Scholarly article on topic 'A Powerdomain of Possibility Measures'

A Powerdomain of Possibility Measures Academic research paper on "Computer and information sciences"

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Abstract of research paper on Computer and information sciences, author of scientific article — Michael Huth

Abstract We provide a domain-theoretic framework for possibility theory by studying possibility measures on the lattice of opens 𝒪(X) of a topological space X. The powerspaces P[0,∞] (X) and P[0,1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valuations by replacing ‘+’ with ‘V’ in their modular law. The functors above send continuous maps to sup-maps and continuous domains to completely distributive lattices; in the latter case they are locally continuous. Finite suprema of scalar multiples of point valuations form a basis of the powerdomains above if 𝒪(X) is the Scott-topology of a continuous domain. The notions of [0,1]- and [0,∞]-modules corresponds to that of continuous cones if addition on the reals and on the module is replaced by suprema. The powerdomain P[0,∞] (D) is the free [0, ∞]-module over a continuous domain D.

Academic research paper on topic "A Powerdomain of Possibility Measures"

Electronic Notes in Theoretical Computer Science 6 (1997)

URL: http://www.elsevier.nl/locate/entcs/volume6.html 12 pages

A Powerdomain of Possibility Measures

Michael Hutli

Department of Computing and Information Sciences Kansas State University Manhattan, KS 66506, USA

Abstract

We provide a domain-theoretic framework for possibility theory by studying possibility measures on the lattice of opens O(X) of a topological space X. The powerspaces F>0 ^(X) and ^(X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valuations by replacing '+' with 'V' in their modular law. The functors above send continuous maps to sup-maps and continuous domains to completely distributive lattices; in the latter case they are locally continuous. Finite suprema of scalar multiples of point valuations form a basis of the powerdomains above if O(X) is the Scott-topology of a continuous domain. The notions of [0,1]- and [0, oo]-modules corresponds to that of continuous cones if addition on the reals and on the module is replaced by suprema. The powerdomain ROoo№) is the free [0, oo]-module over a continuous domain D.

1 Possibility Measures

This extended abstract attempts to recast some of the work done in quantitative domain theory within the traditional domain theory of continuous domains and lattices. We illustrate our approach with the two prime targets of quantitative analysis, the completely distributive lattices [0,1] and [0,oo], Given a dcpo D, there are numerous semantic scenarios in which we are interested in the function space

(1) [D^[0,1]]

(2) [D —[0, oo]].

For example, the former is a natural carrier of meaning for Markov chain processes [1], or quantitative model checks [2], whereas the second space could

1 Reinhold Heckmann, Klaus Keimel and Philipp Siinderhauf make valuable suggestions when I presented preliminary parts of this material at the Comprox II meeting in Darmstadt, Germany, on September 25, 1996.

© 1997 Published by Elsevier Science B. V.

be the carrier of meaning for a cost, or running time analysis. Note that we stipulated that these functions are continuous. That way we ensure that the process of approximating total elements by partial ones in D is consistent with the quantitative evidence provided by some function / e [D —[0,1]]. Even if the notion of partial elements were absent, for example, if D is a flat domain of states of some concurrent system, one still has to resort to continuous functions in order to allow for sound higher-order semantics. To wit, the space [[£> —[0, oo]] —[0,1]] could be the domain which, given a property (j> and some t E [0, oo] returns the possibility of satisfying (j> within time t, where Q has an additional structure of a continuous real-time system. In addition, it should be clear that the internal hom functor [• —[0,1]] is a plausible candidate for marrying the theories of domains [3] and fuzzy sets [4], Possibility measures [6] are a concept used in possibility theory. As such, they belong to the broad field of theories of evidence in Artificial Intelligence and Empirical Sciences, Possibility measures are functions ¡jl\V{X) —[0,1], where V(X) is the power set of a finite set X, such that

for all T C V(X). Possibility theory is motivated by the observation that non-determinism can arise through uncertainty, or through unsharpness of data. The first kind is captured well with standard notions of probability theory whereas the latter seems to fare better with concepts based on fuzzy set theories [4], Unsharpness implies non-specific and therefore set-valued semantics. This brings us into the familiar realm of powerdomains [3],

Joint work done with Eeinhold Heckmann uncovered a natural isomorphism between [D —[0,1]] and the space of all sup-maps ¡jl\o{D) —[0,1], where o(D) is the Scott-topologv of D [5], The corresponding result is true for [0, oo] as well. This provides a reassuring link between the spaces of quantitative meanings in (1) and (2), and spaces of topological possibility measures.

In this paper we develop an algebraic theory of possibility measures in a general topological setting; the power set V{X) is then merely the special case of a discrete topological space. This leads us to defining possibility measures as sup-maps from the lattice of opens O(X) to [0,1],

One notices that the 'type' O(X) —[0,1] of topological possibility measures looks like the one for continuous valuations [7,8], Recall that a continuous valuation is nothing but a strict Scott-continuous map ¡jl E [0(X) —[0,1]] such that ¡jl satisfies the usual modular law of measure theory [9]:

for all U,V E O(X). Clearly, such a property cannot be expected from sup-maps of that same type. However, topological possibility measures do satisfy a similar modular law if we replace '+' with 'V', the binary supremum in

H{U U V) + n(U n V) = n(U) + n(V)

[0, l],2 Any topological possibility measure v must be monotone since it is a sup-map. Therefore, the fuzzy modular law

(5) v{U U V) V v{U n V) = v{U) V is(V)

reduces to v{U U V) = v{U) V v(V). Thus, v satisfies the fuzzy law in (5), Conversely, any strict Scott-continuous function v'\ O(X) —[0,1] which satisfies (5) has to be a sup-map. To summarize, we see that continuous valuations and topological possibility measures share that they are strict maps in [O(X) —[0,1]], but are distinguished by their modular laws (4) and (5),

2 Possibility Measures on Topological Spaces

Definition 2.1 For complete lattices L and M we define L-oM to be the complete lattice of all sup-maps f: L —M, ordered pointwise. For a topological space X with topology O(X) we define f*0oo](X) to be 0(X)—o[0, oo]. Given any continuous map f: X —Y between two topological spaces we define

&«,](/): ¿¡um h

(6) ffo^if) ^P/"1

for all 11, e ff0tOOp0-

Note that the map Fj"0 (/): (X) ^(K) is well defined since the inverse image function /-1: 0(Y) O(X) preserves suprema. Since o(D)^>2 is just the order dual of cr(D), we realize o(D)^>2 as another version of the lifted lower powerdomain. In replacing 2 with [0,oo] we obtain the space P s (I)). which we can therefore think of as a fuzzy (lifted) lower powerdomain. This suggestion gets formal support in the section on free [0, oo]-modules.

Proposition 2.2 The two-sorted function ffOao0 constitutes a functor from TOP, the category of topological spaces and continuous maps, to SUP, the category of complete lattices and maps preserving suprema. This functor restricts to a locally continuous functor from CONT, the category of continuous dcpos and Scott-continuous maps, to CD, the category of completely distributive lattices and maps preserving suprema.

In semantics we are mostly interested in probabilities or possibilities in the range of [0,1]:

Definition 2.3 Let X be a topological space. Then ff01](X) is the set of all H G f^0oo](X) with range contained in [0,1]; the order on ff01](X) is the pointwise one.

Proposition 2.4 The functor f^0oo]Q:TOP SUP restricts to a functor P0 jjQ: TOP —SUP and to a locally continuous functor P0 ^Q: CONT —CD.

2 Reinhold Heckmann and Gordon Plotkin kindly pointed this out to me at the COMPROX II meeting in Darmstadt.

Since completely distributive lattices are L-domains and FS-domains we may solve recursive domain equations in L and FS involving the functors Fj"0 (•) and x](-) by using the standard machinery of [3], The wav-below and wav-way below relation [3] on P r(/)) are induced by the ones in Pu s (I)).

Lemma 2.5 Let X he a topological space. Then ^(X) is a sup-projection of ^(X). In particular, the way-below and way-way below relation on ^(X) are induced by the ones in f^0oo](X). The projection of /j, E f^0oo](X) maps O E O(X) to fj,(0) A 1.

If we define R=i(X) to be those possibility measures // e Pu ^(X) with fj,(X) = 1 plus the constant zero measure then this is a complete lattice closed under all suprema in Fj"0 (X). It would be interesting to find out about the distributivitv, continuity and co-continuity of P=i(X), at least when X is a continuous domain and O(X) its Scott-topologv.

3 Point Valuations as Possibility Measures

Point valuations are a crucial technical tool in the theory of continuous valuations [7,8]. Given a topological space X and x E X, the point valuation r]x\ 0(X) —[0, oo] is defined by r]x(0) = 1 if x E O and r]x(0) = 0 otherwise. As such point valuations are quite crisp in nature but they are nonetheless possibility measures.

Lemma 3.1 Let X be a topological space and x E X. Then the point valuation r]x is a possibility measure, so we have maps r]x'.X —f*0oo](X) and rj'x-X ff01](X) sending x to r]x. Moreover, these maps are Scott-continuous and injective if 0{X) is the Scott-topology on a dcpo X.

Proof. Given open sets U,V E O(X) we have r]x(U U V) = 1 if and only if x e U or x e V; the latter is equivalent to r]x(U) V r]x(V) = 1. Thus, r]x e 0(X)^[0, oo] = P[0tOc](X). The rest follows as in [7,8]. □

What other facts about point valuations carry over from the classical case?

Definition 3.2 Given any a E [0, oo] and ¡jl E ff0tX}](X) we define a * ¡jl by (7) (a * fj.) O = a ■ fi(0) (O E O(X))

just as for continuous valuations.

Since scalar multiplication r a • r is a self map on [0, oo] preserving suprema,3 we see that a * ¡jl is in F^ (X).

Lemma 3.3 Let X be a topological space. The map

(a, [0, oo] x />0oo](X) ^ />0oo](X)

preserves suprema in each coordinate separately. The corresponding statement holds for [0,1] and f*0 ^(X) as well.

3 \V(> set oo • 0 = 0 as in the case of continuous valuations.

The crucial distinctive feature of possibility measures is that they replace the notion of sums of continuous valuations by that of suprema of possibility measures. Summing up possibility measures results in functions that don't preserve suprema in general; just take the sum of rjx and r]y where x and y are incomparable with respect to the specialization order,

4 Simple Possibility Measures

Since addition is not admissible for possibility measures, and since we replaced '+' by 'V' in their modular law, we suggest to define simple possibility measures as finite suprema of scalar multiples of point valuations. Such finite suprema model fuzziness.

Definition 4.1 A possibility measure /j, G f^0oo](X) is called simple if there are finitely many points xi, X2, ■ ■ ■, xn in X and scalar s ai, «2, ■ ■ ■ ,otn [0, oo) such that

(8) (j, = (ai * rjXl) V (a2 * rjX2) V ... V (an * rjxJ.

We call possibility measures of the form a * rjx scalar point valuations.

It is worth pointing out that, unlike continuous valuations on sober spaces [8], scalar point valuations are not characterized by having a two-element image (see equation (10) below). Simple possibility measures are necessity measures [6] in the sense that they preserve infima of opens whose filtered intersection is open again. This follows readily, as in the case for simple valuations [8], since V and multiplication preserve filtered infima in [0,oo],

Lemma 4.2 Any simple possibility measure // G P ^(X) satisfies //((") J-) = l\oer tJ,(0) for all filtered sets T in O(X) whose intersection is open.

We would like to know whether the preservation of filtered open intersections also characterizes possibility measures with finite image, but this seems unlikely.

5 Simple Possibility Measures as a Basis

Now we show that simple possibility measures form indeed a basis of ^(X) and ^(X) if X is a continuous domain and O(X) its Scott-topologv. The proof of that uses results on the structure of function spaces L-oM where L and M are completely distributive. The concrete setting at hand is where L is the completely distributive lattice cr(D), D a continuous domain, and M is the completely distributive lattice [0, oo], respectively [0,1]. We only present the argument for Fj"0 ^(D) since the one for ^(D) is completely similar.

Given ¡j, G Pu s (D) this is just an element of <t(£>)-°[0, oo] and we need to show that it is the supremum of simple possibility measures wav-below it. First, we note that scalar point valuations are maps in L—°M which are well-known in different contexts [10].

Definition 5.1 Let L and M be complete lattices and z E L, y E M. The 'map z /* y: L —M maps the set \.z = {I E L \ I < z} to the zero of M and all other elements to y.

Lemma 5.2 The map z y above is in L—°M. Given a E [0, oo] and x E X, we have

(9) a*rix = (X\Jx}) Generally, for any O E 0(X) we have

(10) \/ a*r)x.

x£X\0

Recall the notion of a step function x \ y E [L —M] which maps all I with x <C I in L to y and all other elements to the zero of M. The following is a straightforward generalization of a lemma in [10]:

Lemma 5.3 Let L and M be complete lattices, x E L and y E M. Then the greatest map preserving suprema below x \y is z /* y, where

(11) z = \J{l E L | x<tl}.

This suggests to finish our argument as follows: Since o(D) and [0, oo] are continuous lattices we know that every ¡j, E [o(D) —[0, oo]] is the supremum of step functions wav-below it [3], In particular, this is true for possibility measures. Those step functions are not possibility measures in general, but the greatest sup-preserving map below such a step function is the supremum of scalar possibility measures by Lemmas 5,2 and 5,3,

Thus, for any x \ y <C ¡jl in [a(D) —[0, oo]] we take z as above and obtain

(12) z /*y<x\yCix

in [a(D) [0, oo]] which implies z y in [o(D) —t [0, oo]]. The latter entails z z71 y <C // in <r(D)-o[0, oo] = Pu s (I)) since the inclusion of Pu s (I)) into [o(D) —t [0, oo]] is Scott-continuous, In particular, we have y * r]a <C ¡jl for all a E d \ z by (10), Since Pu s (i)) is a complete lattice it suffices to show that the supremum of all such z y equals ¡jl. Since ¡jl is the supremum of all step functions wav-below it, we are done as soon as the process of 'taking the greatest sup-preserving map below a Scott-continuous map' preserves suprema. Thus, we need to show that the self-map P on [o(D) [0,oo]], defined by

(13) P(f) = \J{gEa(D)^[0^]\g<f},

preserves suprema. Since o(D) is completely distributive, we can state P explicitly, Every element O in a(D) is the supremum of way-way below elements O'CO [11] (O'CO iff for all O C a{D) with OC[J 0 there is some V E O with O' C V). Now it is routine to verify [10] that

(14) P(f)0 = \J{f(0') | O'CO}

and that P preserves suprema.

Theorem 5.4 Let D be a continuous domain and 0{D) its Scott-topology. Then the set of simple possibility measures in f*0 ^(D) forms a basis of f*0 ^(D). Likewise, the set of simple possibility measures in ^(D) is a basis of ^(D).

In [10] it was also shown that, given completely distributive lattices L and M and f,gE L^M, we have /«Cg in the space L-»M if and only if we have /«Cg in [L —M], Furthermore, the same statement holds for <C instead of C whenever L and M are linear FS-lattices [10], but completely distributive lattices are linear FS-lattices [12], Therefore, we know that <C and <C are induced by the respective relations in [a(D) —[0,oo]], This also applies to F^ ^(D) since this is a sup-projection of ^(D) by Lemma 2,5,

Proposition 5.5 Let D be a continuous dcpo. The way-below relations on P , .(/)) and P s .(/)) are the restrictions of the way-below relation on [o{D) — [0,1]], respectively [o{D) —[0,oo]]; this also holds for the way-way-below relations.

6 Free Modules

Continuous cones are dcpos D with the structure of a commutative monoid (D; © , 0) and a continuous action

(a, d) a * d: [0, oo] x D —D

of [0,oo] on D which interacts with that monoid structure in the expected way [7], Our setting requires that we replace the addition © on D and the addition on [0, oo], respectively [0,1], by suprema. To do so we only need to add the axiom

cf © d = d

to the commutative monoid and think of '+' on reals as the maximum operation, In particular, D is then a complete lattice with © as binary suprema, and we may condense all these conditions to saying that * preserves suprema in each coordinate separately. We phrase this in the language of monoids.

Definition 6.1 We consider the monoids ([0,1]; •, 1) and ([0, oo]; •, 1), where '•'is the usual 'multiplication. An [0,1 }-module is a pair (L; where L is a complete lattice and [0,1] x L —L preserves suprema in each coordinate separately, such that

(15) (ri • r2) *Ll = ri *L (r2 *l I)

(16) 1*1 = 1

for all I e L and all ri, G [0,1]. We define an [0, oo}-module in the obvious and similar way.

Note that [0,1] and [0,oo] are [0, l]-modules with as *[o,i], respectively *[o,oo]■ Also, Pu s (I)) is an [0, oo]-module, and Pu,.(/)) an [0, l]-module

by Lemma 3,3, In any [0,1]-, or [0, oo]-module we must have

(17) 0 * Ll = ± L

for all I 6 L since *l preserves all suprema in its first coordinate. For the rest of this paper we speak of 'modules' if a statement applies to [0,1]- and [0, oo]-modules at the same time. We view modules as algebras (A; ©.4, *a) and morphisms between such algebras (A; ©.4, *.4) and (B; ©B, *B) are Scott-continuous functions /: A —B such that

(18) f(a ©,4 a') = f(a) ®B f(a')

(19) f(a *A a) = oi *B f(a)

for all a E [0, oo], respectively [0,1], and a, a' E A. Since © has to be interpreted as supremum, we see that the first equation merely says that / is a sup-map. Of course, one may apply Frevd's General Adjoint Functor Theorem to secure the existence of free modules [3,13], In [3,13] it has been shown that such an initial algebra ID is a continuous domain if D is a continuous domain to begin with. Thus, we obtain initial, or free, [0,1]- and [0, oo]-modules over a continuous domain D. However, one often would like to have concrete representations of such initial algebras, which validate and strengthen our semantic intuitions. It turns out that the initial algebra for [0, oo] is nothing but (an isomorphic copy of) Fj"0 (D),

We already have a Scott-continuous map r]D: D —^(D) which associates to each x E D its point valuation r]x. So let A be any [0, oo]-module and f-.D—tA a Scott-continuous function. We need to show the existence of a unique morphism of [0, oo]-modules /: P s (I)) A such that

(20) f = foriD.

Since A is an [0, oo]-module it is certainly a complete lattice, so the function

(21) f(p) = Y{a *A f(x) | a * r]x < //}

is well-defined, for a*r]x = /?*% implies a = ¡3 and x = y (the opens separate the points). Since v <C ¡jl < ¡j! implies v <C // we conclude that / is monotone. By Theorem 5,4, ^(D) is continuous; thus, the wav-below relation on Fj"0 ^(D) satisfies the interpolation property [14,3], Using that fact one readily sees that / is Scott-continuous,

Next, we verify one half of the statement that / is a morphism of algebras. For that we need to establish that scalar actions preserve and reflect the wav-below relation in [0, oo]-modules.

Lemma 6.2 Let A be an [0, oo]-module and > ^ 0. Then we have a <C 6 in A if and only if /3 *ao, <C /3 *a b in A.

Proof. The proof works with the scalar action of using that scalar actions are Scott-continuous, □

Lemma 6.3 For the map f above we have f(a * ¡j) = a *A f(p) for all a E [0, oo] and // t P ^jD).

Proof. Since the map ¡j, a * //: ^(D) ^(D) preserves all suprema, we may compute

(22) a *A f(p) = a *A *A f(x) | 7 * r]x < //}

= /(x)) I 7 * Vx < /"}

= \/{(a • 7) *a /(a;)) | 7 * % <

= Vi(« ' ^ *A | ft * (7 *%,)•< ft * /i} by Lemma 6,2 = \/{(ft • 7) *A f(x)) | (a • 7) * < a * //}

= \/{(3 *4 f(x) | (3 * r]x <C a * //} since f3 = a ■ —

= f(a*fJ)

if ft ^ 0, Otherwise, both sides equal ±.4 due to (17), □

We may use the property of / above to show that / o r]n = /■ For that we need to identify certain elements wav-below r]x in P s (I)).

Lemma 6.4 Let a E [0,oo], y E D, and ¡jl E P s (I)) such that

(23) ft < /¿({d E D | y < d}). Then ft * r]y <C jU.

Proof. By Theorem 5,4, ¡j, is the supremum of scalar point valuations f3 * r]x wav-below ¡jl. Thus, fj,({d E D \ y <C d,}) equals \/{/3 \ y <C x, /3 * r]x ■< //}. By our assumption we get a < \J{/3 \ y x, /3 * r]x ■C /i} and all suprema in [0, 00] are directed. Thus, there is some /3 * r]x <C ¡jl with y <C x such that ft < (3. Since Scott-open sets are upper sets it is immediate that ft*% < /3*r]x. So ft * tjy <C ¡jl follows, □

Lemma 6.5 For the maps t)d and f above we have f = / o rjo-

Proof. By Lemma 6,4 we have a * % <C r]x for all ft < 1 and i/«i, Thus, we compute

(24) (/ o r]D)x> {ft *A f(y) | a < 1, y < x}

= (\f{a | ft < 1}) *a (\/{f(y) | y -C x}) as * is Scott-cont, = 1 *.4 /(x) as / is Scott-cont, = №■

Conversely, let ft * % <C r]x. Clearly, a < 1 follows. But we also get y < x; otherwise, y would be in the open D\lx which does not contain x, contradicting a * j]y < j]x. Therefore, a *.4 f(y) < 1 *.4 f(x) = f(x) as / and * are monotone. This implies (/ o r]D) x < f(x). □

The building blocks a * rjx of simple possibility measures are sup-primes in

Lemma 6.6 Let X he a topological space, a E [0, oo] and x E X. Then a*r]x is a sup-prime in f^0oo](X).

Proof. This is evidently so in ease that a = 0, If a ^ 0 then suppose that a * r]x = ¡jl V v in ^(x) (since ^(D) is a distributive lattice we may assume equality). Suppose that ¡jl ^ a * r]x. Since ¡jl < a * r]x there must be some O E O(X) such that fj,(0) < a • rjx(0), which also implies a • r]x(0) = a. Likewise, iff ^ a*?]x then v < a*r]x implies the existence of some O' E O(X) such that v(0') < a ■ rjx(0') and again a ■ r]x(0') = a follows. Thus, x is contained in the open set O DO' and we compute

(25) (/j, v v) (0(~) o') = n(0 n o') V y(0 n o') pointwise supremum

< fj,(0) V v(0') as ¡jl and v are monotone

= a-Vx(OnO')

This contradiction shows fj, = a * j]x or u = a * j]x. □

Lemma 6.7 The map f is the unique morphism of [0,oo]-modules with f o Vd = f ■

Proof. It remains to verify the uniqueness of / and ¡(¡jl V v) = f(p) V /(f). Since / is monotone, it suffices to prove ¡(¡jl V v) < f(p) V /(f) for the latter. By definition, /(// V f) equals the supremum of all a *a f(x), where a * r]x <C ¡jl V f, Since all such elements a * r]x are sup-primes in ^(D) by Lemma 6,6, we may assume that a * r]x < ¡jl without loss of generality. Since / is monotone we obtain

(26) a *A f(x) = a *A f(r]x) as / o r]D = f

= /(« * Vx) as a *A f(Q) = f(a * () for all C < f(M)

</(/') V ./>).

Thus, ¡(¡jl V f) = \J{a *A f(x) | a * rjx <C // V v} < f(p) V /(f).

As for uniqueness, let g\ P s (I)) ^ A be any morphism of [0, oo]-modules such that gor/d = /■ Given ¡j, E Pu s (i)) we know by Theorem 5.4 that ¡jl is the directed supremum of all simple valuations v wav-below it. Such a valuation v is of the form (ai *r]xl)V (a2 * r]X2)... (ak * j]Xk) for some k > 1. In particular, all ai * r]Xi are wav-below ¡jl (1 < i < k). Thus,

(27) (j, = \/{a * rjx | a * rjx < //},

together with the fact that g preserves suprema and scalar multiplication, shows that g = /, noting that gorjo = f- n

Theorem 6.8 Let D be a continuous domain. Then ^(D) is the free [0, oo]-module over D.

Incidentally, given a Scott-continuous function f E [D E], we readily see that Fj"0 (j) is the unique morphism of [0, oo]-modules h: ^(D) Fj"0 (E)

such that t]e ° / = h o t/d-

The proof techniques employed in verifying the univeral property of Fj"0 (D) seem quite general, but there are two spots where, on the face of it, these arguments won't carry over to the case of [0, l]-modules; namely, the proofs of Lemmas 6,2 and 6,3 need fractions and ^ which won't be defined in [0,1] in general. This is clearly unsatisfactory and there is need for abstracting the line of argument presented here. This can indeed be done and recent improvements of the work in [5] have shown that the algebras studied here for L = [0,1] and L = [0,oo] are free not only for L = [0,oo], but for any continuous lattice L.

7 Related Work

It would be a worthwhile project to determine parallels, as well as differences, of our approach to quantitative semantics with work done by others. For example, there is a framework for generalized metric spaces by Mareello Bonsangue et al, [15], Philipp Siinderhauf's work on quantitative V-powerdomains [16], and E.C, Flagg's studies on quantales and continuity space [17], The concept of [0,1]- and [0, oo]-modules fits into the general framework of a powerdomain theory based on semirings [18], of which Eeinhold Heckmann's work on abstract valuations for the Plotkin powerdomain [19] is the most recent example.

References

[1] J.G. Kemeny and J.L. Snell. Finite Markov Chains. Van Nostrand, 1960.

[2] M. Huth and M. Kwiatkowska. Quantitative Analysis and Model Checking. Technical report, Kansas State University. Department of Computing and Information Sciences. To appear in Logic in Computer Science 1997.

[3] S. Abramsky and A. Jung. Domain theory. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 3, pages 1-168. Clarendon Press, 1994.

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