Great Britain. Adoress from September L977: Department of Mathematics, University of. Southern Cal-ifornia, Univensity Pa::k, Los Angeles, California, 90007 by ...
The Univensity of
Warwick
THEORY OF COMPUTATION NO.19
REPORT
TI
ATA
TVPES
BY
JInTIEL.J,LEHMANN AND
Departrnent
of
Computen Science
Unjversity of
Wanwick
COVENTRY CV4 7AL ENGLAND.
11
TcHAEL
B,St',t
yrH
DATA,TYPES
by
Daniel ,J. Lehman#and Michaef B.
Department
of
University of
Smlrth
Computen Science War.wick
Coventry, West Midlands, CV4 7AL
Great Britain
Adoress from September L977: Department
of
Mathematics, University of
Southern Cal-ifornia, Univensity Pa::k, Los Angeles,
California,
90007
Abstract
A Mathematical interpnetation Ttre main noveltSr
is given to the notion of a data .tJrpe.
is in the genenality of the nathematical treattnent
which allows procedural data't5pes and cincular"ry defined data tlpes. What
is
rneant
by data type is p:rettxr close to what any corputen
scientist would r:ndenstand by this mode, clustenr
cl-ass.
terrn or by data stnuctune, tJlpe,
The mathennatical trreatment
is the conjunction
of the ideas of D. Scott on the solution of domain equations (Scott (71), (72) and (76)) and the initiality properry noticed by the ADJ gnoup (ADI
(75),
to the data types of
ADrl
to
ADJ
(77')).
pnoposed
The present wonk adds openations
by Scott and generalizes the data tlpes
procedur,al types and anbitrary ci:rcurar tJpe
The advantages
definitions,
of a mathematical interp:retation of data
tjrpes are
those of mathematical semantics i:: genenal : throwing ]ight on sore
iIl-rndenstood constnucts in h-iglr-leveI pnognarrning languages, easing
the task of wr.iting conr€ct
pnograms and naking possible pr"oofs
cornectness fon pnograns on inplernentations"
'l
of
I" Introduction Ali
pr"ogramming languages have
some have
few,
sorne
comrnnly used ane
have only one basic
: booleans, integers,
Iabe1s, atomso lists. openations
basic data types,
some have many,
data'type.
The most
r:ea1s, procedures,
They generally come equipped
with
sorne
: logical openations fon booleans, a::ithmetical
operations fon intege:rs and reaIs, compositionn evealuation
and
abstnaction for" pnocedunes, a defining facility fon labels,
and
list-rnanipulation pnimitives fon iists.
fn
most data..qpes the user. has
the facitity to denote new objects
by the use of expnessions combining o1d objects by openationso Ai-l these facilities are easy to undenstand. The fi::st non-
tr"ivial fact about
data. B(pes
in centain prograrming
is that, in ce::tain data types
languagesn objects can be defined
and
implicitly;
they ane defined by e:pnessions that contain thein own denotation" This is the facility which is gener"ally ::efe::ned to as |tnecu::sive
definitionf' but which we prefen to cail "circu.l-ar definitiont'. This facility is generally offened for pnoceduna.l- data types only. ln the most advanced
ianguages (caIled extensible) the user
may
define new data ty-pes fnom old ones by the use of eonstnuctol"s. In
68, for example, these constnuctors are gq]}g!, !!!!r ref row, fp$. New ope::ations on data'Wpes may also be defined by
ALGOL
3
defi.ning new procedures.
z.
In
some languages,
like
ALGOL
68, new data types
may
also
be
defined cir"cula::Iy" The following two definitions a::e examples
of such cincula:: definitions in ALGOL 68: ( iE Iabel, nef t.ree. 1eft, € mo.g g = gg! moi.e fun = puoc ( frn ) firn A11 languages known -
to the authors that
lgeg ::ight
)
al-Iow such circulan
definitions of data types put stringent ::estrictions on the genenality of such a facility.
Lehmann
circul-a:: definitions not allowed in
(77) showed that
ALGOL
many
68 are meaningful and
ve::y useful.
The question
of the mathematical
meaning
of cincula:r definitions
inside pr.ocedural data types was finst answened, independently,
by
H. Bekic, D. Pank and D. Scott, who noticed that the funetions defined wene least fi>ooints of rpnotone functionals. The fact that mode-constructons ane functors was mentioned nemank
attributed to Lawvere, and later
and Wand
( 74
been known
).
Cir^cular.
explicit by
a
Reynolds
definitions of sets and languages
had
to algebraists for" a certain time when, in 1969, Scott
gave a pnecise neaning space constructor
to solve
made
in Scott (ZZ), in
to ci::cuLar definitions involving the function-
(his arnow, the ALGOL
dornain equationso
irplicit
Bggg). A general method in Scott (72), was rnade e>glicit 68
by Relmolds. The categorical natune of this rnified construction, only hinted at by Scott, was emphasized by behind the pnesent wor:k
is that the
Wand
(74).
pnoblems involved
The main idea
in defining
oata rypes can best be handled by an exact generalization of the well-understood methods used in studying definitions of objects
within a data tjpe.
This involves genenalizing fuom posets to
categorieso fnom monotone functions to frr::rctons, and fuom least
fixpoints of continuols functions to initial fi>rpoints of continuous frnctors. The ADJ gnoup concentrated on the p::oblem of defining frnctions on data types and insisted that data types do not consist only of a set of elements, st:3uctur"ed in some way (generally a partial onden) but consist also of centain frstctions. They understood the impontance
of initiality
and noticed
rnany-sonted algebnas) but wene unable
initial algebras (on initial to include
procedr::na1 data types
the link with initiaL between our" wonk and
as fo11ows"
We
that certain data t5rpes wene
in thei:r tl:eatment and did not see
fi:3
f 'l,l ) =u n ul'l = !"1 uf cr,
L R ! ui "i ) is an amow in cpoa .
Lenma
12 : Let I be an ardiag:ram in
D
CpOa and
U : | + A_ be
that irR is a limiting then u is a colimlting eone (in
a
comnuting cone such
cone fon IR
in
cpoa).
ar-CPo*r
E_::oof : Dual of Lemma lI. Definition : Let E, be the covariant which sends each pain
enbedding
of
cpoa
in
ur-cpo*
of functions to its left par:t and
E* be the contravaniant embedding of
CpOa
in
r,l-Cpo* which
of fr:nctions to its niglrt pant. cleanly E, and E* ane the identities on objects. sends each pain
Theorem
4 : l)
CPOa
2) E, 3
is
an 0r-qatsg6rSr
pnesenves and
::eflects ar-co-Ifunits
) E* t::ansforms or-co-limits into aroP-linits ,oP-lirnits into ar-co-1imits.
Pnoof
: 1) by Lemrnas 5, ?, 9 and IL. 2) by Lernrns 8 and 9 and Lernma lL. 3) by Lernnas 8 and 10 and Leruaa
19.
12.
and neflects
The next Theonem
in
CPOa
Theonem
wilI be used to plove that frnctons (on bi-functor.s)
ane o;-functor6.
5 : If T is CPOa
x
a.n
endo-functor on
CPOa
+
cPoa) o
CpOa Ca
T presenves a,-colimits . if fon
c * *1 J every sequence = *^o J'o I':_
rl { H :
Proof
=
By Lemmas
r
}
e*
**+16CPO
: rrr'-CPO"
is a fr.rrcton T :
(r ri) L (r r.;R = 1.
a Cnon-full)
subcategory
pnesenves adjunctions
CPOa
- -'L'8 =""3'i 'i l: ""
8, 9 and 11.
CPO-, can be considened as
Tr
U
bi-fr.mctor
+ CPOa.
The
of
its restniction to
following
; if
o-CPO
theonem shows
CPOa
that
the initial T-algebna is also an initial Tr-algeb:ra. It is useful
to
d,raw mone
r.adical initiatity
pnopenties fon the data epes
ci:nculanIy defined by definitions involving only +
and x (at
the
exception of +), as ttrose eonsidered in ADJ (75).
: Let T :
Theonem 6
+
CPOa
CPOa
an a.r-fi:ncton such
be a fi.rrcton and Tr :
that
ErT =
ar-CPo?t+ ar-CPO*
TrE, then T is
an
if 6 is the initial T-algebra then El,0 is the initial Tr-algebna. : T t is an 4v*fplsfoD by hypothesi.s and E, is an arfi:ncton ar-frmcton and
Pnoof
4 =) TtE, = ELT is an ar-fi:ncton. But by Theonen 4 E, r.eflects ercslimits and then T is an r^r-firrctor" Theonem I by
Theonem
assents the existence pnoof
of
Theorem
of g :
TA
+ A the initial T-algebna.
The
1 shows that { is the r:nique arnow in cpoa sueh t}rat
d o Tti = pi*1 fon p : i + A the col_imiting
r 3 r?tTr Tt+;oo. *Tir*..
tttrr.
20.
Ti+1r
cone
fon I
By Theonem
4, E, pnesenves arcolimits and
colimiting cone.
Because
the initial
Ei,!
object in
initial in ar-cPO'!, E,t is the following
Erf i r ,* Ttrh..r? l'', * .... * t'i : -TtL t *TtL
diagnam
of
Theor"em
cPOa
in
isa
is also *
aJ-CPO
:
,ri+1 * "...
-.
lrtrrr,
The pnoof
:Erf+A
1 shows that the (up to isomonphism) initial
is the r:nique arrow 0 : Vie/[ { o TtELIi = Etui*:. Tr-algebna
But Elui+t = EL 0 o E, T u.
TtA + A
of
ar-CPO*
such that
= EL 0 o TfEru.
and ,t=Et0. tr
Remank
:
Ma::kowskyts (7+) r'esu1t on
* the cocompleteness of CPC is
not used. Cinculanly defineo data.types the definition of invol-ves + (r:nion), " (E;ggg) and * (9,)
initial algebras in Lehmann
CPOao
the subcategory of
checked
domain
CPOa whose
whose arrows are pains
ALGOL
68).
and
can be
cLosed under o-col-imits.
definition of data tlpes
and neduces the
solution of comain equations to least upper bornds. Plotkin Snryth have necently shown
be
Scottts
objects a:re continuous lattices
of A-continuous pr.ojections. It
anoth-en
to
equations consisted of considening
that this subcategory is
Scott (ZO) proposes
can be seen
will be expiained in the sequel (see
tZZ) for a preview oriented towards
original (ZZ) solution to
easily
as
which
that these l.u.brs
al.e r,r-col-inits
and
in
a
suitable category of adjr:nctions which is equivalent to the subcategory
of
CPOa
considened oniginalJ.y by Scott (72)"
21.
Ex€inple Lehmann
5 : Dom and a mone genenal notion of a data-t54pe. (76) defined a catego.r'y Don the objects of wh-ich ane
orcategories and the arnows of which a:re adjr.rnctions with identiry
nnit.
Dan Ls an ar-category
ar-frtrctor.s.
and X, *, + and p can be defined to
is a furl subcategony of Dom closed unden a.rco-lirnits, X, tr and + . The eornespondents of rheonerns 4 be
cPoa
and 5 hold and nelate Dorn and
and,
stnict ar-functons. bm
ar-Cat, the category
provLdes
of
ar-categonies
a more genenal notion of
a
data typen useful when the powerset constnuctor, ou non-deterministic pnocedu:raI Qpes are allowed.
we
shall
now proceed
ci::cu1a:r type
to give a nurnben of
examples showing how
definitions do indeed define initial algebras.
The
algebnaic aspects will be stnessed : a cincular t5rpe definition
not only define a pantiari-y
ondened
set but also
sorne
fi:nctions
does on
this set.
Exanrpfe
6:
Ttre
natr:ral
numbers.
naturel numbens, or even the integens, are generally thought to be a basic data'tJpe. I{e shall now demonstrate how they can be T'l:e
cir"culanly defined" our treatment is equivarent to Lawverers (64), as reponted in Maclane-Binkhoff (62) (pp. 62-70).
Let oul: r:nder.lying category be Set, I be the one-point set (it is a terminal object in Set and aLso a generaton) , r the .;nique eiement
of J, + be the co-pnoduct (disjoint urion) and
22.
iet
T be the functor defined
if
e isaset
if
f :
"1* "2
by
:
Te:J+e is a function Tf :
Tea+ Te,
is the fu:action defined hy (Tf)(u) = ft if a e ea r if aeJ In category theonetic notation Tf = I, 1 5 Clea::Iy T is a fr.:nctots : Set + Set. It is indeed the firrcton type Tt = { OrS } nank (O) = O, nank (s) = 1 descnibed afte:: Definition 6" T is an orfuncton, as all firnctons associated with sinilanity tlrpes, as explained associated with the simiLanity
in
E>hism,
the fr:nction
zeno anci ne/l/
to n + 1 = suc(n).
Lemma
O
is an initial
T-algeb::a,
property of g cha::acterizes, up to
+ suc z 1+ N + N which
gsnds,
lef to
1 : 0 + suc : 1 + N + N is the initial T-algebra.
Proof :Suppose {:J+B+B Suppose q, : y'/ + B. If (f) o " (O + suc) = 0 o (Ir+ cr) then (z) a (o) = ( o o (0 + suc)) (r) = ( 0 "(tr+ a)) (r) = 6 (r) (S) o (n + 1) = ( o o(O + suc))(n) = ( O .(rr+ a)(n) = {(o(n)) Conversely if o ve::ifies (Z) and (3) suc)(r) = c (o) = 6 (r) u o(,O + suc)(n) = cr (n + 1) = 0(cr(n)) u o(0 + suc) = 0 o(Ir+ c). c
o(O +
23.
and
By
the induction nule fon the natuml
nr-unbens
frnction a : il + B ver:ifying (2) and (3)
thene exists exactly
and
this
pnoves
one
the initiatity
ofO+suc:L+N+fl. The
point we want to
make hene
defined by : tr = 1 + N or in union (t"fr, natunal).
The
is tiut the tlpe /[
ALGOL
68
can-
be ci:ncularly
notation mode natrr:ral =
initial ffurpoint involves not only
the
set fl but also the constant 0 and the function successor" The induction ::ule fon natural nunbens which is vital" fon pr:oving ptopenties of
prog:rams
manipulating natural numbens is nothing else than an initial-iry propenty.
fn other wo::ds the fact that the rnique o : // + B inplied by
Theor"em 1
is a total fturction is the main tool in prrcving that centain fi:nctions are total-, in contr"adistinction with the more general pantial fi:nctions which may be defined cincularly continuous rirnctionals
Fo:: othen
inside
[
/il
+ B] by means of arbitnar:y
: [ff + BI +:[/t/ + B] "
cincurarly defined data types the initiality
pnopenty may often
be used Cirectly in place of an ihclrlction principle. fn othen cases it is
the main tool in pnoving the connectness of an induction principle. question of the exact relation between initielity tro:f
pd in
Soni-r'nn
Lemma
1.
nr.unbens may
The usual
r:.'' be neadil-y transfonrned to fit this if m=0 n+m=r suc(n + mr) if may be
will
be
q
A-1 useful fr:nctions on natu::a1
with the help of
and induction
The
obtained the following way.
24.
be defined f::om 0 + suc
definition of addition for fr^amework.
m = suc(mr)
example
Let T = trf cr: * f) and u: T(Iffdlll).+ by : ,lt = TN + lf (suc o f). Then thene
[N+N]be defined
is a unique u : d *t il+ryl :naking the foll-owing
diagnam comnste.
N t$t J.+N
i" ! r.,.'[,
rr*o
[ff*fl] S L+[N+Nl The commutation
of the
above diag::am
is equivalent to
a(O) = If
and a(suc n)
a(O)(m) = m
and c(suc n)(m) =
=
:
Suc o cCn)
Suc (a(n)(m)).
o is the addition" More genenally any
function defjned by a pnimitive necunsive
is the unique annor{ rnaking a sfunilan
diag::arn
scheme
commute" Ce::tain such
diagnams, howeven, define functions which a::e not pnimitive necursive"
Theonem
2 assents that 0 + suc is an {g.omonphismrits invense is obviously
nuii*pr"ed:
N+7tN
defineciby (nutt+pned) (O) =rand
hull- + pned) (n+1) : The p::oof
As a
of
fheorem 2 shows how
to define pned in terms of
further example we shaII show that
.v may be defined as
n
suc.
even the equal_ity pnedicate on
the rnique anrow f:rom an initiar algebna.
25.
= I + I/ + fl + S. The initial T-algebra is 0 : f + N + N + N x /y' = (J.+N)(I+W1 + d x lV defined by d = Co+suc) r(olsuc).
Let
TS
Let 3= {tr:ue, false} andrl,: J +N+ /[+B+Bbe definedby 0(r) = tnue, $(nr) = OCnZ) : false and 4(b) = b" (o+suc)xCo*suc)
NxN
7+N+N+NxN
I
I
I
I
I Ir+ Irr+ I,,+ c
lrwtv
I
+
lL
b
L+N+N+B
a is the r:nique o(OrO) =
ar.tsow
: I x il + B
such that:
tnue, c(n+1rO): c(0rm+1) = false and
c(n+1rm+1)
= c(nrm).
a is the equality pr.edicate.
Examole
7 : Context-fnee
languages.
All- l-east fi:points methods previously
special cases of the presented
mone gene::al
rr^sed
in
Computen Science ane
category-theo::etic initial
fixpoints
hereo In partieular the charaetenization of context free
languages as
least solutions of a set of equations
can
be ca:rried
th:rough.
Let ) be an alphabet (not necessarily finite), then P(x*), ttrc set of aII languages oven X, ondened by inelusion, standand methods, an 0r-categor..yo Let
is a complete lattice and, by
V* be an alphabet of non-terrninals
(not necessanily finite) and p be a fi:nction : V* -r E where E is the set a-:. expr"essions built fnorn V*, f,, concatenation and uniono 0n P(ft') concatenation and i:nion ar"e additive Cthey pneserve arbiilrany 1.u.bts)
zo.
and so they ce::tainly ar?e &r-. fr:nctors
I rt lVnt T : P(f")' "l * Pcf,")r N' ,',
lVu
I
and the
i,, r - A of subsets of PCI -. rt-) lv*l-ulle
Exarnple
8 : Context-fnee
p clear"ry defines
an al-fwrctor
:
initial T-algebra TA c A is the 1east
such
that
TA
c A.
By theonem 2 TA = d.
g?ammars.
A much mone intenesting example concerfns context-free grarmars (as opposed
to languages). A context-fnee
can be viewed as an rrr-endo-furcton
to
example T above, but when
granman
with n non-terminals
T : setn + Setn, in a manne:: similar:
u (of sr$sets of r't) i"
nepLaced
by
+
(aisjoint union) and . (concatenation of subsets of x't) by r (pnoduct). Fo:: example
the context-fnee
gramrnar
: s + alasafassal
can be ]-ooked
at as the functon : T : ),S. {a} + ta} x S x' {a} +' {a} x S x S r' {a} The initial T-algebna g : TA + A consists of a set A isomorphic to {a} + ta} * A x' {a} +' {a} x A x e x' {a}, venifying the initiality property. A is isomorphic to the set of all parse treeso $ constructs and 0-1 decomposes parse trees. Note her.e that the fuaction fnontier : fr. : A * f,* which assigns to each parse tnee the
panse tnees
wond generated
is not one-to-one
multiplicity of
w.
27.
and fo:r wexft
: lrr-1{w)l is
the
3.
Data types as aLgebnas
we now want
to
make
our thesis pnecise: a data type is an object in
a suitable category of domains but
Dom
(CpOa wil-l- do
or othen categonies could be
for alL applications
considened) equipped
hene
with centain
operations.
Def.initionJ: A type t consists of a natural numben n>o and n pai::s of functons (SirT.), SirTi , CPOa + CpOa, i=lr...rn Definition 2: A data type D of type t:(nrS. rT. ) consists of an or-CPO D (with light notational ambiguity) and n (6-continuous)
fr:nctions d. 'l- :
S.D l_
+ T.D. t_
rn practicat applications the functors s. and r. used are always "polynomial'r (ttrey are built f:rom pnoducts and sums onry) and indeed
a data type is a domain equipped with a finite
numben
of functions.
Definijion 3: A homonorphism from D, to D, of type t is a fr.nction f : Dl * D2 such that the following diagr.am commute: o-.1 -+ S,D. tl-
s.r 11
J
"i"2
T.Dl_l_
i r.r . 2 'i"2 I
i,
9i
wirl be used in section 6 to study inplernentations. For the moment just notice that we have defined a categony of Homomorphisms
.iata types of type t.
28.
?asic Sata_jg>gs The::e
is no neal pnobrem in defining the basic data types of
pnog::amming languages
The data CPO
t]pe
boorean
as objects in
for
CPOa
usual
equipped lr-ith cer:tain openations.
example wouLd werl be rrrde:rstood as the
Bool repnesented below:
ffue false ^u, / Bool
with the constant tnue, the unary openation (-true=faIse, -g"1se=tnue, ;I=I) and the binary ope:ration
equipped
v
(rv.L=1v1pue=truevl-=falsevl=lvfalse=I, ttueVtrue=tr:uevfalse=falsevtrue=true,
falsevfalse=false ). A11 othe:: connectives: Tha nrrac+-lnn ^nr1ld u
De rrue i:-nsteaci
Ar+r$ etc..e
be'aised I qfesu whether nrls Lr.cl. !s
of r).
cED
be defined from the above thnee"
we wq: shourd DlrLrulq
not ItuL (lgJ.J.lle define rvtnue IvIrue to
This latten pnoposal is conceivable but would
contraci-ct the operational meaning that some want to give to i
J_
a non-terminating computation ).
of the general-ity of cincuia:: definitions of data types and the extended facilities available to define functions on data types we think that it is possibre (though we in no way suggest that it should be the Seeause
case
in
pnognamming languages)
to
use only a very few basic data qpes:
7 the one point domain, and 2 the two points domain for: exanpre. .T
I
.I
.I 2
29.
rgr
Type-constnuctors As explained
the
1n
Section 3 a Qqpe-constnucton is an ar-endo-fi:nctor in
catego::y of domains' we
shalI list
of the
below some
most interesting
ones.
@ : cPOaxcPOa+cPOa, cornesponds
to the categorical pnoduct in
ur:cpo't
if II is the categonical product in or-cpor' ). If A and B ane cpots AEB consists of all pains of objectso the fir"st one in A, the (E- oQ=1o(ErxEr)
second one
in
in
cpoa then
If f:A+Ar
B ondened componentwise.
f6g=(rlngl, fnt*l
and g:&+Bt are arrolrs
for (aIIb)(xry)=(ax,by).
o-functon is easily checked with the herp of
Theonem
That @ is an
Sr,and so the
o-continuity of all fr:nctons to be defined now. There ane sone obvious arrows attached to @: P],@B*A, P2:A6B+B. A
srightly differ:ent
pnoduet
will- be needed for^ defining stacks and 1ists.
x:
cPoarcPOa*cPoa conresponds
(the
anr"ows
to the catego::icaI product in
ur-cpo**
are very stnict o-continuous frnctions, those functions which
to bottom only the bottom element). If A and B are cpors then AxB consists of only those couples (arb) with aeA and beB such that u*tA arrd
send
b*r, oo a=IO
anci
b=rB It is easily
g:b'tBt are annows in
CPOa
seen
to be a cpo. If
then fxg is the nestniction of fgg to AxB.
This non-standand pr"oduct has al-neady been used by M.
thesis. The obvious
+:
cPoaxcPoa->cPOa. J.r.
in i,r-cPO .
A+B
f:A+A? and
Gondon
in his
anrows pa:AxB+A and pr:AxB+B nay be defined.
conresponds
to the categonicar
is the coalesced
sum
of
ir:B+A+B and d:A+B+Bool_ may be defined.
an
A and
B.
sum
in
rrr-cpo*, and
Arrows ir:A+A+B,
Q: cPOaxcPOa+ cPoa is the separated sum. itri2 and d may be defined as above.
+:
CPOaxcPOa+ cPOa
is the fi:ncton space functo::.
+(AnB)
is
[A-+B]
in
oun notation.
App: [A-+B]QA+B, Abst: [AxB+C]-+[fi+[l+f ]l and Y: [a*A]+A may be defined. /ts was mentioned above a power constnuctor P, or more precisely a
number"
such constnuctor.s have been studied; they
pnesent work.
The nexl pa::agnaph exemplifies cincula::
will not be used in the
of
definitions of data types. Fon example
it will be sho'nrn how, given a data type A, it is possible to define the
data
type Stacks of A (StackA). It is only in the next section that it will be shown how these
definitions
amount
to
rnaking Stack
a
t5ape
constructor.
C!ncu1a.r1y defined data tfpes Thi.s panagnaph, and
is to
show
that
the next one, pnoceed uniguely by examples. Their
many usual data types ane indeed defined
circularly,
purpose
and that
cinculan definitions define not only a centain domain but also centain operations on
it.
i) Simple data types Those are anci
the data types built
f::om
basic data types by type constnuctons
cinculan definitions.
Natunal
In the
numbens
preceCing section the natunal numbers were defined circula::Iy as
initial T-algebna for" a fr:nctor T on Set. For see-.is pnefenable CFOa. By
to define
them as an
initial
computen science purposes T-a1gebr"a for-
an
it
a functor T on
analory with the case of Set one coulpect: one bottom element and atl finite non-empty binary tnees
the night leaves of which ane uniabetled (they nepr"esent empty iists)
and
the lef-t leaves of which are eithen r:rrlabelled or. label1ed by a non-bottom element
of A. If the tnee consists of only
one 1eaf
it has to be considered
a r ight leaf.
ExampJ-es:
() the empty list
. . - /\
(a)
q.
.
(o)
./l (a' (b) ) = (a'b
' - /\ o' i, b.': /\/\
it /.'\
)
(a).(b)) = ((a), l)
a";'
ci:
2+(A+LispA)xlispA+LispA
cl(r)=.r_ ,
is
cl(r)=O,
such that: ci(a,,Q,)=(a.4)
2+(A+LispA ) xl,ispA=2+ (Rx1,1stA+LispAxLispA ) =( 2+AxlispA )rLispAxlispA
which
is obviously desinable.
9ina:rv tgees Labei-ied binary trees suggest the following
definition:
BtreeA = A+AxBtreeAxBtneeA. The ::eader may car:e
to
check
that the initial- algebra consists of
aii flnite binary trees a1l- the nodes of which el-ements
of A,
and one bottom
element.
(cb : A+AxBtreeAx!13s.A'+BtneeA) and tnees.
ane iabelled by non-bottom
The annows obtained construct
decompose ("b-1:Bt::eeA+A+AxBtrreeAxBtreeA)
Had we chosen
on
a
to solve T = A+AgIdI
domain containing
Tr"e.e-s
finite
and
we would. have defined. an
infinite taberled binary
initiat
algeb::a
tnees.
and_E€stF
A iabell-ed tree consists eithen node and
of a single labelled
A+AxF
I=
z+TxF
of a labelled
a fonest. A for:est is eithen empty or" consists of a tnee and
fo:est. ?rees ancj forests are defined by the pair: of T=
node on
a
equations:
It will be shown in Section 4 that forests could equivalently be defined as stacks of tnees , after having defined tr€es as either a single labelled node
or a labelled
defineo by:
node and a stack
E
of t::ees. In other
wonds T and F may be
A+A Stack(T)
F= Stack(T) whene the mUtUallrr nr'n^,r'i2n rt-€inition
'r Composite- data twes v ,i.y data types are not simply
haS been eliminated.
2
of t5pe constructons
built up f::om basic data Wpes with the
and ci::cular
definitions but defined,
by -'irst defining a sirnple data type and then type and possiSly for^getting composite data t5pe
which
some
will be the
some new
indir:ectly,
fgrrctions on this data
of the oid finctions.
sane as
somewhat
help
The canr.ier
of
that of the simple data type
it is built but the operations will be diffenent.
T\uo examples
a
from
will
be analysed he::e: Queues and Ar::ays. Qqerres-
Qr,o,res a:re
of the
same
type as Stacks but the push and pop pnocedunes intenact
36.
in a diffe::ent
in the sense that the
way
fi::st
which has been pushed on
erement popped
(and not the
last
one as
alre a ve::y useful and intenesting data t5rpe which seems
attention of p::evious nesearchers.
The obvious idea
witt be the
one
in Stacks). Queues to
have escaped the
is to define
Queues firom
Stacks by defining new pop and top fr:nctions and then forgetting about the
old ones. Stacks of A wene defined as two inrrense functions:
cs:
the initial algebna
2+AxStackA+StackA
cs-f:
and
StackA-+Z+AxStackA the empty+topxpop arnow.
define dq: StackA+2+AxStackA by:
We may
dq(s): if empty(s) or empty(pop(s)) ttren push(top(s The above
),
p2(dq(pop(s
)))
to
ichain in [C-+C] wrth col-imit r:Q-rF, . g = FO-j9"F1-:l-r Tne
translation of cr by s is the following infinite r*_______)
diagram:
i_r
l
1*_ Jrv*
t
"!
,'l!
vv
- ",-1 |
UUI
,
KK'
v
t-qni !'n€ I '^'^i
ittr
d?
i
F:- ('TLi...
Fnf ' 4'4 rl
oi.l \" !t/,
rirr;-=-\ij-
nnr KKI v I
!
i
'
i
nn+
'rt'?ut'L,
v t
i
Tr together with the fact ihat colimits in [C-tC] are computed ^::.::tra:.;e j t,r're coiimits of the rows are as indicated in the diagran. Let
By
Leruma
us ....iove ihal- the arrows q;{"
mediate between the successive conest
:is :.:dicr,.teo. In tre iirst piace, L ithe uniilue arrow from -l to Frirnecj-a;es triviaii-y. Nos, for any n ) 1' k Z O' the foilowing
=
is the carrier (codomain) of an initial solution. Then, interchanging the roles of XrZ, we trave (inp notation syrametrical to that used previously,
whose
interpretation should be clear): F, is
AX.?,,
4:tz'> = d()z.F(7.&)rl(,\z.r(z&)> is the caryier of an initia 1 solution. Since all initial algebras for 4.1G) 4'e
X" is Z, G' is ),2.t(zrz)r
Eurd
isomorphic, the result follows.
Corollary 2. BtreeA i ForA .
;|. Apply Corollary 1, with F(X'Z) as 2 + A xX x The mere existence
of an isomorphism
itseif a ver)r interesting result. closer inspecti.on reveals
between BtreeA and ForA
However, as we
much more
Z.
shall
see
in Section 5,
a
than this: it gives us the "representation
of forests by binary tree6rr. Likewise for: Corollary>)
= d. TI c((a, ). Since the back square and the bottom rectan,qie commute, h, dq is an arrow from cso
to d*. since the front
arl affow from cso
resul-t is proved.
square and top rectangre commute, e*
to d{. By initiality
o is
of cso these arrows are equal;
the
5" Inpiernentation of
crata types'
to prove that certa:.i: openations on data type D' iype DN enabLe him to sj-rnulater represent or implemer't those of a data Typioally, D will have been defined'eabstractlyrr, say by a circular defj'nit'ion' sornetlmes one wants t,o be abie
prograrnming while D,is morerrconcretet'(rnore like what is usually availabl-e in cited as Ianguages). The:.epresentation of stacks and queues by arrays may be
that exanples. An int,eresting exanple that does not quite fit this pattern is of the representation of forests by binary trees. Here the representation
involves, as it were' a reduction of rrstar-height't: in the definition of the Btrees data type Forests there occurs already a paraneterized data type' while
is sefined via a purely polynonial functor (see Sec.3t ot
end
of Sec'4)'
These exanples are weli-known, an
