A simple formalism for solving domain equations

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Mathematik und

Informatik

Informatik-Berichte 57 – 11/1985

Klaus Weihrauch

A simple formalism for solving domain

equations

A SIMPLE FORMALISM FOR SOLVING DOMAIN EQUATIONS

KLAUS WE I'HRAUCH

ABSTRACT

The purpose of the theory of domains is to give models for spaces which are useded for denotational semantics of programming languages (Scott and Strachey [1],

Stoy [2], Gordon [3], Tennent [4] ).The kinds of spaces which are needed involve spaces of higher functional types and spaces defined recursively. Several strate- gies have been proposed for finding suitable spaces which satisfy the desired recur- sion equations (Scott [5], [6], [7], Sanderson [8]). lt has been suspected already for a lang time that the theory of domains itself can be applied for solving recur- sive domain equations. Scott [7] remarks that domains can be made into a domain and outlines roughly his ideas. In this paper a computable "super-cpo" IT is defined

s

which essentially consists of all "constructive domains". Domain equations and domain

11isomorphisms¹¹ correspond to fixed point equations on IT _s which can be solved _. effectively by methods well known from cpo theory. The formalism is very simple and describes adequately naive constructions for solving domai~ ~quations.

1.

PRELIMINARIES

lt is assumed that the reader is familiar with the basic theory of algebraic cpo's (see e.g. Egli and Constable [9], Stoy [2], Weihrauch [10]). Let B be the

(algebraic) basis of the algebraic cpo IT= (D,=,.l). Remember that the partial order (B,=) is determined by IT and that IT is determined uniquely up to isomorphism by the partial order (B,=). A partial order (X,~) is called b-complete (fb-complete), iff

u

Y exists for every non-empty bounded (non-empty, finite, and bounded) subset Y~ X. If IT is a cpo with basis order (B,=) then IT is b-complete, iff (B,=}

is fb-complete. In this paper b-complete algebraic cpo¹s will be called domains. If 01 and U

2 are doma ins then the sum [U

1 + IT

1], the product [TI"

1 ^x 0

2], and the cpo [D₁-D₂^] of the continuous functions from 0₁ to 0₂ are domains.

For algebraic cpo¹s with denumerable basis, computability can be introduced by defi- ning a numbering of the basis (cf. Weihrauch and Schäfer [11], Weihrauch [10]). A constructive algebraic cpo is a tuple D= (D,=,.l,ß)' such that (0,=,.l) is an algebraic cpo and ß :IN--- B is a (possibly partial} numbering of the algebraic basis. Computa- bility of elements and of functions can be defin'ed directly. Here, we shall use the

concept of representation (Weihrauch and Schäfer [11], Kreitz and Weihrauch [12], Weihrauch (10],[13]). Let IB=INJN be Baire's space. The computable elements of IB

are the total recursive functions. A function r :IBn ___ IB is continuous (computable), iff there is a procedure (cornputable procedure) by which every initial segment w of r(p1, .•• ,pn) can be determined from initial segments w

1 of p

1, . . . , and wn of p for all (p , ... ,p )Edom(r). Then r is continuous, iff it is continuous w.r.t.

n 1 n

Baire's rnetric, and it is computable if it is the restriction of a "partial recursive functional operator¹¹ (Rogers [14], § 9.8). The standard representation 6 :IB---D of a constructive cpo IT= (D,!:,.L,B) is defined by

ö(p) = x: ~X:= {ß(i) ^j i+ lErange(p)} is directed and x = UX Let D. be a constructive cpo with standard representation 6.(i= 0,1,2). Then

i i

xE D is computable, iff c5 (p) = x for some computable pElB, and D

1 ^x 0

2- D

0 0 0

is continuous (computable), iff there is a continuous (computable) r :IBxIB---IB

suchthat f(c5

1(p

1),o

2(p

2))=c5

0f(p

1,p

2) forall p

1Edom{<5i) and p₂Edom{o₂).

A constructive cpo (O,!:,.L,ß) is called computable, iff {(i,j) j ß(i)!: B(j)} is recursively enurnerable (r.e.).

2, ^{THE CPO} D _s

lt is our aim to study the class DOM of the constructive domains, i.e. of the

0

constructive b-complete algebraic cpo's. A constructive domain will be cal1ed reaursive, iff the "essential" properti'es of the basis can be determined by computable functions.

Definition 1 (recursive domain)

Let D= (O,!:,.L,ß) be a constructive domain (i.e. ITEOOM ).

0

(1) Define rel(D) := (A,p) as follows: A:= dom(B),

p : = { ( i ,j) E A X A ! ß ( i ) !: ß ( j)} .

(2) D is called a reaursive domain, iff (A,p) := rel(U) satisfies the following properties.

(a) Ais recursively enumerable.

(b) There is a computable function \ :IN²---IN with

f1(i,j) = (0 if (i,j)Ep, 1 otherwise) for all i,jEA.

(c) There is a computable function f

2 :IN---IN with

f2(n) = (0 if Sen is bounded, 1 otherwise) for a 11 n wi th ^e _n-^{c A}

(d) There is a computable function f

3:IN---IN such that sf 3(n)=LJßen for all n with e c A for whi eh ß e i s bounded.

n- n

(By e we denote the numbering of the set {A:=.IN I A finite} defined by

e-¹(A) := H2i I iE A}). By Property (2) (a), for a recursive domain there is a total numbering ß¹ of B which is many-one equival=nt to s. For ß' there are total- recursive functions f

1,f

2, and f

3 corresponding to the usual definition of recur- sive domains. Therefore, Definition 1 (2) is not too general. If (A,p)= rel(D) then

(A,p) is a pre-order which informally can be obtained from the partial order (B,=) by ¹¹splitting¹¹ each point bE B into its ß-numbers. Vice versa, factorization of (A,p) by the equivalence relation i ~ j: ~ (ipj and jpi) yields the partial order (B,=)

(up to isomorphism). We shall transfer concepts from partial orders to pre-orders in a natural way. Let A= (A,p) be a pre-order and Xs_A. The meanings of ¹¹X is bounded"

and ¹¹X is directed" are obvious. X is called saturated, iff

(VaEA)(vbEX)(apb > aE X). Define Sup X := {aE AI X is bounded by a, and apb foreveryupperbound b of X}, Min(A):=Min(p):={aEA\(vbEA)apb}. Finally (A,p)

i s ca l l ed fb-comp l ete, iff Sup Y

*

⁰ for every finite, bounded, and non-empty Y :=. A.

For any constructive domain D, the relation (A,p) = rel(D) contains all the infor- mation which is essential from the constructive point of view. We shall call IT

1 and IT2 constructively equivalent or constructively indistinguishable, iff

rel(D

1)= rel(D

2). The large class DOM

0 of all constructive domains factorizes into a ¹¹small¹¹ set of constructive equivalence classes, for each of which we shall define a standard representative. Domain equations will be solved within the class DOM of these standard representatives. For every DE DOM , (A,p) is an fb-complete pre-order

0

with minimum. On the other hand for every fb-complete pre-order (A,p) where A~IN with minimum there is some DE DOM with rel(D)= (A,p). We shall define the standard

0

representative by ¹¹d-completion¹¹ of (A,p) explicitly.

Definition 2 (d-completion)

Let D~:= {(A,p) \ A:=.IN,ps_AxA, (A,p) For each (A,p)E D' the d-completion

s

is an fb-complete pre-order with Min(A,p)4= 0}.

(D,=,~,ß) is defined as follows:

D .- {ys_A I y is p-directed an p-saturated}

~ := Min(A,p)

X=Y :~X:=.Y for all x,yED,

dom(ß) := A, ß(i) .- {kE A \ kpi} for i E A , B : = range ( ß) .

Define a set DOM and a surjective mapping cpl: D'-DOM by cpl(x) := the

s

d-completionof x. Define r,;:DOM-DOM by r,;(D):=cpl(rel(D)).

0

- 4 -

The completion cpl has the following properties.

Lemma 3

( 1) cpl(x) is a construcitve domain for all xE D', i.e. DOMc DOM.

s - 0

( 2) c: (IT) = IT for any IT E DOM.

(3) c:(0

1) = c:(0

2) <::;=> IT

1 and IT

2 are constructively equivalent.

By Lemma 3, for each DE DOM , ,(D)= cpl(re1(D))E DOM can be considered as a standard

0

representative of the constructive equivalence class of IT. The comp1etion function cp1 is a bijection from D' to DOM, the set of representatives. For studying the

s

set DOM it suffices to consider the set D' = rel(DOM). The next definition which

s

introduces a partial order on D := D' u {(0.~)} is the crucial one.

s s

Definition 4 (the super cpo IT)

s

Define IT := (D ,!::,i,s ), where B is a total numbering of B , as fol1ows.

s s s s s

(1) (2) (3) (4) (5)

D _s

.-

^D_s ^{1 u {(0,0)}}

i _s

.-

^(0,0)

B

.-

^{(A,p)ED I A finite}

s s

S is a canonic~l numbering of

.s B

s

For (A,p), (B,o)ED define

. s (A,p) !:: (B,o) by the following conditions:

(a) As_B,

(b) p = on(AxA), (c) Min(p) sMin(o) , (d) (E bounded in er (e) SupP(E) sSup

0(E)

;;:, E bounded in p) for every finite non-empty for every finite non-empty Es_A.

Es. A ,

By Condition (5), (A,p) !:: (B,0) iff (B,a) can be obtained from the pre-order (A,p) by adding vertices and edges provided that the following restrictions hold:

- no new edge may be added for i ,j E A (by (b)),

- if A=l=0 then no "strict1y smaller minimum" may be added (by (c)),

- i f EsA, E=1=0, is not bounded, no upper bound for E may be added (by (d)), - if EsA, E=1=0, is bounded (in A), then no "strictly smaller" upper bound may

be added (by (d)).

- 5 -

The conditions guarantee that we shall be able to determine a minimum, to decide ipj, to decide boundedness of a finite subset and to determine a least upper bound of a finite and bounded subset of A from a finite portion of an approximation of

(A,p) by a directed set of basis elements.

As an example consider the recursive domain D= (P ,c,0,e). We have rel(D)= (IN,p),

w -

where p = { ( i ,j) ¹ ei-'= ej}. The fo 11 owi ng sequence on B ^s i s i ncreas i ng. ( Here {n1, .•. ,nk} denotes the number mEIN with e = {n , ... ,n })

m 1 k

{1,2,3}

/ 1

~

{ 0, 1}

/ "'

^{{0,1} {0,2} {1,2}}

{O} ^{O} {1}

J}X{l}X{L

1

0

""/

⁰

^~'~

⁰

Its least upper bot,1nd is the relation (:Il\l,p).

The proof of the following theorem is laborious but essentially straightforward (see Weihrauch 1985).

Theorem 5

D = (D ,=,.l ,ß) is a constructive algebraic cpo such that {(i,j)Jß

5(i)= ß

5(j)}

s s s s

is recursive. Especially, ö is a computable cpo. D is, however, not b-complete,

s s

i.e. D is not a constructive domain.

s

Forany x=(A,p)ED

5,X4=(0,0), therearefunctions \•···,f

4 andanumber n₀ such that the following properties hold (cf. Def. 1):

- n E Min(A,p) ,

0

- f1 enumerates A , - f2 decides (i ,j)E p

f3(m)= (0 if em is p-bounded, 1 otherwise) whenever - f₄(m)ESupP(em) if emS:.A and em is p-bounded

e ^c A m-

The data n ,f , ... ,f are necassary and sufficient to handle the pre-order xE D

0 1 4 s

and hence the constructive domain cpl(x) adequately.

- 6 -

The following theorem is the main justification for the sophisticated definition of the partial ord~r ^c

5 We only state it informally. A precise formulation and a proof can be found in Chapter 3.7 of Weihrauch [10].

Theorem 6

Let c5 :IB---D be the standard representation of the cpo ti Then (1) and (2)

s s s

hold.

(1) There are "computable operators" r , ... ,r

4 such that n := r (p),

0 0 0

f1 := r

1(p), ... , f

4 := r

4(p) satisfy the above properties, if c5

5(p)= (A,p).

(2) There is a "computable operator¹¹ r such that p:= r(n

0 ,f

1, ••. ,f

4) satisfies

c55(p)= (A,p), if t.he above properties hold for (A,p)E D

5 and n

0 ,f

1, •.. ,f

4.

Especially, if c5

5(p)= x, then the number n

0 and the auxiliary functions f

1, .•• ,f

4

for handling x can be determined effectively from pEIB. Since computable operators map computable functions into computable functions, we obtain the following important Corollary.

Corollary 7

Let IT be a constructive domain. Then IT is a recursive domain, iff rel(D) is a computable element of the cop

IT.·

_s

Let D

1 and 1\ be domains. lt is known that the spaces [0₁ + 0_2],

[1\

^{x 0}_{2 ], and}

[D1

-o

₂] can be constructed ¹¹effectively¹¹ from 0₁ and 02 and that they are

11effective¹¹ if IT

1 and 0

2 are ¹¹effective^11• Our approach admits to formulate and prove this rigorously. (In the following <,> denotes Cantor¹s pairing function.)

Definition 8

(1) Define a mapping sum: D ^x D - D as follows:

s s s

(A,p) =: sum((Al,pl},(A2,P2)) where

A := <1,A

1> u<2,A

2> u {O}

p := {(<1,i>,<1,j>) ¹ (i,j}Epl}u{(<2,i>,<2,j>) ¹ (i,j)Ep2}U{(O,x) ¹ XEA}.

- 7 -

(2) Define a mapping prd: D x D - D as follows:

s s s

(A,p) := prd((A1,p1),(A2,p2)) where

A := <A1,A2>

p := {(<\,i?,<j1,j2>) ^J (\,j1)Ep1" (i2,j2)Ep2}

(3) Define a mapping fct: D x D - D as follows:

s s s

For

(A,p) := fct((A 1,p

1),(A 2,p

2)) where

A := {n I en=t=0 "ens_<A₁,A2> "(vE~en,E=t=0)

(TT1E p

1-bounded ~ TT

2E p

2-bounded)} , (m,n)Ep: ~ (v<a,b>Ee) bp2Sup {d [<c,d>Ee ACP a}

m p

2 n 1

for all m,nE A. (We define Sup (0) := Min(p 2)) p2

constructive domains Dl,D2EDOM we. call [01+02]

.-

cpl s um ( re l (D

1 ) , re l (D

2) ) the s tandard . swn ,

[01 ^X 0

2]

.-

cpl prd(rel(D

1),rel(D2)) the s tanda.rd product ,

[01-02] ^.

-

cpl fct(rel (0₁) ,rel (02)) the standard function domain .

Lemma 9

The functions sum, prd, and fct from Definition 8 are welldefined. They are computable w.r.t the computable cpo IT.

s

Again the proof is laborious but straightforward. Let 0. = (D. ,=,~,S.)E DOM and

l l l.

Ei:= (Di,=,~)(i= 1,2). Let [E₁+I2] be the commonly used sum of the domains E¹ and E

2, and let D= (D,=,~,S) := [IT

1 + IT2]. Then [E

1 + E

2] and (D,~,~) are iso- morphic cpo's. Therefore, Definition 8 extends the non-construcitve definition of the sum. The corresponding remarks hold for the product and the function space. We only mention that the injections for the sum, the projections for the product, and the eval and the Curry functions for the functions space can be computed from the arguments x and y.

3,

SOLVING DOMAIN EQUATIONS

By applying the fixed point theorem for cpo's to the cpo IT _s we are now able to solve domain equations in DOM. A firstclass of equations is covered by the next theorem.

- 8 -

Theorem 10

Let 9. : D - D be

1. s s D -continuous functions, define f.: DOM- DOM by

s 1.

f. (D) : = cp 1 g. re 1 (D).

1. 1. Then there is a constructive domain UE DOM with IT = [ f

1 (D) + f

2 (D) J • If 9

1 and 9₂ are IT -cornputable then there is a recursive domain satisfying the above equation.

Notice that TI solves an equation and not only an isomorphism. Examples for f

1 and f 2 are: IT t - -ITO, U t - -IT, Ö t - - [IT+ IT], IT t - - [IT ^x IT], IT t - - [IT -IT], etc.

(see Def. 8, Lemma 9). The proof is easy.

Proof

Define 9: Os-Os by g(x) := sum(g

1(x),g

2(x)). Then 9 is continuous. By the fixed point theorem x := Fix(9)=U 9i(.1) is the least fixed point of 9.

0

Since g(.1) ~ x and 9(.1) *.l, x =1=.1, hence D:= cpl(x ) exists. Then

0 0 0

D= cpl(x

0 )= cpl 9(x

0)= cpl sum(g

1(x

0 ),9

2(x

0

)) = cpl sum(rel f

1(ö.),rel f

2(D)) =

= [f

1(TI)+fi(D)]. If 9

1 and 9

2 are computable, then 9: Os-Os is computable and x is computable, hence D= cpl(x) is a recursive domain by Corollary 7.

0 0

Q.E.D.

As an example let x

1 := ({3},{(3,3)}), 9

1(x) := x

1, 9

2(x) := x for all x, D1 := cpl(x

1). We want to solve the fixed point equation x= g(x)= sum(x

1,x).

We know that x = u 9i(.1) is a fixed point. The sequence .1,g(.1),9²(.1),9³(.1), ...

0

can be represented as follows:

<2 ,<2 ,<1,3>>>

"'

<2 ,<1,3>> <2,<1,3>> <2,<2,0>>

"" "" ^/

<1,3> <1,3> <2,0> <1,3> <2,0>

.l s

~

0

~ /

0

·"'

⁰

^/

Then (A,p) := x =

u

9i(.1) = Fix(g) can be represented by the followin9 9raph.

0

etc.

- 9 -

<2,<2,<1,3>>> <2,<2,<2,0>>>

~

/

<2,<1,3>> <2,<2,0>>

~ /

<1,3> <2,0>

~ /

0

Let TI:= (D,=,.L,ß) := cpl(x ). Then by Definition 2, B(i)= {jE A/jpi} for all iE A,

0

B= range(ß), D= Bu {d}, where d= {0,<2,0>,<2,<2,0>>, ... }. D satisfies the equation D= Ci\+ D]. Further examples are given in Weihrauch [10].

A similar slightly weaker theorem can be proved for x and - instead of +. Since prd(x,y) = .L and fct(x,y) = .L if x= .L or y= .L, we need some additional assumption which quarantees that cpl(Fix(g)) exists.

Theorem 11

Let g. : D - D be D -conti nuous

l. s s s functions such that g. (.L) ⁼¹⁼ .l, define

l.

f. :.DOM - DOM by · f. (D) := cpl

l. l.

have solutions in DOM.

( 1) D = [ f

1 (0) x f 2 (D) ] ( 2) T5 = [ f

1 (D) - f

2 (IT) J

gi rel(D) (i= 1,2). Then the following equations

If g

1 and g

2 are computable then in both cases there are recursive domains as solutions.

The proof corresponds to that of Theorem 10. Examples for f

1 and f

2 are constant functions and sums.

Homogeneous equations like D= [D -D] cannot be solved this way. Consider the proof of Theorem 10. For solving the equation x= fct(x,x) we would define g(x) := fct(x,x).

Since fct(.l,.L)= .L, the fixed point theorem yields Fix(g)= .L, but cpl(~,ß) does not exist. Suppose there is some Y* .l such that Y= g(y). Then x := u gn(y) is the

0

least fixed point of g which is greater or equal to y. In this case D:= cpl(y) satisfies the equation 0= [D-0]. The function fct from Definition 8, however, does not guarantee the ex i s tence of some y ⁼¹⁼ .1 with y

=

fct(y ,y).

- 10 -

We shall now present a method for solving domain isomorphisms like

o~

^{[IT- IT].}

We use the fact that the Standard function domain [IT - IT 1 may be replaced by

1 2

any (recursively) isomorphic domain

r.

Definition 12

Let 0. = (0. ,'=,L,ß.) be a constructive domain with basis B. (i= 1,2). 0

1 and IT

1. 1. 1. 1. 2

are isomorphic, iff there is some bijection 1: B

1- B

2 with a'= b ~ 1(a)'= 1(b) for all a,b EB

1• IT

1 and IT

2 are reaursively isomorphic, iff there is a bijection 1: B

1- B

2 with a'= b ~ 1(a)

=

1(b) such that the numberings tß

1 and

s

2 of B2 are many-one equivalent.

By the following lemma there are sums, products and function spaces with special properties.

Lemma 13

For nEIN let X:= {(A,p)ED

i

<a,b>EA ~ a::;n}.

n s

( 1)

(2)

(3)

For any n ~ 1 there is a computable function sum : D x D - D such that

n s s s

range(sumn)s Xn, cpl sumn(rel(D

1), rel(0

2)) is recursively isomorphic to [01 + 0₂], and x '= sumn(x,y) if XE Xn-l' (Correspondingly with sum~ and y '= sum¹ (x,y).)

n -

For any n~ 1 there is a computable function prd : D x D - D such that

n s s s

range(prdn)sXn, cpl prdn(rel(IT

1), rel(IT

2)) is recursively isomorphic to [01 ^x 0₂], and x

=

prdn(x,y) if x E Xn-l and y =t= L. (Correspondingly with prd ^I and y

=

^{prd 1 ( x ,y).)}

n n

For any n ~ 1 there is a computable function fct : D x D - D such that

n s s s

range(fctn) sXn' cpl fctn(rel(i\), rel(IT

2)) is recursively isomorphic to [01- 0

2], and Y= fctn(x,y) if yEXn-l and x=t=L.

For the proof the facts that 0

1 and 0

2 can be embedded into [0 1 + 0

2] and [01x0

2] and that 0

2 canbeembeddedinto [IT

1-!Y

2] areused. Insteadof a complete proof we only define as an example a function prdn. For (A

1,p

1),

(A2,p

2)ED

5 define (A,p)= prdn((A

1,p

1),(A

2,p

2)) as follows. Let (A',p') := prd((A

1,p

1),(A

2,p

2)). Define a function q: A ' - IN by al E <{l, ... ,n- 1}, IN>

- 11 -

Let A:= range(q) and a¹p¹b^{1 ~ :} q(a¹)pq(b^{1) .}

Then the function prd has the desired properties. In each of the three cases

n

the computable translation functions for the numberings of the basises which deter- mine the isomorphism (cf. Definition 12) can be computed from x and y. We show by two examples how ¹¹homogeneous¹¹ domain isomorphisms can be solved.

Example 14 Problem: Let 0

0 and D

1 be recursive domains such that x

0 := re1(0

0 ) and

x1 := rel (D) have the following graphs:

X 0

Find a recursive domain O which is recursively isomorphic · to [D x 0

1] such that IT can be embedded into 0.

0

Solution: Notice that 0= <0,0>, 1= <1,0>, 2= <0,1>. Define g: D - D by

s s

g(x)= p~d

3(x,x

1). Then g is computable and x

0 1= g(x

0 ) . by Lemma 13. Let

x:=Ugi(x ), then x is smallest yED with x '=Y and g(y)=y. Since g is

0 S 0

computable, x is a computable element of D. The following diagram shows the first

2 s

three elements of the sequence x

0 ,g(x

0 ),g (x

0 ) , . . . .

X :

0

g(x ):

0 <3,<2,5>>

2

<3,<1,5>>

1

<3,<0,5>>

0

C

2

shortly b

1

a 0

- 12 -

<3, <c, 5->>

g2(x ) :

0

0

Let IT:= cpl (x). Since x ^!:: x, IT can be embedded into IT, and since

0 0

O= cpl(x)= cpl g(x)= cpl prd

3(x,\)= cpl prd

3(rel(IT), rel(IT

1)), IT is recursively isomorphic to [IT ^x IT

1] by Lemma 13. Notice that the embedding and the isomorphism can be determined effectively.

Example 15 Problem: Let D

0 , 0

2 be a recursive domains. Find a recursive domain !r such that 00 can be embedded into IT and IT is recursively isomorphic to [[!r+ U)x[D-D]).

Solution: Let (A

0 ,p

0

) := rel(0

0) . Define 0

1 := cpl(x

1), where x

1 := (A

1,p

1),

A1 := {<0,a> ¹ aE A

0} , p

1 = {(<0,a>,<0,b>) ¹ ap

0b}. Then 0

0 and 0

1 are recursively isomorphic, and x

1 EX (see Lemma 13). Oefine g: 0 - 0 by

0 S S

g(x) := prd~(sum(x,x

2),

^fc~1(x,x)). Then g is computable and x₁ ^!:: fct₁(x₁,x₁^)!:: g(x₁⁾ by Lemma 13. Let y:= U gi(x

1). Then y is computable, x

1 ~ y, and y= g(y). There- fore, 0 can be embedded into 0:= cpl(y), 0 is a recursive domain and recursively

0

isomorphic to [[O+ 0

2] x [0-0)) by Lemma 13.

The formalism admits to show that the solutions in Theorems 10 and 11 andin Examples 14 and 15 can be determined concretely as well as the isomorphism and embeddings in the last two cases. For technical reasons we have used numbers iEIN and functions pEIB= INlN as names for objects. In practice it is more convenient to use words and w-sequences of words as names. This change, however, does not effect the essentials of the approach. Some more details of the formalism can be found in

Chapter 3.7 of the author's book [10).

The method for solving domain equations suggested in this paper depend~ on a rather simple formalism which can be well justified by intuition, and the examples show that the constructions of solutions are very concrete.

- 13 -

REFERENCES

[1] Scott, D.S., Strachey, C.: Toward a mathematical semantics for computer languages; Technical monograph PRG-6, University of Oxford (1971).

[2] Stoy, J.E.: Denotational semantics, the Scott-Strachey approach for programming language theory, MIT Press, Cambrindge Massachusetts (1977).

[3] Gordon, M.: The denotational description of programming languages, an introduction; Springer Verlag, Berlin (1979).

[4] Tennent, R.D.: The denotational semantics of programming languages, Comm. ACM 19, 437-453 (1976).

[5] Scott, D.S.: Continuous lattices; in: Lawvere, F.W. (ed.), Toposes, algebraic geometry and logic, Springer Verlag, Berlin (1972).

[6] Scott, D.S.: Dato types as lattices; SIAM J. on Comp. 5, 522- 587 (1976).

[7] Scott, D.S.: Domains for denotational semantics; A paper prepared for ICALT ¹82 Aarhus, Denmark (1982).

[8] Sanderson, J.G.: The lambda calculus, lattice theory and reflexive domains;

Mathematical institute lecture notes, University of Oxford (1973).

[9] Egli, H., Constable, R.L.: Computability concepts for programming language semantics; Theoretical Computer Science 2, 133-145 (1976).

[10] Weihrauch, K.: Computability; Springer Verlag, Berlin, to appear (1986).

[11] Weihrauch, K., Schäfer, G.: Admissible representations of effective cpo¹s, Theoretical Computer Science 26, 131- 147 (1983).

[12] Kreitz, C., Weihrauch, K.: Theory of representations, Theoretical Computer Science 28, 35 - 53 (1985).

[13] Weihrauch~ K.: Type 2 recursion theory, Theoretical Computer Science 28, 17- 33 (1985).

[14) Rogers, H.Jr.: Theory of recursive functions and effective computability:

McGraw Hill, New York (1967).