Electronic Notes in Theoretical Computer Science 6 (1997)

URL: http://www.elsevier.nl/locate/entcs/volume6.html 21 pages

Tensor Products and Powerspaces in Quantitative Domain Theory

Philipp Sünderhauf1

Department of Computing

Imperial College London SW7 2BZ, England P.Sunderhauf@doc.ic.ac.uk

Abstract

One approach to quantitative domain theory is the thesis that the underlying boolean logic of ordinary domain theory which assumes only values in the set {true,false} is replaced by a more elaborate logic with values in a suitable structure V. (We take V to be a value quantale.) So the order C is replaced by a generalised quasi-metric d, assigning to a pair of points the truth value of the assertion x C y.

In this paper, we carry this thesis over to the construction of powerdomains. This means that we assume the membership relation € to take its values in V. This is done by requiring that the value quantale V carries the additional structure of a semiring. Powerdomains are then constructed as free modules over this semiring.

For the case that the underlying logic is the logic of ordinary domain theory our construction reduces to the familiar Hoare powerdomain. Taking the logic of quasi-metric spaces, i.e. V = [0, oo] with usual addition and multiplication, reveals a close connection to the powerdomain of extended probability measures.

As scalar multiplication need not be nonexpansive we develop the theory of moduli of continuity and m-continuous functions. This makes it also possible to consider functions between quantitative domains with different underlying logic. Formal union is an operation which takes pairs as input, so we investigate tensor products and their behavior with respect to the ideal completion.

1 Introduction

Quantitative domain theory [16,2,15,6] refines ordinary domain theory by replacing the qualitative notion of approximation by a quantitative notion of de-

1 Partly supported by the Deutsche Forschungsgemeinschaft and by the E.P.S.R.C. project Foundational Structures for Computer Science at Imperial College.

© 1997 Published by Elsevier Science B. V.

gree of approximation. So the order C is replaced by a distance function d and the value of d(x, y) represents the truth-value of the assertion x C y (cf, the introduction of [6]), The present paper chooses the setting of V-domains [6], where d takes its values in a value quantale [3]. Section 2 collects all necessary preliminaries on V-domains from [6].

The main goal of this paper is to develop a theory of powerdomains. A powerdomain construction is supposed to give formal sets of elements in order to model the outcome of certain nondeterministic computations. In previous constructions for quantitative domains (see e.g. [17,18,4,2]) the underlying logic of the membership relation is classical logic: Applying them to the one-point space yields a one or two point domain, depending on whether they allow the empty set or not. Also, these powerdomains have representations as collections of certain subsets. In the present paper we take a different approach: We assume that the underlying logic of "e" is the same as that of it takes its values in V instead of {true,false}.

Having a V-valued membership relation demands a richer structure on the powerdomains. If, for example, V = N U {oo} then we get a bag or multiset powerdomain. Hence one would want to have operations such as "start with the set M and take all its members five-fold" or "add two copies of x to M", So in general we want an operation x m • x for m e V and x an element of the powerdomain. This has two consequences: (a) We need a multiplication on V to capture the maps x m, • (n • x) and (b) we cannot expect these operations to be nonexpansive and are forced to leave familiar territory.

That is because quantitative domains are usually considered with nonexpansive functions. Apart from the problem (b) above, this seems quite a restriction as quantitative domain theory is also supposed to be a generalisation of the theory of metric spaces, where many other interesting classes of uniformly continuous functions are studied (e.g. Lipsehitz-, Holder-continuous), In Section 3, we introduce the notions of modulus of continuity and m-continuous function to overcome this problem and get the result that these functions are well-behaved with respect to the ideal completion. This framework makes it also possible to consider functions between quantitative domains for different value quantales.

The operation of formal union is in the centre of interest in theory of powerdomains. It takes pairs of elements as input, so for a systematic account it is necessary to investigate products. The familiar product of posets has two different manifestations in quantitative domain theory: the categorical product with sup-distance and the tensor product, where the distance between two pairs is the sum of the distances between the components. These two constructions coincide in the case that the addition on the value quantale is the maximum, making the theory of quasi ultrametric spaces particularly well-behaved. This also shows that tensor products provide the more general framework. In Section 4, we develop the necessary tools in order to handle formal union as a function of type X®X —X. We are interested in lifting

the operation to the ideal completion. Even though, unexpectedly, it is not necessarily true that Idl(X®F) = Idl(X) ® Idl(F), we are able to prove that the desired lifting is always possible.

In Section 5 we finally turn our attention to powerdomains. As mentioned in (a) above, we want to have operations x m • x and a multiplication on the value quantale V. Hence we introduce V-semirings which are value quantales carrying the additional structure of a semiring, Powerdomains are then V-modules: Modules with additional compatibility conditions for the distance function. Using the tools provided in Sections 3 and 4 we show that this module structure may be lifted to the ideal completion. This is used in Section 6 for the construction of the free algebraic V-module over an algebraic V-domain, The construction resembles the powerdomain constructions of ordinary domain theory, where finite subsets of compact elements are suitably preordered and the ideal completion is performed.

We conclude this paper by investigating the construction for certain special choices of the semiring quantale V. For the case of ordinary domain theory (for a technical reason, we have to take V = {0, l,oo}) our powerspace construction specialises to the Hoare powerdomain. More interesting is the case of V = [0,oo], where our construction is closely related to the construction of the domain of all continuous valuations [10], If an ordinary algebraic domain D is thought of as a [0, oo]-domain with two-valued distance function, then the powerdomain of extended probability measures of D (also known as free domain cone) is closely related to the free V-module over D. (In [19] we show that it is even possible to embed ordinary domains in V-domains in such a way that the free domain cone and the free [0, oo]-module coincide.) Similar considerations apply to the powerdomain of possibility measures introduced in [9], where [0,oo] is taken with the maximum operation as addition.

2 A primer on V-domains

Let us briefly introduce the basic notions of quantitative domain theory as developed in [6]. This theory goes back to ideas of Smyth [16] and Lawvere [12]. The first concept is that of a value quantale, which the reader might think of being the extended non-negative reals [0, oo]. It is a structure (V, <, +) where (V, <) is a completely distributive complete lattice and + an associative, commutative binary operation on V which preserves arbitrary infima. Moreover, 0 oo is required to hold, where 0 is the least and oo the largest element of V. Also, 0 is demanded to be an identity for + and the set of all elements well above 0 is supposed to be filtered. Here an element x is well above y (abbreviated as x y y), if y > inf A implies that x E fA It is well-known that complete distributivitv is equivalent to the condition x = inf {y E V \ y y x} for all x E V [14]. Value quantales are studied in detail in [3]. We will repeatedly use the fact that addition is continuous, i.e. that p y a + b implies the existence of a' y a and b' y b such that p y a' + b'. This is a consequence of

the following lemma.

Lemma 2.1 A function f: V —V preserves arbitrary infima if and only if it is monotone and continuous in the sense that p y f(x) implies the existence ofy y x such that p y }'{y).

Proof. If / preserves arbitrary infima, it is clearly monotone. Moreover p y f(x) = f(m{yyxy) = infyyxf{y) implies that there is y y x such that p y

Now suppose that / is monotone and satisfies the above condition. Then /(inf A) < infaeAf(a) by monotonieitv. For the reverse inequality, suppose p y /(inf A) then there is y y inf A with p y f(y). But y y inf A implies the existence of a E A with y > a. So p y f(y) > f(a) > infaeAf(a), hence the assertion follows, □

Examples of value quantales are (C, <,max), where (C, <) is any complete chain. Of our special interest are the cases C = {0,1}, C = {0,loo}, C = [0, oo] and C = [0,1], For the latter two, it is also possible to consider (truncated) addition of real numbers as +,

For the following definitions, it is necessary to fix the value quantale V. A distance function on a set X is a function d:XxX —V such that (1) d(x, x) = 0, (2) if d(x,y) = 0 = d(y,x) then x = y, and (3) such that the triangle inequality d(x, y) + d(y, z) > d(x, z) holds for all x,y,z E X. The order derived from d is defined by x □,/ y id' d(x, y) = 0, A V-poset is a pair (.X,d), where X is a set and d a distance function. This notation is chosen as V-posets may in fact be interpreted as generalized partially ordered sets. To do so, think of V as the set of truth-values, of 0 as true, of + as logical conjunction &, and as the relation p > q as entailment p b q. The values of the distance function d may be thought of as the "truth value" of the assertion ux C y". In this setting, the triangle inequality corresponds to the law of transitivity, d(x,x) = 0 reveals reflexivitv and the second assumption is antisymmetry. This logical interpretation is studied in greater detail in [6]. A function /: X —Y between V-posets is nonexpansive if d(x, y) > d(f(x), f(y)) holds for all x, y E X. The category of V-posets and nonexpansive maps is denoted by VPOS,

We denote with Be(x) = {y E X \ e y d(x,y)} the open e-ball around x. The opposite V-poset of (X, d) has distance d^(x, y) = d(y, x) and is denoted by JY"_1.The value quantale V itself is a V-poset with distance function

d{p, q) = q^p = inf {r E V | p + r > q},

which is defined through the function Ax.x^p which is the lower adjoint of Ax.p + x, i.e we have

p + r > q r > q^p.

Note that the order Cd derived from this distance is the opposite of <,

An ideal on the V-poset X is a nonexpansive map cp: X^1 V such that

(1) there is an x such that oo ip(x) and (2) whenever €\ y <p(x\), e2 >~ <£>(£2) and S y 0, there is an x such that S y <p(x), y d{x\,x), and €2 y d{x2,x). Each element x of X induces an ideal, [x], defined by

W (y) = d(y,x), y EX.

These ideals are called representable. An element a of X is the supremum of the ideal <p, denoted by a = \J p>, if for all b E X, we have d(a,b) =

Mipr v (d(x, 6)^99(2;)^ , The V-poset (X, d) is directed complete or a V-domain. if every ideal <p on X has a supremum,

A subset U of a V-domain X is Scott open if for all ideals <p on X, if V V e U, then there are e y 0 and x E X such that e y ip{x) and Be(x) C U. The Scott topology is the collection of all these subsets and denoted by ax- A function /: X —Y is Scott-continuous, if it is continuous with respect to the Scott topologies on X and Y. For nonexpansive maps there is an alternative characterisation of Scott-continuitv: If <p is an ideal on X, then the direct image of under /, denoted by /(99), is the function from Y to V defined by

(1) /(99)(y) = mf (ip{x) + d{y,f{x)j), for y e Y.

This defines an ideal on Y. Now / is Scott-continuous iff \J f(ip) = /(V^9) holds for all ideals 99 on X. We denote the category of V-domains with nonexpansive Scott-continuous maps by VDOM,

An element a in a V-domain X is compact if d(a,\J 99) > 99(0) for all ideals 99 on X. An ideal 99 is compactly generated, if

ip(x) = inf {(p(k) + d(x,k) | k compact}

holds for all x E X. (I.e., if 99 is the direct image under the subset inclusion of its restriction to the compact elements.) A V-domain is algebraic, if for all x E X there is a compactly generated ideal 99 on X such that x = \J tp. The full subcategory of VDOM comprising all algebraic domains is denoted by VALG.

Let Idl(X) denote the set of all ideals on X. A distance function on this is given by

d((p,xp) = sup d((p(x),xp(x)).

Via Lx-X Idl(X) with ix(x) = [x], a V-poset X is isometrieally embedded in Idl(X), The latter is directed complete, it is the ideal completion of X. Even more, Idl(X) is an algebraic V-domain and {i?£([a;])}e^o^ex is a base for its Scott topology. The assignment Idl(/) = /, with / as in (1) above, makes Idl a functor VPOS —VDOM, Moreover, the usual universal property holds: For any V-domain Y and any nonexpansive map J: X ^ Y there exists a unique Scott continuous and nonexpansive map f:ld\(X) Y such that / o rjx = /. Every algebraic V-domain is isomorphic to the ideal completion of the subset of its compact elements.

Finally, we should note that directed completeness, the Scott topology

and Scott-continuitv may also be characterized using nets. This approach was initiated in [16] and is developed in [15,2],

A net (xi)iei on X is forward Cauchy, if for all e y 0, there is i E I such that whenever i < j < k we have e y d(xj,xk). A point x E X is the directed limit of the net, denoted by x = limJeJa;i, if d(x,y) = m{ieI supj>id(xj,y) holds for all y E X. In Section 8 of [6] it is shown that a V-poset is directed complete iff all forward Cauchy nets have a directed limit, that a nonexpansive function between V-domains is Scott-continuous iff it preserves directed limits and that a set O C X is Scott-open iff it is open with respect to the e-ball topology and whenever limja^ e O for a forward Cauchy net (xi)ieI, then there is i El such that B£(xj) C O for all j > i. In addition, it is readily seen that an element k E X is compact iff d(k, limj xi) = liir^ d(k, Xi) holds for all forward Cauchy nets {xi)i£i and that a space is algebraic iff every element is the directed limit of a forward Cauchy net of compact elements,

3 Moduli of Continuity

Definition 3.1 Suppose V and W are value quantales, A modulus of continuity between V and W is a function m\ V —W with

• m{p + q) = m{p) + m{q)

• rn( 0) = 0

• m(inf A) = infae4 m{a)

for all p,q E V and A C V. If (A,dx) is a V-poset and (Y,dY) is a W-poset then a function /: X —Y is called m-continuous, if

m(dx(x,y)) > dY(f(x),f(y))

holds for all x, y E X.

In particular, a function is Id-continuous if and only if it is nonexpansive.

Lemma 3.2 If m is a modulus of continuity then m{p^q) > m{p)^m{q).

Proof. We have m(p^q) + m(q) = m((p^q) + q) > m(p) as (p^q) + q > p. Thus the assertion follows, □

Theorem 3.3 Every m-continuous f: X —Y has a unique Scott-continuous m-continuous extension /:Idl(A) —Idl(F).

Proof. We define / by

(2) f{v){y)=inix(m{v{x)) + d{y,f{x))).

Let us first check that this defines an ideal. As oo >~ m(0), there exists by continuity (Lemma 2,1) p y 0 with oo y m{p). Pick x E X with p y <p(x) and set y = f(x). Then oo >~ m(ip(x)) > f(ip)(y). Now assume that S y 0 is given and that £i y f(ip)(yi) holds for i = 1,2, Pick 5' y 0 such that

S y rn(S'). There are Oj with et y m(y(oj)) + d(iji, f(a,i)). Hence we might pick e'^Si such that ^ m{e'j) + Si, e'i y and Si y d(yi, f(a,i)). As

is an ideal, we can find x E X such that e'i y d(a,i,x) and S' y ip(x). Then £i y m(e$) + Si > m(d(ai,x)) + d{yi}/(a*)) > d(f(ai),f(x)) + d{yi} f{a,i)) > d(yi,f(x)). Moreover S y m(S') > m(ip(x)) > f(ip)(f(x)). This proves that f{<p) E Idl(F).

To see that this function extends / we calculate

/(N)(y) = int(m(d(x, a)) + d(y, f(x)))

>M(d(f(x),f(a)) + d(y,f(x)))

> inf d(y,f(a))

= d(y,f(a))

=[f(om.

Picking x = a in the first line reveals that in fact equality holds. Now let us check m-continuity.

, fW) = SUP Qnf (m,(i/j(x)) + d(y, f(x))) inf (m{(p{a)) + d(y, /(a))) )

= sup sup ( inf (rn(ip(x)) + d(y, f(x))j — (d(y, f(a)) + rn((p(a))))

y&aex V r s v ' '

< sup sup ( inf ((rn(ip(x)) + d(y, f(x)))—d(y, f(a))) -=-m(y(o)))

y&a<EX V r s v ' '

< sup sup ( inf (rn(ip(x)) + d(f(a), f (x)))-=-m((p(a)))

y&a<EX V r s v ' '

< sup (mf (m(il>(x) + d(a, x))^j -=-m(y(o)))

< m(sup (mf (xl>(x) + d(a, )

= m(sup (^>(0)^99(0)^)

= m,(d((p, -0)),

Next Scott-continuitv, By Theorem 23 of [6], the Scott topology on Idl(X) has as a base the sets of the form B£([x]), where e y 0 and x E X, and similarly for Idl(y). So assume e /_1(Se([y])), i.e. e y d([y],f(cp)) = }{<p){y). Then there are e' y f(ip)(y) and S y 0 such that e y e' + S. We have seen above (when we proved that f(ip) is an ideal) that this implies the existence of x E X with e' y d(y, /(x)) and S' y (p(x), where S y rriS'. We claim that 99 e By ([a;]) and f(Bgi([x])) C B£([y]). The first is true as S' y ip(x) = d([x],ip). To verify the subset inclusion, observe first that (as above) r] E By ([a;]), i.e. S' y rj(x) implies S y rriS' > mrj(x) > f{rf){f{x)) = d([f(x)],f(rj)). Then we have e' y d(y,f(x)) = d([y],[f(x)]). Using the triangular inequality, we get e y e' + S > d([y], [f(x)]) + d([f(x)],f(n)) > d([y],f(n)), thus f(n) E B£([y]). This completes the proof of Scott-continuitv.

Finally uniqueness. This follows from the fact that / is prescribed on the set of representable ideals. This set is dense in the ideal completion with respect to the supremum of the Scott topology and the dual Alexandroff topology (the topology induced by dThis join topology is Hausdorff and so the extension is unique by continuity [5, Lemma 22 in conjunction with Theorem 29], □

Uniqueness of this extension has a somewhat surprising consequence with respect to the concrete description in (2), Suppose J: X —Y is m-eontinuous and n: V —W is any modulus of continuity such that m < n (pointwise). Then / is certainly n-eontinuous and thus has a unique n-eontinuous lift. But the m-eontinuous lift is n-eontinuous, too, so these two coincide. Thus

(3) mf (m{(p{x)) + d(y, f(x))) = inf (n(<p(x)) + d(y, f(x)))

for all y E Y and tp E Idl(A) in this case. This observation will be used in the proof of Theorem 5,4,

3.1 Multiplication by natural numbers

For a natural number n E N and p EV define the 'product' np by

np := p + p + ,,, + p. "--'

n times

This multiplication is distributive, monotone, and preserves 0, Hence, in the light of Lemma 2,1, continuity is the only thing missing so that the map p i—Y np: V —V is a modulus of continuity. Unfortunately, it need not hold. Even though it is true that if p y q+q, there are q', q" y q such that p y q'+q", we in general may not choose q' = q".

Definition 3.4 The value quantale V has n-refining addition, if whenever p y nq, there is q' y q with p y nq'.

Corollary 3.5 Suppose X and Y are V-posets, and n E N. If the addition on V is n-refining, then every n-continuous f: X —Y has a unique Scott-continuous n-continuous extension /:Idl(A) —Idl(F).

4 Tensor products of V-domains

Using the logical interpretation outlined in Section 2, the coordinate-wise order on the set XxY translates to the distance function

d((x, y), (x', y')) = d(x, x') + d(y, y').

We denote the arising space by X®Y and call it the tensor product. In fact, this tensor product turns VPOS into a symmetric monoidal closed category, where the function space is the set of nonexpansive maps equipped with the sup-distance (Proposition 3,3 of [21]),

In order to investigate the behavior of tensor products with respect to directed completeness, we have to consider ideals. For V-posets X and Y, we define a function J:Idl(A®F) ^ Idl(A) ® Idl(F) with J($) = ($ i, $2) where

Lemma 4.1 The function J is well defined and satisfies + >

d(J($), (I.e., it is 2-continuous).

Proof. Let 7r: X ® Y —X denote the canonical projection on X, which is nonexpansive. Then

Specialising a = x in this formula and nonexpansiveness of $: (A ® F)-1 —V gives #($)(a;) = infheY$(x,b), thus #($) = $1 and so the function J is well-defined, Now 7T is nonexpansive, and so

and similarly for d($2,^2). This yields + > 1,^1) +

Going in the other direction, it is also easy to construct a map: If <p e Idl(A) and tj) e Idl(F) define <p + ip by (cp + ip)(x, y) = 99(2;) + V'(y)-

Lemma 4.2 A: + V> is a nonexpansive function from Idl(A) ®

Idl(F) to Idl(A®F). Moreover, A and J are inverses of each other.

Proof. Let us first check well-definedness of A, i.e. that A(<p, ip) is an ideal on X 0 Y. Pick p y 0 such that 00 y p + p. Then there are x e X, y e Y with p y ip{x) and p y xp(y). Hence 00 y ¡p{x) + xp(y) = A((p,xp)(x,y). If £i y p(xi) + r (//,-) for i = 1,2, then there are ef, e\ with ei y ef + ef, e'f y ip(xi), and e\ y ip(yi), If S y 0 is given, then for 5' y 0 with S y 5' + 5' there are a E X and bey such that 5' y (p(a),ip(b) and e'f y d(xi,a), and £J >~ d(yi,b). Then S y cp(a) + ip(b) and ^ d((xi,yi), (0,6)), Thus A(cp,ip) is indeed an ideal.

To see nonexpansiveness, we calculate

If $ = (p+ip, we get with the above notation $1(2;) = infheY((p(x)+ip(b)) = ip{x) + inf|,ey xp(b) = <p(x) and similarly $2 = -0, So J o A = Ididi(A-)®idi(r)-

Seeing A o J = IdMi(x®y) is considerably harder. Assume $ e Idl(A®F) and p y A o J($)(a;, y) = in{aeX,beY($(0, y) + b)), Then there are a e X and b e Y with p y $(0, y) + $(2;, b). Pick pi,p2, $ such that p y pi + p2 + S,

$i(a;) = inf $(x,b) and $2(y) = inf $(0,

b& a£X

= inf ($(0, b) + d(x, a)).

a£X,b£Y

Pi y <&(x,b), P2 y $(a,y), and S y 0, As $ is an ideal, there exists (x',y') E X®Y such that S y <fr(x', y'), pi y d((x, b), (x', y')), and p2 y d((a, y), (x', y')). Then

p y S + pi + p2

> y') + d((x, b), (x', y')) + d((a, y), (x', y'))

> $(V, y') + d(x, x') + d(y, y')

Thus A o J($) > Now suppose p y <&(x, y). Pick S y 0 and p' y $(a;, y) such that p y p' + 5 + 5. There is (a,b) EX® Y with S y $(o,6) and p' y d((x,y), (a,b)). Then p S + S+p' > $(o, b) + d(x, a) + $(o, b) + d(y, b) > $(a;, b) + $(o, y) > infaex,6ey($(°, v) + = AJ{<&){x, y). Thus the two

functions are inverses of each other, □

One would now expect to be able to prove that the function J is actually nonexpansive, i.e. that Idl(X) ® Idl(F) = Idl(X®F), But in general this is not true as the following example shows.

Example 4.3 Take V = u + 2 = {0,1, 2,,,,, u, oo} with the order 0 < 1 < ,,, < u < oo. Addition is usual addition of natural numbers extended to V by setting p+u = u+p = oo for p 0 and p+oo = oo+p = oo for all p E V, Take X = lo and Y = {0,1} where d(a,b) = a^b on X and d(0,1) = d(l,0) = 1 on Y. Then Y is directed complete and Idl(F) = Y. The space X is not directed complete and Idl(X) = u + 1, But Idl(X) ® Idl(F) is not complete.

To see this, consider the ascending sequence (0, 0), (1, 0), (2, 0),____ Being

ascending, it is clearly a forward Cauchv sequence and ought to have a limit. This could only be {lo, 0), the pair of limits of the projection sequences. But d((u, 0), (0,1)) = d(u, 0) + d{ 0, l)=w + l = oo#u; = lim„ d((n, 0), (0,1)), so this is not the limit.

So Idl(X) ® Idl(F) need not be complete. The following theorem shows, however, that this is the only thing this space falls short of being the ideal completion of X®Y.

Theorem 4.4 Suppose that the addition on V is 2-refining and X and Y are V-posets. Then every nonexpansive function f: X ® Y —Z, where Z is directed complete, has a unique ami(x) x cidi(Y)-<?z-continuous nonexpansive lift f: Idl(X) ® Idl(F) —yZ. If Z is a W-domain and m/.V —W is a modulus of continuity, then the same holds with lm-continuous' replacing 'nonexpansive''.

Proof. The function / has a unique m-eontinuous Scott-continuous lift / to Idl(X®F), By Theorem 23 of [6], the Scott-topologv on Idl(X) has as a base the collection of all e-balls around representable ideals, and similarly for Y and X ® Y. Hence A and J constitute a homeomorphism between (Idl(X) ® Idl(F), <Tidi(A") X(JHi(y)) and (Idl(X®F), o-ldi(X0r)). Thus / := foA is a continuous lift of / which is m-eontinuous as A is nonexpansive. It remains to show unieity. Suppose g\ Idl(X) ® Idl(F) —t Z is another such lift. Then

g o J: Idl(A®F) —tZ is a continuous, (rn + m)-continuous lift of /, So is /, hence / = g o J by Theorem 3,3, So f = f o A = g o J o A = g. □

Lemma 4.5 J/Idl(A)®Idl(F) is complete, then its Scott-topology is the product of the Scott-topologies on the factors.

Proof. If Idl(A) ® Idl(F) is directed complete, then the limit of a Cauehv net is the pair of the limits of the projection nets. Hence the result follows by the characterisation of the Scott topology in terms of nets, □

Theorem 4.6 Suppose that the addition on V is 2-refining. J/Idl(A)®Idl(F) is directed complete, then it is isomorphic to Idl(A®F). The isomorphisms are given by A and J.

Proof. In this case, there is a unique nonexpansive Scott-continuous lift J': Idl(A®F) ^ id!(.V)::-:id!(n of the function i: (x, y) ^ ([ai], [y]) to Idl(A® Y). Also, there is a unique 2-eontinuous Scott-continuous lift of i. Both J and J' qualify, hence J = J' and the Theorem is proved, □

This treatment of tensor products leaves some open questions, A further investigation is not in the scope of the present paper.

Problem 4.7 Give (necessary and sufficient) conditions on V or on X and Y so that Idl(A)®Idl(F) is directed complete. Find full subcategories o/VDOM and VALG closed under

5 V-modules

Definition 5.1 For a value quantale V, let Vo := V \ {oo} be the set of its finite elements. A semiring quantale is a value quantale (V, <,0,+) with an additional multiplication •: V0 x V0 —V0 (not necessarily commutative) and a specified finite element 1^0 such that (V0,0,1,+, •) is a unital semiring. Moreover, we extend the definition of the multiplication to V0 x V by setting m ■ oo = oo for m e Vo and demand that for all m e Vo, the map x mx: V —V preserves all infima, (We usually write rnn in place of m • n.)

It is part of these axioms that the addition on V is finitary in the sense that for all p,q E V we have that p + q = oo implies p = oo or q = oo, (In other words, (oo ypkoo>-q)^oo>-p + q.) The logical interpretation of this axiom is very intuitive. It states that if oo is the truth-value of pkq, than it is the truth-value of one of p, q. This means that oo behaves like false in classical logic.

As for addition, preservation of infima implies continuity of multiplication by Lemma 2,1: If p y mx then there is y y x such that p y my. In particular taking x = 0 reveals a law of repeated subdivision: If p y 0 and oo y m, then there is r >~ 0 such that p y rn/r. The requirements on the multiplication imply that x mx is a modulus of continuity for oo y m. We will say that

a function f: X —Y is Lipschitz continuous if there is m E Vo such that / is m-continuous, i.e. such that m, • d(x, y) > d(f(x), f(y)) holds for all x,y E X.

Examples of possible multiplications are the usual multiplication on [0, oo] for both the usual addition and max as +, and the multiplication of {0,1} for V = ({0,1, oo}, <, max).

In the remainder of this paper V is assumed to be a semiring quantale.

Definition 5.2 A V-module (X, d; +, •, 0) consists of a V-poset (X, d) with a scalar multiplication • : Vo x X —X, an addition + : I x I 4 I, and a special element 0 E A' satisfying the usual algebraic axioms of modules:

(4) x + y = y + x

(5) x + (y + z) = (x + y) + z

(6) x + 0 = x

(10) (m + n) • x = m ■ x + n ■ x

(11) m, ■ (x + y) = m, ■ x + m, ■ y

(12) (mn) • x = m ■ (n • x)

for all x,y,z E X and m,n E Vo- Moreover, we demand for x,y,z E X and rn EVo the following:

(13) d(x,y) > d(x + z,y + z)

(14)md(x,y) > d{mx, my)

(15) m < n => d(mx, nx) = 0

A morphism of V-modules X and Y is a nonexpansive map /: X —Y such that f{x + y) = fix) + fiy) and /(mi) = m • fix) hold for all x,y E X and m E Vo-

If (X, d) happens to be an algebraic V-domain and if addition and scalar multiplication with a fixed scalar are Scott-continuous (i.e., + is oxXOx-vx-continuous and Xx.rn ■ x is ox-ox-eontinuous) for all scalars m E Vo, we speak of an algebraic V-module. (See Lemma 5,3 below for continuity of the function Arn.rn • x.) Morphisms of algebraic V-modules are required to be Scott-continuous,

Using the triangle inequality, it is easy to see that the requirement (13) on d is equivalent to demanding that +: X 0 X X is nonexpansive. Axiom (14) is m-eontinuity of multiplication with the fixed scalar m, viewed as a function of type X X. The condition in (15), finally, is monotonieity of scalar multiplication with a fixed point x E X. This implies in particular that 0 is the least element of X. In conjunction with the other axioms, however, we are able to deduce that the function rn rn ■ x: V X is also Lipsehitz-continuous.

(8) (9)

m • 0 = 0

0 -x = 0

1 • x = x

Lemma 5.3 If (X, d; +, 0) is a V-module then

(n—m) • d(x, 0) > d(nx, mx) holds for all x e X and m,,n eVq.

Proof. Using m + (n—m) > n, we calculate

d(nx, mx) < d(nx, (m + (n—m))a;) + d((m + (n—m))x, mx)

Theorem 5.4 If X is a V-module thenldl(X) is an algebraic V-module. IfY is an algebraic V-module and f: X —Y is a morphism of modules then its lift f: Idl(A) —Y is a module-homomorphism, too.

Proof. The zero is given by [0] = d(-,0). Addition is a nonexpansive map X 0 X X. Hence there exists a unique Scott-continuous nonexpansive lift to Idl(A) ® Idl(A) by Theorem 4,4, Explicitly, it is given by

For fixed m e V0, the map x ^ mx: X ^ X is m-eontinuous. So there exists by Theorem 3,3 an m-eontinuous lift to Idl(A), Explicitly, we have

These lifts are unique with the property of being Scott-continuous, This proves validity of the module equations (4)-(12). Each side of an equation, take e.g. eommutativity x + y = y + x, defines a Scott-continuous and nonexpansive (resp, Lipschitz-continuous) lift of the corresponding function on X, X ® X, or even X 0 X ® X (for associativity). The equation holds on X and so both sides are lifts of the same function and thus coincide.

It remains to consider the inequalities involving the distance. Axioms (13) and (14) hold since addition lifts to a nonexpansive map and multiplication by m to an m-continuous one. Finally (15), Suppose n < m We have to show that d(n(p,m(p) = 0, ie, that (mip)(y) < (nip)(y) holds for all y E X. So fix y e X and pick p y (nip)(y). As n < m, the function Ax.nx is in fact m-continuous, and so we have

Hence there is x e X with p y m,(p(x) + d(y,nx). As d(nx,mx) = 0 by assumption, we conclude p y mip{x) + d(y,mx) > (mip)(y). So the assertion follows.

The homomorphism property for the lift of a nonexpansive module homo-morphism holds for the same reason as the equations (4)-(12): Both sides of

= d((n—m)x + mx, 0 + mx)

< d((n—m)x, 0)

< (n—m)d(x, 0)

to get the result.

f(x + y) = f(x) + f(y) and /(mi) = rnf(x), respectively, define functions which uniquely extend to the ideal completion and thus coincide, □

6 A free construction

In order to be able to construct the free V-module over a V-domain X, we need an additional piece of information. What should the distance d({|a;|},0) be? It turns out that the concept of a bottom predicate is the appropriate answer.

Definition 6.1 A pointed V-poset is a triple (X, d, b) where (X, d) is a V-poset and b : X —V is a function (the bottom-predicate) such that for all x, y E X, we have that b(x) > d(x, y).

A map / : (X, d, b) —(X', df, b') between pointed V-domains is strict, if b(x) > b'(f(x)) holds for all x E X.

Example 6.2 1) Note that the constant function b(x) = oo is a bottom-predicate for any V-domain (X,d), As all functions are strict in this sense, ordinary V-domains form a full subcategory of pointed V-domains with strict maps,

2) If X has a least element (wrt Cd) _L, then b(x) = d(x,±) is a bottom-predicate, A nonexpansive function between two such spaces is strict iff it preserves the least element. This example includes all V-modules since 0 is their least element as observed above.

Now we are going to construct the free V-module over a pointed algebraic V-domain (X,d,b), where V is a semiring quantale. Denote by X0 the set of compact elements of X. We define F0(X0) to be the free algebra with respect to the signature of V0-modules (i.e. (0,2, (l)y0)) over X0. (Recall V0 = V\ {oo},) Furthermore let 9 be the congruence on X0 defined by the module equations (4)-(12). The generators of F0(X0) are denoted by {|a;|}, where x E Xq.

By induction, we define a (partial) distance function on this set. For x,y E X and r, s E V0 we set

(16) d0(r{\x\}, r{\y\}) = rd(x, y)

(17) dQ(r{\x\},s{\x\}) = (r^s)b(x)

Moreover, for cp, cp', ip E F0(X0) we define

(18) d0{<p + ip,<p' + ii>) = d0{<p,<p') (19) do(Tl> + <p,xl> + <p') = do(<p,<p')

Let R C F0(X0) x F0(X0) be the domain of d0. This enables us to define a distance on F0(X0). For E F0(X0), we write [(p,ip] for the set of all sequences a = ip[, p>2,..., p>n, <p'ri) with ((p,p>i) E 9, (p>'n,ip) E 9, and the

property that (<^¿,99-) G R and (99-, G 9 for all i. For such a sequence a we set

Now we define

d(tp,i/j) = inf w(a).

a£[tp,ip]

It is easy to see that this function satisfies the triangle inequality. Moreover, it is constant on 0-elasses and so defines a distance on the set

Po(X) := F0{X0)/e

which is the free Vo-module over A0 (in the sense of universal algebra). For simplicity, we will denote elements of Pq(X) by their representatives and use 9 as equality relation.

Proposition 6.3 Pq(X) is a V-module.

Proof. We have to verify that the distance function satisfies (13)—(15). If a = (<Pu ...,<Pn, v'n) e [<P, 4>], then a + C := (<pi + C, Vi + C, ■ ■ ■, <p'n + C) G [cp + (,ip + (]. As w(a) = w(a + (), we conclude d(ip + C, ip + C) = inf¡i^^^QwiP) < infae[¥,^] w(a) = d((p,tp), i.e. (13). Similarly for (14): A path a E [99, ip] gives a path m, • a E [mip, mip] and so we get md(ip, ip) > d(m<p>,mxp). Finally (15): If n < m, then there is a sequence a E [nip,mip] involving only 0-steps of the form nr{|a;|} ^ mr{\x\} with distance 0, hence d(mp, mip) = 0. □

Lemma 6.4 {| • |} : X0 —Pq(X) is an isometric embedding.

Proof. As ({|x|}, {|y|}) G [{M},{|y|}] holds, it is clear that d({\x\}, {\y\}) < d(x, y). The task now is to proof that there is no better path from {|x|} to {]y|} than this. We do this by showing that if a E [99, {|y|}], where 99 = ^¿=1 '"¿{W}? then w(a) > J2i=i rid(xi, y)- The proof is by induction on the length of a. The statement is clear for length 0. Suppose a = (ipi, ip[,..,, ipn, ip'n) with (<Pn> M) e e and = E*=i Then (tp2, ip'2,..., ip'n) E [ip[, M], thus

(20) w(a) > d0((pi, ip[) + ^ nd(xi, y)

by induction hypothesis. Depending on the ii-step (pi ip[, we have to distinguish two cases. Let us assume without loss of generality that it is the summand ri{|a;i|} which is changed in this step.

Case 1:

If 991 = ri{|w|} + Ei=2 '"¿{W} then do(ipi,ip[) = rid(w,xi) and thus

k k w(a) > rid(w, xi) + nd(xi, y) + ^ rtd(xi, y) > nd(w, y) + ^ ridixh V)

i=2 i=2

bv virtue of (20) and the triangle inequality. Case 2:

liipi = then d0((pi, (p[) = (s-ri)b(xi) > (s-ri)d(xi,y)

and thus

k k w(a) > (s-ri)d(xi, y) + nd(xi, y) + ^ rtd(xi, y) > sd(xi, y) + ^ rtd(xi, y)

i=2 i=2 by (20) and the fact that (s — ri) + ri > s, □

Lemma 6.5 Suppose f : X —F is nonexpansive and strict, where X is a pointed V-poset and Y is a V-module. Then the unique lift to a module homo-morphism f : Pq(X) —Y (which exists by universal algebra) is nonexpansive and strict.

Proof. The lift is given by f(Y^i=iri{\xi I}) = rif{xi)- Let us start by cheeking the two clauses in the definition of d0. For (16), we calculate

d(/> W), />M)) = d(rf(x), rf(y))

<rd(f(x)J(y))

< rd(x,y)

= d0(r{\x\},r{\y\}).

Then (17) yields

d(f(r{H)J~(4H)) = d(rf(x), sf(x))

< (r—s)d(f(x), 0) by Lemma 5,3 = (r-s)b(f(x))

< (r—s)b(x) as / is strict = 4HM},s{|a;|}).

Now suppose a = (tpi, ip[,..., ipn, (p'n) e [99, ip]. Then, by the above and the triangle inequality, w(a) > d(f(<Pi),^ d(f((p),f(ip)). Hence

Strictness is clear as / preserves 0 and is nonexpansive, □

Taking the ideal completion, we finally arrive at our goal. We define P{X) = Idl(-PoPO)- This is an algebraic V-module by Theorem 5,4, Moreover, X is embedded into P(X) by Idl({| • |}),

Theorem 6.6 PX is the free V-module over X in the category of pointed algebraic V-domains with strict nonexpansive Scott-continuous maps. In other words, for every strict nonexpansive Scott-continuous function X —Y, where X is pointed algebraic and Y is an algebraic V-module, there is a unique extension f : P(X) —Y to a nonexpansive Scott-continuous homomorphism of V-modules.

Proof. We construct / in two steps. First, the restriction of / to the compact elements of X lifts as a nonexpansive module homomorphism to Pq{X) by

Lemma 6,5, Then this lifts by Theorem 5,4 to a module homomorphism on Idl(.FbPO) = P(X). Uniqueness is guaranteed by the uniqueness of both these lifts, □

7 Examples

7.1 Ordinary domains

To recover ordinary domain theory, we take V = {0,1, oo} with addition 1 + 1 = 1, so that V0 = {0,1} which is the simplest possible case. If (D, C) is a dcpo, then we get a V-domain by setting

{0 x C y oo x g y.

The bottom predicate is defined by

b(x) = oo Va; e D.

Cauehv nets in these spaces reduce to directed subsets and their limits to least upper bounds. So an element is compact iff it is compact in the ordinary sense and such a space is an algebraic V-domain iff the original dcpo is an algebraic domain. Moreover, Idl(D, d) is the usual ideal completion of (D, C) with 0-oo-valued distance as above.

As there are no non-trivial scalars for scalar multiplication, being a V-module reduces to containing an element 0 and carrying a commutative, associative operation + such that

(21) x + 0 = x

(22) x = x + x

(23) x C y =>- x + z Q y + z

(24) 0 Q x.

Using (23), it is immediate that (24) may be replaced by a; Q x + y, and so we see that this is exactly the theory of the free inflationary semilattice or Hoare powerdomain. Indeed, our construction of the free module coincides exactly with the construction of the Hoare powerdomain for algebraic domains: the ideal completion of the set of finite subsets, suitably (pre-)ordered [1].

7.2 Domain cones

We take as value quantale the extended non-negative reals V = [0,oo]. The semiring structure on Vo = [0, oo) is the restriction of usual addition and multiplication of reals. Again, ordinary dcpo's can be viewed as V-domains with the above definition of a distance. The requirements for the distance in the definition of V-modules reduces for these spaces to monotonieitv of addition and 0 being the least element. Thus such a space is a V-module exactly if it is a domain cone [20], Moreover, algebraic V-modules correspond to algebraic

domain cones i.e. algebraic cones with Scott-continuous operations. This reveals a connection to the free domain cone, a.k.a. powerdomain of extended probability measures.

The universal property of the free domain cone C(X) over X gives us a cone morphism C(X) —P(X). But even more, the two constructions are very alike: A simple valuation V(1 , rar]a corresponds to the element , ra{|o|}. The clauses in the definition of the distance correspond to the 'elementary steps' for comparison of simple evaluations which can be extracted from the splitting lemma (see [11,20]), In fact, Pq(X0) is exactly the set of simple valuations with coefficients from X0. As C(X) is its rounded ideal completion and P(X) its ideal completion, we get that C(X) is a retract of P(X), where both embedding and retraction are cone homomorphisms.

Note: This is superseded by the results of [19], There, the distance on D is chosen to be 0-1-valued rather than 0-oo-valued. The free [0, oo]-module over such an algebraic domain is isomorphic to its free domain cone, equipped with a suitable distance,

7.3 Possibility measures

Similar considerations as for the domain cone apply for the powerdomain of possibility measures investigated in [9], Here, one has to take V = [0, oo] with max as + and the usual multiplication. Taking V = [0,1] U {oo} makes it also possible to capture the possibility measures with value not greater than 1,

8 Future work

It seems possible to develop a representation theory for the powerdomains presented in this paper. This will be representation in terms of second order predicates (see [8]) and has to be based on a theory of integration. It will specialize to the representation of the Hoare powerdomains as domain of all Scott-closed subsets for the case of ordinary posets. For V = [0, oo] and V-domains with simple 0-1-valued distance this representation theory will specialize to the results concerning the free domain cone and powerdomain of possibility measures as discussed in Section 7,

Another line of thought is an abstract setting for algebraic structures on V-domains, accompanied by a construction of free algebras for arbitrary signatures as in Chapter 6 of [1], Instead of prescribing inequalities for the algebras, one would have equations for the distance. The equations for our example would be

• d(x, y) = d(x + z,y + z)

• d(mx, my) = md(x, y)

• d(0,x) = 0

• d(x, 0) = b(x)

For a proposed theory of the Smyth powerdomain, one would replace the last two by d{0, x) = b(x) and d(x, 0) = 0, For a Plotkin powerdomain, they would be dropped entirely.

The treatment of tensor products in this paper leaves many open questions, foremost the question formulated as a Problem 4,7: When is the tensor product of two V-domains a V-domain? Are there any categories of V-domains closed under formation of tensor products. What about symmetric monoidal closed categories?

The notion of modulus of continuity introduced here opens a whole new world to quantitative domain theory, where traditionally only nonexpansive maps are considered. So, for example, one could find a notion of contraction in this setting and proof a fixed point theorem,

9 Related Work

Eeinhold Heckmann develops in [7] a theory which relates powerspace constructions in ordinary domain theory with semirings (see also [13]), In fact, he constructs powerdomains as modules over certain semirings, A precise connection between the results of these papers and the present work, however, has yet to be worked out.

Acknowledgement

I would like to thank Bob Flagg for many inspiring discussions about V-domains and value quantales. Also, I am indebted to Gordon Plotkin for useful comments on notational issues and to the anonymous referees for many valuable suggestions. Finally, I would like to thank the members of the Department of Mathematics and Statistics of the University of Southern Maine in Portland for their hospitality during my stay in Portland in the academic year 1995-96.

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