efir
stge
neric
p ro
gram
min
glib
rary
inou
rdo
mai
nre
a-so
ners
.Its
sim
plic
ityal
low
edfo
rea
syin
tegr
atio
nan
dim
plem
enta
tion
offu
nctio
n-al
itysu
chas
top-
leve
leq
ualit
yan
da
form
of
rew
ritin
g.H
owev
er,
func
tiona
lity
such
asda
tage
nera
tion
or
fold
ing
cann
otb e
expr
esse
din
Uni
plat
e.
1.3.
3M
ulti
rec
Mul
tirec
[22]
isa
rece
ntly
deve
lope
dge
neric
prog
ram
min
glib
rary
.It
has
the
uniq
ueab
ility
tore
pres
ent
mut
ually
r ecu
rsiv
eda
taty
pes
ina
fixed
-poi
ntvi
ew,
ther
efor
eal
low
ing,
for
inst
ance
,th
ede
finiti
onof
gene
ricfo
lds
onm
utua
llyre
-cu
rsiv
eda
taty
pes.
Itis
base
don
the
r epr
esen
tatio
nof
the
patte
rnfu
ncto
rof
afa
mily
of
type
s,us
ing
type
fam
ilies
.C
onsi
der
apo
ssib
ler e
pres
enta
tion
for
the
dom
ain
of
prog
ram
min
gin
asi
mpl
efu
nctio
nal
lang
uage
:
data
LExp
r=
L am
bda
Stri
ngLE
xpr
|L
Var
Stri
ng||
A pp
lyLtE
rixngpr
LExp
r||
AF pixp
LlyE LxE
pxr||
FL iInx
tLIEn
tx||
LL Ient
tD Ien
tclLE
xpr
data
Dec
l=
Stri
ng:=
LExp
r|
Seq
Dec
lD
ecl
Toill
ustra
teth
ege
neric
r epr
esen
tatio
nus
edby
Mul
tirec
,w
esh
owth
epa
ttern
func
tor
ofth
isfa
mily
ofm
utua
llyre
curs
ive
data
type
s:
data
AS
T::
∗→
∗w
here
LtaE Axp
SrT::
:A:∗S
→TL ∗E
wxphre
Dec
l::
AS
TD
ecl
type
inst
ance
PF
AS
T=
(C
Lam
bda
(KS
tring
:×:I
LExp
r):+
:CC
LLVama
rb((K
KSS
ttrriinn
gg) :
:+:
CA
pply
(ILE
xpr
:×:I
LExp
r):+
:C
Fix
((IILLE
Exxppr
r):+
:C
LInt
(KIn
t)
:+:
CLe
t(I
Dec
l:×
:ILE
xpr)
):.
:LE
xpr
:+:
(C
Ass
ign
(KS
trin
g:×
:ILE
xpr)
:+:
CSe
q((IK
D Setc
ril:×
:IILDE
exclp)r
):.
:D
ecl
Her
e,A
ST
isan
auxi
liary
data
type
defin
edto
r epr
esen
tth
efa
mily
.In
itspa
ttern
func
tor
PF,
sum
s(:
+:)
and
prod
ucts
(:×
:)en
code
the
choi
cebe
twee
nal
tern
ativ
e
8C
HA
PTE
R1.
GE
NE
RIC
PR
OG
RA
MM
ING
FO
RD
OM
AIN
RE
AS
ON
ER
S
cons
truct
ors
and
mul
tiple
argu
men
tst o
aco
nstru
ctor
,r e
spec
tivel
y.C
onst
ruct
ors
are
enco
ded
b y
C,c
onst
ant
type
s(o
utsi
deth
efa
mily
)b
yK
and
recu
rsiv
eoc
cur-
renc
es(ta
gged
with
the
recu
rsiv
et y
pe)
b y
I.A
grou
pof
alte
rnat
ives
belo
ngin
gto
asi
ngle
data
type
ista
gged
with
:.:.
Als
one
cess
ary
are
the
conv
ersi
onfu
nctio
nsto
and
from
the
orig
inal
data
type
sin
toth
ege
neric
view
:
inst
ance
Fam
AS
Tw
here
from
LExp
r(L
ambd
as
e)=
L(T
ag(L
(C(K
s:×
:I(I∗
e)))
))fr
om..
.
toLE
xpr
(L(T
ag(L
(C(K
s:×
:I(I∗
e))
))))
=L
ambd
as
ettoo
...
Not
eth
atth
ere
curs
ive
occu
rrenc
esne
edto
b ew
rapp
edin
the
I ∗co
nstru
ctor
be-
caus
eof
the
high
erki
ndof
theI
cons
truct
or.
We
used
Mul
tirec
inou
rse
cond
vers
ion
of th
ege
neric
dom
ain
reas
oner
sto
in-
crea
seth
eex
pres
sive
ness
.W
ith
Mul
tirec
we
can
writ
ege
neric
fold
san
dda
taty
pege
nera
tion,
asw
ella
san
impr
oved
r ew
ritin
gm
echa
nism
.H
owev
er,
Mul
tirec
can-
not
repr
esen
tpa
ram
etric
data
type
ssu
chas
Equ
atio
n,or
data
type
sw
ithlis
tssu
chas
Exp
r(S
ectio
n1.
3.1)
3.Th
isre
duce
sth
enu
mbe
rof
dom
ains
onw
hich
we
can
use
gene
ricp
rogr
amm
ing.
1.3.
4R
egul
ar
The
Reg
ular
libra
ryis
the
unde
rlyin
gm
echa
nism
of
the
rew
ritin
gfra
mew
ork
de-
scrib
edby
Van
Noo
rtet
al.
[18]
.It
can
bese
enas
asi
mpl
ifica
tion
of
Mul
tirec
,re
pres
entin
ga
sing
lety
peat
atim
e:
type
inst
ance
PF
Logi
c=
CIm
pl(I
:×:I
)--
impl
icat
ion
: +=:C
CIEm
qpuliv
((II:×
:II))
--eq
uiva
lenc
e::++
::CC
CEoqu
nji((II
:×:I
I))--
conj
unct
ion
(and
)::++
::CC
DCoisn
j((II
:×:I
I))--
disj
unct
ion
(or)
:+:
CN
otI(
--ne
gatio
n(n
ot)
:+:
CV
ar(K
Str
ing)
--va
riabl
es:+
:C
TU
--tru
e:+
:C
FU
--fa
lse
inst
ance
Reg
ular
Logi
cw
here
from
(p:→
:q)
=L
(I p
:×:I
q)fr
om..
.
toL
(I p
:×:I
q)=
p:→:
qttoo
...
3As
we
men
tione
dp
revi
ousl
y,w
ew
ant
totre
atlis
tsas
asp
ecia
lca
se.
We
coul
dre
pre-
sent
ada
taty
pew
ithlis
tsin
Mul
tirec
b y
cons
ider
ing
a(m
onom
orph
ic)
list
data
type
asbe
ing
part
of
the
fam
ily,
but
this
chan
ges
the
sem
antic
so
fth
ege
neric
f unc
tions
.F
orex
ampl
e,th
eal
gebr
asfo
rth
efo
ldb
ecom
ele
ssna
tura
l.
1.4.
QU
ALI
TY
CH
AR
AC
TE
RIS
TIC
S9
The
patte
rnfu
ncto
rre
pres
enta
tion
issi
mpl
erth
anin
Mul
tirec
,si
nce
we
nolo
nger
requ
ireta
ggin
gof
recu
rsiv
eoc
curr
ence
s.R
egul
ardo
esno
tsu
ppor
tpa
ram
etric
data
type
sor
data
type
sco
ntai
ning
lists
.H
owev
er,
we
h av
ein
vest
igat
edth
epo
ssib
ility
of
chan
ging
the
libra
ryto
add
sup-
port
forp
aram
eter
s[1
0]an
dfu
ncto
rco
mpo
sitio
n(o
fwhi
chda
taty
pes
with
lists
are
asp
ecia
lca
se).
Bot
hex
tens
ions
seem
poss
ible
and
dono
tle
adto
muc
hin
crea
sed
code
com
plex
ity.
Afte
rh
avin
gus
edM
ulti
rec,
we
real
ized
our
dom
ain
reas
oner
sw
ould
bene
fitm
ore
from
supp
ortin
gpa
ram
eter
san
dda
taty
pes
with
lists
than
from
supp
ortin
gfa
mili
es.
Mor
eove
r,th
eR
egul
arlib
rary
isea
sier
tou
se.
1.4
QU
ALIT
YC
HA
RA
CT
ER
IST
ICS
The
ISO
9126
inte
rnat
iona
lst
anda
rdin
trodu
ces
aqu
ality
mod
elfo
rth
eev
alua
tion
ofs
oftw
are
qual
ity.
This
mod
ellis
tssi
xch
arac
teris
ticar
eas
ofi
mpo
rtanc
e.W
eus
eth
isse
tof
char
acte
ristic
sto
stru
ctur
eou
rdi
scus
sion
onth
equ
ality
char
acte
ristic
sof
our
dom
ain
reas
oner
s.
Fun
ctio
nalit
y:W
eha
vesp
litoff
seve
ral
com
pone
nts
from
the
indi
vidu
aldo
-m
ain
reas
oner
sto
b ere
plac
edby
gene
ricco
mpo
nent
s.Th
isha
sm
ade
itcl
ear
whi
chfu
nctio
nalit
yis
com
mon
toal
ldom
ains
,an
dw
hich
func
tiona
lity
issp
ecifi
cto
ap
artic
ular
dom
ain.
Exa
mpl
esof
gene
ricco
mpo
nent
sar
eth
ef o
ldfu
nctio
n,fu
nctio
nsfo
rre
writ
ing
and
mat
chin
g,fu
nctio
nsfo
rge
nera
ting
arbi
trary
term
s,et
c.S
plitt
ing
off
gene
ricco
mpo
nent
sle
ads
toan
incr
ease
dse
para
tion
of
con-
cern
s:im
prov
emen
tsin
rew
ritin
gfu
nctio
nalit
yar
eso
lved
inth
er e
writ
ing
com
-po
nent
and
lead
toim
prov
emen
tsin
r ew
ritin
gfo
rev
ery
dom
ain.
Furth
erm
ore,
the
func
tiona
lity
oft
hege
neric
com
pone
nts
can
now
beus
edby
othe
rap
plic
atio
nsas
wel
l.In
addi
tion
toth
ege
neric
com
pone
nts
ther
eis
dom
ain-
spec
ific
func
tiona
l-ity
,fo
rex
ampl
efu
nctio
nalit
yfo
rde
term
inin
gth
ese
man
tics
of
anex
pres
sion
inth
edo
mai
n.O
urfir
stge
neric
varia
ntoft
hedo
mai
nr e
ason
ers
u se
dU
nipl
ate
for
the
gene
ricfu
nctio
nalit
y.U
nipl
ate
allo
wed
usto
spec
ifyre
writ
ing
and
trave
rsal
sge
neric
ally
.H
owev
er,
itssi
mpl
icity
limits
itsfu
nctio
nalit
y.In
Uni
plat
eon
eca
nnot
writ
ege
neric
func
tions
that
b ui
ldva
lues
from
scra
tch.
Inbo
thM
ultir
ecan
dR
egul
arw
eca
nw
rite
gene
ricQ
uick
Che
ck[4
]A rb
itrar
yin
stan
ces,
whi
char
eus
edto
gen-
erat
eex
erci
ses
and
test
data
,w
here
asus
ing
Uni
plat
ew
eca
nnot
.U
nipl
ate
also
cann
otde
fine
the
gene
ricfo
ld,
orev
ento
p-le
vel
equa
lity.
Bot
hM
ultir
ecan
dR
eg-
ular
supp
ort
the
defin
ition
of
thes
efu
nctio
ns.
Mul
tirec
supp
orts
fam
ilies
of
(pos
sibl
ym
utua
llyr e
curs
ive)
data
type
s.W
eha
vefa
mili
esof
data
type
sin
som
eof
our
dom
ain
reas
oner
s,bu
tcu
rren
tlyw
edo
not
need
acce
ssto
the
recu
rsiv
est
ruct
ure
of
afa
mily
of
type
s:in
gene
ral,
we
can
focu
son
asi
ngle
type
and
treat
allt
heot
her
occu
rrin
gty
pes
asco
nsta
nts.
For
inst
ance
,fo
rth
edo
mai
nre
ason
erof
syst
ems
of
linea
req
uatio
nsde
scrib
edin
Sec-
tion
1.3.
1w
eon
lyha
vere
writ
eru
les
for
the
Equ
atio
nsan
dno
tfo
rLi
near
Sys
tem
s.H
owev
er,
havi
ngsu
ppor
tfo
rfa
mili
esof
data
type
sm
ight
b ede
sira
ble
inge
nera
l.
10C
HA
PTE
R1.
GE
NE
RIC
PR
OG
RA
MM
ING
FO
RD
OM
AIN
RE
AS
ON
ER
S
We
coul
dde
scrib
eou
rdo
mai
nfo
rsi
mpl
efu
nctio
nal
p ro
gram
min
gb
yth
eLE
xpr
and
Dec
lda
taty
pes
show
nin
Sec
tion
1.3.
1.N
eith
erU
nipl
ate
nor
Reg
ular
can
rep-
rese
ntth
ere
writ
eru
lefo
rin
linin
gin
this
dom
ain,
Let
(x:=
e)(L
Var
x )
e,si
nce
itre
quire
sm
atch
ing
of
b ot
hex
pres
sion
san
dde
clar
atio
ns.
Cur
rent
ly,
our
dom
ain
for
sim
ple
func
tiona
lpr
ogra
mm
ing
treat
sle
tde
clar
atio
nsas
synt
actic
suga
r,an
dw
eth
eref
ore
dono
tne
edth
eab
ove
mut
ually
recu
rsiv
eda
taty
pes.
Usi
ngM
ultir
ec,
how
ever
,w
ould
allo
wfo
rsu
cha
chan
gein
the
futu
re.
The
curr
ent
vers
ion
ofM
ultir
echa
sno
supp
ort
for
type
varia
bles
.Th
edo
mai
nre
ason
erfo
rsy
stem
sof
equa
tions
uses
para
met
eriz
edda
taty
pes,
asw
eha
vese
enin
Sec
tion
1.3.
1.W
eex
pect
supp
ort
for
para
met
ricda
taty
pes
tob e
avai
labl
ein
Mul
tirec
soon
,bu
tfo
rno
ww
eca
nnot
use
iton
this
p ar
ticul
ardo
mai
nre
ason
er.
For
Reg
ular
,th
eex
tens
ion
nece
ssar
yto
supp
ort
p ar
amet
ricda
taty
pes
ism
ore
orle
ssst
raig
htfo
rwar
d,an
dde
scrib
edin
deta
ilb
yH
esse
link
[10]
.
Rel
iabi
lity:
The
gene
ricdo
mai
nre
ason
ers
are
mor
ere
liabl
eth
anth
eor
igin
aldo
-m
ain
r eas
oner
s.O
ner e
ason
isth
atth
ege
neric
dom
ain
reas
oner
sm
ake
mor
eus
eof
libra
ryco
de,
whi
chis
gene
rally
mor
ere
liabl
eth
anda
taty
pe-s
peci
ficco
de.
Fur
-th
erm
ore,
the
orig
inal
dom
ain
r eas
oner
sco
ntai
ned
som
ead
h oc
,an
dso
met
imes
dang
erou
s,so
lutio
ns,
whi
chha
vebe
enre
mov
edin
the
gene
ricdo
mai
nre
ason
ers.
An
exam
ple
of
ada
nger
ous
solu
tion
isth
ew
ayin
whi
chm
etav
aria
bles
wer
ein
trodu
ced
and
used
inth
eor
igin
aldo
mai
nre
ason
ers.
Tode
scrib
ere
writ
ing
r ule
son
the
dom
ains
,th
eva
riabl
esof
the
dom
ains
them
selv
esw
ere
u se
das
met
avar
i-ab
les.
This
was
impl
emen
ted
usin
ga
type
clas
s:
clas
sM
etaV
ara
whe
rem
etaV
ar::
Int→
ais
Met
aVar
::a
→→
Maa
ybe
Int
The
inst
antia
tion
for
the
dom
ain
of
logi
cex
pres
sion
s,fo
rin
stan
ce,
r epr
esen
ted
met
avar
iabl
esas
varia
bles
with
apa
rticu
lar
nam
e:
inst
ance
Met
aVar
Logi
cw
here
met
aVar
n=
Var
("_"
++
show
n)
isM
etaV
ar(V
ar(’
_’:s
))|¬
(nul
ls)
∧a
llis
Dig
its
=ar
r( e’tu
_r’n
:(sr)e)a
|d ¬¬s)
(is
Met
aVar=
Not
hing
Oth
erdo
mai
nsha
dsi
mila
rin
stan
ces.
This
led
toth
epo
ssib
lyda
nger
ous
situ
-at
ion
inw
hich
ado
mai
n-le
vel
varia
ble
coul
db e
conf
used
with
ar e
writ
ing
r ule
met
avar
iabl
e.A
dditi
onal
ly,
this
only
wor
ksfo
rdo
mai
nsw
hich
have
ano
tion
of
varia
bles
.Th
eab
ove
solu
tion
ispa
rtof
the
dom
ain
reas
oner
sth
atus
eU
nipl
ate.
Usi
ngei
ther
Mul
tirec
orR
egul
ar,
we
repr
esen
tm
etav
aria
bles
atth
ege
neric
stru
ctur
ere
p-re
sent
atio
nle
vel.
The
type
syno
nym
Sche
me
tr e
pres
ents
anal
tern
ativ
e(in
dica
ted
byth
esu
m:+
:)be
twee
nth
et
type
and
am
etav
aria
ble:
1.4.
QU
ALI
TY
CH
AR
AC
TE
RIS
TIC
S11
type
Sche
me
t=
KM
Var
:+:
PF
tty
peM
Var
=In
t
The
r ew
ritin
gfu
nctio
nsno
wop
erat
eon
Sche
me
tin
stea
doft,
guar
ante
eing
that
nom
etav
aria
bles
can
esca
pefr
omth
er e
writ
ing
mec
hani
sm.
Sche
me
tad
dsa
new
case
toth
epa
ttern
func
tor
of
t ,na
mel
ya
cons
tant
case
for
inte
gers
.Th
isal
solif
tsth
ere
stric
tion
that
the
dom
ain
has
toh
ave
som
eso
rtof
var
iabl
eco
nstru
ctor
:ev
ery
type
tfo
rw
hich
we
h av
ea
patte
rnfu
ncto
rca
nno
wbe
exte
nded
toSc
hem
et.
Usa
bilit
y:F
rom
the
p er
spec
tive
of
ap
rogr
amm
erw
hois
addi
nga
new
dom
ain,
cust
omiz
ing
our
dom
ain
r eas
oner
sis
muc
hea
sier
usin
gge
neric
prog
ram
min
g.W
eon
lyne
edto
spec
ifyth
eab
stra
ctsy
ntax
for
the
dom
ain
and
itsse
man
tics-
depe
nden
top
erat
ions
(par
sing
and
pret
ty-p
rintin
g,re
writ
ing
rule
s,as
soci
ativ
ean
dco
mm
utat
ive
oper
ator
s,an
dth
est
rate
gies
).W
ege
tfu
nctio
nalit
ysu
chas
fold
ing,
top-
leve
leq
ualit
y,r e
writ
ing
with
type
-saf
em
etav
aria
ble
exte
nsio
n,tra
vers
als
and
exer
cise
gene
ratio
nfo
rfre
e.W
ithge
neric
prog
ram
min
gw
eal
soge
ta
h ig
her
degr
eeof
com
posa
bilit
y.If
we
have
spec
ified
the
dom
ains
of
ratio
nals
,flo
ats
and
mat
rices
,th
enco
mbi
ning
them
toob
tain
mat
rices
of
r atio
nals
orm
atric
esof
float
s,w
ithth
eco
rrec
tse
tof
rew
rite
rule
s,re
quire
sno
effo
rt.
Effi
cien
cy:
Gen
eric
prog
ram
ste
ndt o
bele
ssef
ficie
ntt h
anth
eir
hand
writ
ten
coun
terp
arts
.C
ompi
lers
dono
tal
way
spe
rfor
mth
ene
cess
ary
optim
izat
ions
to
rem
ove
the
gene
ricr e
pres
enta
tion
t ype
san
dco
nver
sion
func
tions
from
the
com
-pi
led
code
[16]
.W
hile
this
has
b ee
nsh
own
tob e
poss
ible
inat
leas
tth
eap
proa
chof
Alim
arin
ean
dSm
etse
rs[1
],in
the
gene
ral
case
itre
mai
nsan
open
prob
lem
.R
odrig
uez
Yaku
shev
[21]
perfo
rms
asi
mpl
eb
ench
mar
kof
gene
ricpr
ogra
m-
min
glib
rarie
s.Th
etw
oU
nipl
ate
func
tions
test
edar
eap
prox
imat
ely
five
times
slow
erth
anth
eir
hand
writ
ten
coun
terp
arts
.F
orM
ulti
rec,
perfo
rman
ceva
ries
be-
twee
n1.
7an
d47
times
slow
er.
Rew
ritin
gin
Reg
ular
was
benc
hmar
ked
byV
anN
oort
etal
.[1
8],
reve
alin
gth
atit
is,
inge
nera
l,th
ree
tofo
urtim
essl
ower
than
type
-spe
cific
rew
ritin
g.M
ore
r ece
ntly
,M
agalh
˜a e
set
al.
[16]
show
that
stim
ulat
-in
gin
linin
gin
GH
Cca
ndr
amat
ical
lyin
crea
seth
ep
erfo
rman
ceof
Reg
ular
,bu
tno
tof
Mul
tirec
.H
owev
er,
effic
ienc
yis
gene
rally
not
apr
oble
mfo
rou
rdo
mai
nr e
ason
ers.
Sin
ceth
ete
rms
r epr
esen
tex
erci
ses
tobe
wor
ked
with
byhu
man
s,th
eyte
ndto
besm
all.
Hen
ce,
nope
nalty
isno
ticea
ble
for
usin
gan
yof
the
gene
ricve
rsio
ns.
Nev
erth
eles
s,th
isco
uld
pose
apr
oble
mfo
rot
her
larg
eap
plic
atio
ns,
such
asab
-st
ract
synt
axtre
etra
vers
als
onco
mpi
lers
.
Main
tain
abili
ty:
Mai
ntai
ning
the
code
of
the
new
dom
ain
reas
oner
sis
easi
er:
not
only
isth
ere
less
code
tom
aint
ain,
butt
heco
deis
also
mor
ecl
early
stru
ctur
ed.
Initi
ally
,w
ede
velo
ped
ana
ive
gene
ricA
rbitr
ary
gene
rato
rth
atdi
dno
tta
kete
rmsi
zeorc
onst
ruct
orfre
quen
cyin
toac
coun
t.L
ater
we
impr
oved
the
impl
emen
tatio
n
tota
keth
ese
into
acco
unt.
Sin
ceth
ege
nera
tor
isge
neric
,it
ison
lyde
fined
once
,
12C
HA
PTE
R1.
GE
NE
RIC
PR
OG
RA
MM
ING
FO
RD
OM
AIN
RE
AS
ON
ER
S
and
allt
hedo
mai
nsim
med
iate
lybe
nefit
from
the
adva
ntag
es.
W it
hha
ndw
ritte
nin
stan
ces,
we
wou
ldha
veha
dto
goth
roug
hea
chdo
mai
nim
plem
entin
gth
esa
me
impr
ovem
ent
ever
ywhe
re.
Inpa
rticu
lar,
repe
titiv
eco
defra
gmen
tslik
eA
rbitr
ary
and
CoA
rbitr
ary
in-
stan
ces,
fold
s,et
c.le
adto
copy
-and
-pas
tep
rogr
amm
ing,
whi
chca
ngi
ver i
seto
subt
leb
ugs.
By
usin
gge
neric
func
tions
we
can
mov
ea
lot
oft
estin
gto
the
gene
ricpr
ogra
mm
ing
libra
ry(a
sw
ell
asre
duci
ngit)
.W
elo
oked
atth
enu
mbe
rof
line
sofs
ourc
eco
defo
rde
finin
gth
elo
gic
dom
ain,
toge
ther
with
esse
ntia
lfu
nctio
nalit
ysu
chas
top-
leve
leq
ualit
y,ge
nera
tion
of
data
and
rew
ritin
g.Th
isis
asi
mpl
ebu
tre
pres
enta
tive
way
of
com
parin
gth
edo
mai
nre
ason
ers.
The
logi
cdo
mai
nis
not
chos
enbe
caus
eit
isp
artic
ular
lysu
ited
toim
prov
emen
t,bu
tsi
mpl
ybe
caus
eit
ison
eof
our
earli
erdo
mai
ns,
sow
eha
vea
non-
gene
ricdo
mai
nre
ason
erfo
rit.
Our
initi
al,
non-
gene
ricve
rsio
nco
nsis
ted
of
496
lines
of
code
,in
clud
ing
type
-spe
cific
code
for
azi
pper
and
hand
writ
ten
trave
rsal
sov
erth
edo
mai
n.Th
eU
nipl
ate
vers
ion,
whi
chus
edge
neric
trave
rsal
san
dr e
writ
ing,
cons
iste
dof
282
lines
of
code
.U
sing
Mul
tirec
,w
eca
nr e
mov
e
som
eof
the
rem
aini
ngty
pe-s
peci
ficco
defo
rr e
writ
ing,
and
get
thef
old
and
data
gene
ratio
nfu
nctio
nsfo
rfre
e,re
duci
ngth
eto
tall
ine
coun
tto
214,
less
than
50pe
rce
ntof
the
orig
inal
line
coun
t.Th
er e
duct
ion
with
Reg
ular
isth
esa
me,
sinc
eth
ese
libra
ries
have
very
sim
ilar
char
acte
ristic
s.Th
enu
mbe
rof
lines
of
sour
ceco
deis
not
nece
ssar
ilya
repr
esen
tativ
em
etric
,bu
tw
eth
ink
that
inou
rca
seit
faith
fully
expr
esse
sth
eso
urce
code
r edu
ctio
nw
eac
hiev
ed.
Inge
nera
l,a
data
type
with
nco
nstru
ctor
sw
illb e
defin
edin
nlin
esof
code
.U
sing
U n
ipla
te,
we
still
had
tow
rite
the
arbi
trar
y,co
arbi
trar
y,to
ple
vel
equa
lity,
f old
,an
dun
ipla
tefu
nctio
ns.
Eac
hof
thes
eis
gene
rally
defin
edby
patte
rn-m
atch
ing
onth
edo
mai
nda
taty
pe,
ther
efor
eth
eyre
quire
nlin
esea
ch.
Add
ition
ally
,w
eha
dto
defin
eth
eM
etaV
arin
stan
ce.
W it
hM
ultir
ecor
Reg
ular
,ea
chof
the
func
tions
whi
chre
quire
dn
lines
requ
ireon
elin
eon
ly,
and
ther
eis
nolo
nger
aM
etaV
arcl
ass.
We
only
need
toad
dfo
urlin
esof
code
toin
voke
the
Tem
plat
eH
aske
llge
nera
tion
of
gene
ricre
pres
enta
tions
.Tu
rnin
gou
rto
olin
toa
gene
ricpr
ogra
mha
sal
som
ade
the
code
inth
eto
olm
uch
mor
eco
nsis
tent
.W
eha
dto
bepr
ecis
eab
out
our
deci
sion
sin
the
gene
ricco
de,
soth
atal
lge
neric
func
tiona
lity
isco
ncep
tual
lyex
actly
the
sam
efo
rth
edi
ffere
ntdo
mai
ns.
F ur
ther
mor
e,th
ere
isno
wa
sing
leim
plem
enta
tion
for
man
yfu
nctio
nsw
hich
p re
viou
sly
had
one
impl
emen
tatio
npe
rdo
mai
n.Th
eon
lyne
gativ
eim
plic
atio
nin
mai
ntai
nabi
lity
is,p
ossi
bly,
the
incr
ease
dus
eof
adva
nced
Has
kell
p ro
gram
min
gte
chni
ques
inth
eco
de.
The
initi
al
vers
ion
of
the
dom
ain
reas
oner
sw
asw
ritte
nb
yre
lativ
ely
inex
perie
nced
Has
kell
prog
ram
-m
ers.
As
the
num
ber
of
dom
ains
grew
,m
aint
enan
ceof
the
proj
ect
was
hand
edov
erto
anex
perie
nced
Has
kell
prog
ram
mer
,w
hoin
trodu
ced
the
gene
ricve
rsio
nus
ing
U n
ipla
te.
The
vers
ion
u si
ngM
ultir
ecw
ould
prob
ably
not
h av
ebe
enpo
s-si
ble
with
out
clos
eco
ntac
tw
ithth
elib
rary
crea
tors
.Fo
rR
egul
ar,
we
h av
eha
dto
deve
lop
som
efe
atur
esou
rsel
ves
toin
crea
seth
eda
taty
pesu
ppor
tin
the
libra
ry.
We
susp
ect
that
only
ase
ason
edH
aske
llp
rogr
amm
erw
ithb
road
expe
rienc
ein
gene
ricpr
ogra
mm
ing
can
mai
ntai
nou
rcu
rren
tdo
mai
nre
ason
ers.
How
ever
,bi
g-
1.5.
CO
NC
LUS
ION
S13
ger
prog
ram
sna
tura
llyb
ecom
em
ore
com
plex
,an
dou
rdo
mai
nr e
ason
ers
now
offe
rm
uch
mor
efu
nctio
nalit
yth
anin
the
initi
al
vers
ion.
Nev
erth
eles
s,ca
reha
sto
beta
ken
toav
oid
prol
ifera
tion
of
gene
rics-
rela
ted
code
thro
ugho
utth
een
tire
code
b as
e.M
ultir
ec,
for
inst
ance
,in
trodu
ces
the
con-
cept
sof
‘fam
ilyty
pe’
and
‘ind
exty
pe.’
Inm
ost
othe
rge
neric
p ro
gram
min
gli-
brar
ies,
ther
eis
asi
ngle
gene
ricty
pe,
like
the
ain
Uni
plat
ea.
InM
ulti
rec,
we
have
Fam
ϕix
,with
ϕre
pres
entin
gth
efa
mily
and
ixa
parti
cula
rin
dex
onth
isfa
mily
.Th
isca
uses
prop
agat
ion
of
para
met
ers
inty
pecl
asse
s,an
din
crea
ses
the
over
all
com
plex
ityof
the
code
inth
edo
mai
nre
ason
ers.
For
inst
ance
,in
stea
dof
anex
er-
cise
onex
pres
sion
s(E
xerc
ise
Exp
r),
with
Mul
tirec
we
have
anex
erci
seon
the
ex-
pres
sion
type
inth
efa
mily
of
data
type
sfo
rex
pres
sion
s(E
xerc
ise
Exp
rFam
Exp
r).
How
ever
,fo
rm
any
dom
ains
the
fam
ilyofd
atat
ypes
cont
ains
asi
ngle
type
.Th
ere
isno
addi
tiona
lge
nera
lity,
but
the
fam
ilyty
peha
sto
b epr
opag
ated
thro
ugh
the
entir
eco
de.
This
redu
ces
the
over
all
r ead
abili
tyan
dm
aint
aina
bilit
yof
the
code
.
Por
tabi
lity:
The
orig
inal
dom
ain
r eas
oner
sw
ere
deve
lope
din
Has
kell
98[2
0],
whi
chm
ade
thei
rpo
rtab
ility
good
.W
hen
we
mov
edto
gene
ricdo
mai
nre
ason
-
ers
we
first
u se
dan
embe
dded
impl
emen
tatio
nof
Uni
plat
e.Th
isim
plem
enta
tion
was
asu
bset
of
the
Uni
plat
epa
ckag
eav
aila
ble
onH
acka
geD
B,
whi
chw
asen
-tir
ely
Has
kell
98,
soit
did
not
influ
ence
port
abili
ty.
Then
we
used
Mul
tirec
and
Reg
ular
,w
hich
intro
duce
depe
nden
cies
ona
num
ber
of
adva
nced
exte
nsio
nsof
Has
kell
(suc
has
type
fam
ilies
)on
lyav
aila
ble
inG
HC
,th
em
ain
com
pile
rfo
rH
aske
ll.Th
ese
exte
nsio
nsar
elia
ble
toch
ange
whe
nne
wve
rsio
nsoft
heco
mpi
ler
are
r ele
ased
,w
hich
mea
nsw
em
ight
have
toad
apt
our
code
.A
sa
cons
eque
nce,
port
abili
tyis
wor
se.
1.5
CO
NC
LUS
ION
S
The
use
of
gene
ricp
rogr
amm
ing
libra
ries
inou
rdo
mai
nre
ason
ers
has
give
nu s
incr
ease
dfu
nctio
nalit
y,re
liabi
lity,
usab
ility
and
mai
ntai
nabi
lity.
This
isdu
eto
acl
eare
rse
para
tion
of
conc
erns
,a
decr
ease
dnu
mbe
rof
lines
of
code
,hi
gher
com
-po
sabi
lity
and
mak
ing
itsi
mpl
erto
add
ane
wdo
mai
n.U
sing
gene
ricpr
ogra
m-
min
gpo
ssib
lyle
ads
tow
orse
perfo
rman
ce,
but
this
isno
tno
ticea
ble
for
user
s.U
sing
Mul
tirec
and
Reg
ular
we
depe
ndon
mor
ela
ngua
geex
tens
ions
and
can
only
com
pile
with
GH
C.
We
have
also
seen
that
usin
ga
sim
ple
gene
ricp
rogr
amm
ing
libra
ry(U
nipl
ate)
give
su s
aco
nsid
erab
leim
prov
emen
tin
our
code
qual
ity,
but
itev
entu
ally
be-
com
esa
limita
tion
inits
elf.
The
r ew
ritin
gm
echa
nism
,fo
rin
stan
ce,
requ
ired
the
dom
ains
toha
veva
riabl
es,
and
we
coul
dno
tw
rite
gene
ricp
rodu
cers
.M
ovi
ngto
mor
eco
mpl
exge
neric
prog
ram
min
glib
rarie
slif
ted
thes
ere
stric
tions
,bu
tm
ade
usgi
veu
pon
com
pile
rp
orta
bilit
y.W
ebe
lieve
that
the
use
of
gene
ricp
rogr
amm
ing
and
p ar
ticul
arly
of
Reg
ular
will
also
enab
leu s
toim
prov
eth
eea
seof
use
and
func
tiona
lity
of
the
dom
ain
reas
oner
sfu
rther
.W
ear
ecu
rren
tlylo
okin
gin
tore
writ
ing
expr
essi
ons
taki
ngin
to
14C
HA
PTE
R1.
GE
NE
RIC
PR
OG
RA
MM
ING
FO
RD
OM
AIN
RE
AS
ON
ER
S
acco
unt
the
asso
ciat
ivity
and
com
mut
ativ
ityof
(som
e)b
inar
yop
erat
ors.
This
isus
eful
for
two
reas
ons:
itre
duce
sth
enu
mbe
rof
rule
soft
hedo
mai
n4an
dit
sim
pli-
fies
reco
gniz
ing
the
appl
icat
ion
of ru
les
toa
term
b y
the
user
.S
ince
p re
tty-p
rinte
dex
pres
sion
sal
mos
tne
ver
cont
ain
unne
cess
ary
pare
nthe
ses,
au s
eris
gene
rally
un-
awar
eof
asso
ciat
ivity
.Fo
rin
crea
sed
r elia
bilit
yin
the
trave
rsal
s,w
ew
ant
tous
eth
ege
neric
zipp
er[1
2].
Ther
eis
anim
plem
enta
tion
of
azi
pper
for
Mul
tirec
u si
ngty
pe-in
dexe
dda
taty
pes
thro
ugh
type
fam
ilies
.Th
isim
plem
enta
tion
can
easi
lybe
conv
erte
dto
Reg
ular
.Th
isw
ould
repl
ace
the
curr
ent
unty
ped
appr
oach
usin
ga
list
of
inte
gers
tore
pres
ent
alo
catio
nin
ada
taty
pe.
We
also
p la
nto
look
into
the
gene
ricse
lect
ion
prob
lem
inth
eex
erci
seas
sist
ant:
the
user
shou
ldb e
able
tose
lect
asu
bexp
ress
ion
and
appl
ya
r ule
toth
atpa
rtof
the
expr
essi
on.
Apa
rtfr
omso
me
dom
ain-
spec
ific
info
rmat
ion
r ega
rdin
gth
ew
ayte
rms
are
p re
tty-p
rinte
dan
dw
hich
term
sar
ese
lect
able
,th
ispr
oble
mca
nal
sob e
solv
edw
itha
gene
rical
go-
rithm
.Th
eus
eof
gene
ricpr
ogra
mm
ing
tech
niqu
esin
our
dom
ain
reas
oner
spr
ovid
esus
with
ap
iece
of
softw
are
of
b et
ter
over
all
qual
ity.
The
adva
ntag
escl
early
out-
wei
ghth
edi
sadv
anta
ges,
and
the
effo
rtin
vest
edin
totra
nsla
ting
toa
gene
ricap
-
proa
chpa
yo
ffw
hen
addi
ngm
ultip
lene
wdo
mai
ns.
W e
expe
ctot
her
proj
ects
po-
tent
ially
tobe
nefit
from
the
appl
icat
ion
of
gene
ricpr
ogra
mm
ing
tech
niqu
esto
o.H
owev
er,
not
all
proj
ects
are
suita
ble
for
this
.In
our
case
,w
eha
vese
vera
ldo
-m
ains
whi
chre
quire
sim
ilar
func
tiona
lity.
Eac
hdo
mai
nis
repr
esen
ted
b y
one
orm
ore
data
type
s,on
top
of
whi
chth
efu
nctio
nalit
yis
impl
emen
ted.
Oth
erpr
ojec
tsw
hich
man
ipul
ate
seve
ral
data
type
s(e
.g.
stru
ctur
eed
itors
),or
perh
aps
asi
ngle
,la
rge
data
type
(e.g
.co
mpi
lers
),ar
eal
sow
ell-s
uite
dfo
rth
eap
plic
atio
nof
gene
ricpr
ogra
mm
ing
and
shou
ldbe
nefit
inth
esa
me
way
asou
rdo
mai
nre
ason
ers
did.
Ack
now
ledg
emen
ts:
This
wor
kha
sb
een
parti
ally
fund
edb
yth
ePo
rtugu
ese
Foun
datio
nfo
rSc
ienc
ean
dTe
chno
logy
(FC
T),
via
the
SF
RH
/BD
/359
99/2
007
gran
t,an
dby
the
Net
herla
nds
Org
aniz
atio
nfo
rS
cien
tific
Res
earc
h(N
WO
),th
roug
hits
proj
ect
on‘R
eal-l
ifeD
atat
ype-
Gen
eric
Pro
gram
min
g’(6
12.0
63.6
13).
W e
than
kA
ndre
sL¨o
hfo
rhi
she
lpw
ithus
ing
Mul
tirec
and
Erik
Hes
selin
kfo
rhi
sw
ork
onits
exte
nsio
n.A
lex
Ger
des
prov
ided
feed
back
onan
early
vers
ion
oft
his
pape
r.H
ar-
riePa
ssie
ran
dA
rthu
rva
nLe
euw
enco
ntrib
uted
tool
der
vers
ions
of
the
exer
cise
assi
stan
ts.
We
also
than
kth
ean
onym
ous
refe
rees
for
thei
rhe
lpfu
lfe
edba
ck.
RE
FE
RE
NC
ES
[1]
A.
Alim
arin
ean
dS
.S
met
sers
.O
ptim
izin
gge
neric
func
tions
.In
D.
Koz
enan
dC
.S
hank
land
,ed
itors
,M
PC
,vo
lum
e3
125
of
Lect
ure
N o
tes
inC
ompu
ter
Sci
ence
,pa
ges
16–3
1.S
prin
ger,
2004
.
[2]
R.
Bac
khou
se,
P.Ja
nsso
n,J.
Jeur
ing,
and
L .
Mee
rtens
.G
ener
icpr
ogra
mm
ing—
anin
trod
uctio
n.In
AFP’
98,
volu
me
1608
of
L N
CS,
page
s28
–115
.S
prin
ger,
1999
.
4Spe
cifying
x+
0x
shou
ldb
een
ough
toex
pres
sth
ene
utra
lel
emen
tof
addi
tion.
The
extra
r ule
0+
xx
shou
ldno
tb
ene
cess
ary.
1.5.
CO
NC
LU
SIO
NS
15
[3]
P.B
oon
and
P.D
rijve
rs.
Alg
ebra
enap
plet
s,le
ren
enon
derw
ijzen
(alg
ebra
and
appl
ets,
lear
ning
and
teac
hing
,in
Dut
ch).
http
://w
ww
.fi.u
u.n
l/p
ub
lica
tie
s/lite
ratu
ur/
65
71.p
df,
2005
.
[4]
K.
Cla
esse
nan
dJ .
Hug
hes.
Qui
ckC
heck
:a
light
wei
ght
t ool
for
rand
omte
stin
go
fH
aske
llpr
ogra
ms.
InP
roce
edin
gsoft
he5t
hA
CM
SIG
PLA
NI n
tern
atio
nal
Con
fer-
ence
onF
unct
iona
lP
rogr
amm
ing,
I CFP
2 00
0,p
ages
268–
279.
AC
M,
2000
.
[5]
A.
Coh
en,
H.
Cuy
pers
,E
.Rei
nald
oB
arre
iro,
and
H.
Ste
rk.
Inte
ract
ive
mat
hem
atic
aldo
cum
ents
onth
ew
eb.
InA
lgeb
ra,
Geo
met
ryan
dS
oftw
are
Syst
ems,
p ag
es28
9–30
6.S
prin
ger,
2003
.
[6]
A.G
erde
s,B
.Hee
ren,
J.Je
urin
g,an
dS
.Stu
urm
an.
Feed
back
serv
ices
for
exer
cise
as-
sist
ants
.In
E C
EL20
07:
P ro
ceed
ings
of th
e7t
hE
urop
ean
Con
fere
nce
one-
Lear
ning
,20
08.
A ls
oav
aila
ble
asTe
chni
cal
repo
rtU
trech
tU
nive
rsity
UU
-CS
-200
8-01
8.
[7]
G.G
ogua
dze,
A .G
onz
a´le
zP
alom
o,an
dE
.Mel
is.
Inte
ract
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