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ELSEVIER Electronic Notes in Theoretical Computer Science 155 (2006) 605-615

www.elsevier.com/locate/entcs

Probabilistic Observations and Valuations (Extended Abstract)1

Matthias Schroder Alex Simpson

LFCS, School of Informatics, University of Edinburgh, UK

Abstract

We give a universal property for an "abstract probabilistic powerdomain" based on an analysis of observable properties of probabilistic computation. The universal property determines an abstract notion of integration satisfying the usual equational laws. In the category of topological spaces, the abstract probabilsitic powerdomain is given explicitly by the space of continuous probability valuations with weak topology. Thus our abstract notion of integration coincides with the usual integration for probability valuations. We end by discussing how our approach might adapt to provide "abstract effect spaces" for other computational effects.

Keywords: Domain theory, probabilistic powerdomain, probability measures, integration, computational effects

1 Introduction

Topological spaces provide a mathematical notion of datatype, with open sets correponding to "observable properties" of data [8]. In particular, Sierpinski space S = T} acts as a result space for "observations", where T represents a computation that halts, and ± represents one that loops. The Sierpinski topology arises naturally: {T} is open because termination is observable, whereas {±} is not open because nontermination is not observable.

In the context of probabilistic computation, termination occurs with some probability A E [0,1]. Thus it is natural to replace S with [0,1] as test space,

1 Research Supported by an EPSRC Research Grant "Topological Models of Computational Metalanguages" and an EPSRC Advanced Research Fellowship (Simpson).

1571-0661/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.entcs.2005.11.075

where A represents a computation that halts with probability A and loops with probability 1 — A. The sensible observable properties in this case are the sets

{[0,1]} U {(A, 1] | 0 < A < 1} .

These sets determine the topology of lower semicontinuity (equivalently Scott topology) on [0,1]. We write I< for this space, and consider it to be our basic space of probabilistic observations

In this abstract we explain how such probabilistic observations can be used to induce an abstract characterization of a probabilistic powerdomain Pprob(X) over an arbitrary topological space X. The characterization gives a universal property for Pprob (X), which can be used to define integration and establish its basic properties, independently of any concrete construction of Pprob(X). Nevertheless, Pprob(X) can be described explicitly: it is the space of continuous probability valuations over X, endowed with the weak topology. It follows that our abstract theory of integration for Pprob (X) coincides with the established theory for valuations.

2 Abstract Probabilistic Powerdomains

The probabilistic powerdomain Pprob (X) should model a notion of probabilistic process outputting values in X. Our aim is to characterize Pprob(X) in terms of its expected properties without specifying details of its construction.

A minimal requirement on a topological space Y for it to model a sensible collection of "probabilistic processes" is that the collection of such processes should be closed under fair probabilistic choice. Thus for processes £ Y

there should be a process 0 £ Y representing the process that tosses an unbiased coin and then, depending on the outcome of the toss, continues either as process or as process Moreover, since there is a uniform (computable) mechanism of going from the pair to the process 0

the operation 0: Y x Y ^ Y should be (jointly) continuous.

Henceforth, we call a structure (Y, 0), where Y is a topological space and 0: Y x Y ^ Y is continuous, a choice algebra. A homomorphism h: (Y, 0) ^ (Y', 0') between two choice algebras is a continuous function h: Y ^ Y' satisfying h(x 0 y) = h(x) 0' h(y).

The observation space I< carries a natural choice algebra structure:

AI©A2 = ^(AI + A2) ,

and, henceforth, when we write (I<, 0), we always mean 0 to be as defined above.

The notion of choice algebra only makes weak requirements on a space of probabilistic processes. For example, no equational properties are required of the 0 operation. Also, the existence of 0 alone only guarantees that A models unbiased two-way probabilistic choices, whereas many other forms of probabilistic choice can arise. Nevertheless, our simple notion of choice algebra is sufficient to next ask:

When does a choice algebra (Y, 0) constitute a reasonable space of probabilistic processes outputting values in X?

We answer this question by placing two further requirements on A Requirement 1 There is a distinguished (continuous) map X —Y.

The intuition is that 5(x) e Y is the deterministic process that outputs the value x with probability 1.

Requirement 2 For every map A"-I< there exists a unique homomor-

phism h: (Y, 0) ^ (I<, 0) such that the diagram below commutes.

This condition can be motivated as follows. First, given f, which performs an observation on X, we can use f to perform an observation on any ^ £ Y, by simply running ^ and applying f to any resulting value x £ X output by The so-induced observation h on Y is a homomorphism, because the probability of termination accumulates according to the probabilistic choices made during the execution of Also, by definition, h(5(x)) = f (x), so the diagram commutes. It remains to justify the uniqueness requirement. This expresses that the only way of performing an observation h on any ^ £ Y in such a way that probabilistic choices in ^ are respected (i.e. so that h is a homomorphism) is by performing an observation f on the resulting values in X of

Definition 2.1 An abstract (probabilistic) choice structure over X is given by a choic holds.

a choice algebra (Y, 0) together with a map X Y such that Requirement 2

The notion of abstract choice structure suffers from the same weaknesses as the notion of choice algebra. However, we can use it to define a "complete-

ness" property for choice algebras. This will guarantee completeness in two senses. First, the operation 0 will satisfy all the expected equational properties. Second, the space will be "complete" enough to interpret all possible forms of probabilistic choice.

Definition 2.2 A choice algebra (A, 0) is said to be complete if, for every abstract choice structure X —(Y., 0) and map X ——- A there exists

a unique homomorphism (Y, 0)-

commutes.

(A, 0) such that the diagram below

Note that it is immediate from the definition of abstract choice structure that the choice algebra (I<, 0) is complete.

The definition of completeness directly formalizes A being a space of probabilistic processes that is complete enough to interpret all forms of probabilistic choice. Explicitly, it says that any program f mapping values in X to probabilistic processes in A, extends uniquely to a choice-respecting program h translating probabilistic processes over X to probabilistic processes in A.

We next state a sequence of results about complete choice algebras. The first result is technical, but important. It states a fundamental property needed in the proofs of several of the subsequent results.

Proposition 2.3 (Parametrization) If (A, 0) is a complete choice algebra then, for every abstract choice structure X-► (Y, 0) and map ZxX -!-* A,

there exists a unique continuous Z x Y argument, such that:

A, homomorphic in its right

idz x 5

Proposition 2.4 If (A, 0) is a complete choice algebra then the topological space A is sober.

Proposition 2.5 The forgetful functor CCA ^ Top (where CCA is cate-

M. Schröder, A. Simpson /Electronic Notes in Theoretical Computer Science 155 (2006) 605-615 609

gory of complete choice algebras and homomorphisms) creates limits.

Thus, for example, I = [0,1] with the Euclidean topology is a complete choice algebra, because it arises as an equalizer

IT ^^.Tr.vTr,-^^ - TT,

(Al,A2)^Al®A2

of homomorphisms.

The next proposition shows that completeness does indeed have equational consequences. In fact, complete choice algebras inherit their equational theory from (I<, 0).

Proposition 2.6 If (A, 0) is a complete choice algebra then the following equations hold:

if* (T^ ry* - I~y*

x 0 y = y 0 x (x 0 y) 0 (z 0 w) = (x 0 z) 0 (y 0 w) .

The above proposition states that (A, 0) is a midpoint algebra in the sense of [2].

Proposition 2.7 If (A, 0) is a complete choice algebra then the space A carries a unique continuous map +: I x A x A-► A (where I has the Euclidean

topology) satisfying:

x +0 y = x xx | a xx xx x +A y = y +(1-A) x X +A (y +A' z) = (x + a(i-a') y) +AA' 2

such that x+iy = x®y. Thus A is a "convex space77 ■with x+yy expressing the convex combination (1 — A)x + Ay. Further, every homomorphism of complete choice algebras is affine (i.e. preserves convex combinations).

It is possible to also show that A has uniquely determined (continuous) countable convex combinations.

Definition 2.8 The abstract probabilistic powerdomain over X, if it exists, is

given by an abstract choice structure X-► CP-pTOb(X), 0) where the choice

algebra (Pprob(X), 0) is complete.

The abstract probabilistic powerdomain is characterized up to isomorphism in two complementary ways:

(i) X —(VpTOb(X), 0) is final amongst abstract choice structures over X.

(ii) X —(VpTOb(X), 0) exhibits (VpTOb(X), 0) as the free (i.e. initial) complete choice algebra over X.

Theorem 2.9 The abstract probabilistic powerdomain over X exists, for every topological space X.

An equivalent statement is that the forgetful functor CCA ^ Top has a left adjoint.

We shall briefly discuss the proof of Theorem 2.9 in Section 4.

3 Abstract Integration

We have motivated Pprob (X) as an abstract space of "probabilistic processes" over X. Alternatively, one can think of it as an abstract space of "probability measures" over X, where "measure" here is, for the moment, to be understood in an intuitive rather than technical sense. In this section, we pursue this direction, by developing a theory of integration relative to the "probability measures" in Pprob(X).

For any complete choice algebra (A, 0) and continuous f: X ^ A, we write J f for the unique homomorphism such that the diagram below commutes, and we use standard notation such as / f d^ for (/ f )(^) etc.

(PprobPO,®) ■ 04,0)

This definition gives us a notion of integration with respect to abstract probability measures in Pprob (X), for functions taking values in any complete choice algebra A. For example, we obtain Euclidean-valued integration by taking I for A, and lower semicontinuous integration by taking I< for A.

Many of the expected properties of integration fall out straightforwardly from the universal property of Pprob (X). It is not necessary to know any concrete description of Pprob(X).

M. Schröder, A. Simpson /Electronic Notes in Theoretical Computer Science 155 (2006) 605—615 611

Proposition 3.1 Using the convex space structure of A (Proposition 2.7),

J (x ^ a) dß = a

J((1 - A)/ + Ag) dß = (1 - A)^/dß) + A(|gdß) .

Proposition 3.2 (Monotonicity) If / Ç g pointwise in the specialization order on A then J / dß Ç f g dß.

Proposition 3.3 (Monotone convergence) If {/d}deD is a directed set of continuous functions from X to A then

sup / /d dß = / (sup /d) dß ,

deD 7 7 deD

using the sobriety of A (Proposition 2.4) to find the suprema.

Lemma 3.4 For topological spaces X, Y, there is a unique continuous map ®: Pprob(X) x Pprob(Y) ^ Pprob(X x Y) that is a bihomomorphism (i.e. a ho-momorphism in each argument separately) and satisfies 5(x) ® 5(y) = 5(x,y).

Proposition 3.5 (Fubini) For any continuous X x Y-A,

/ / /(x,y) dvdß = / /(x,y) dßdv

JxeX JyeY JyeY JxeX

= /(x,y) d(ß ® v) ,

J(x,y)eX xY

using the operation ® from Lemma 3.4 in the third integral.

4 Probability Valuations

In this section we give a concrete presentation of the space Pprob (X).

Definition 4.1 A (continuous) probability valuation on a space X is a continuous function v: O(X) ^ I< (where O(X) is the lattice of open sets of X endowed with the Scott topology) satisfying:

(i) v(0) = 0

(ii) v(U) + v(V) = v(U U V) + v(U n V) (modularity)

(iii) v(X) = 1 .

We write Vi(X) for the set of probability valuations on X, and we give it the weak topology (cf. [4]), which has subbasic opens

{v | v(U) > A}

generated by open U C X and A £ [0,1). Define A Vi(A) by:

i 1 if x £ U i(x)(U )= I

[ 0 if x / U

and Vi(X) x Vipf) — Vi(X) by:

(vi © v2)(U) = vi(U) © v2(U) .

Theorem 4.2 For any topological space X, the structure X —^ (Vi(A'), ©) is an abstract probabilistic powerdomain over X.

Of course, Theorem 2.9 is an immediate consequence of Theorem 4.2.

The proof of Theorem 4.2 is quite involved. We mention only that the

full Axiom of Choice is used to prove that X-► (Vi(A'), ©) is an abstract

choice structure — specifically, it is used to show the uniqueness condition of Requirement 2. We remark that a similar argument provides a positive solution to Problem 1 of [4]. Details will appear in a full version of this extended abstract.

One consequence of Theorem 4.2, is that our theory of integration for abstract probabilistic powerdomains in Section 3 coincides with the the established theory for (probability) valuations, as developed in [5]. For example, Proposition 3.5 is closely related to [5, Theorem 3.17]. (Proposition 3.5 is more general in applying to integration over functions valued in any complete choice algebra, but less general in being restricted to probability valuations.) One advantage of our treatement is that the properties of integration follow straightforwardly from the universal property of the abstract probabilistic powerdomain.

We mention two other consequences of Theorem 4.2, which make use of the literature on valuations. These show that, for good classes of spaces, the abstract probabilistic powerdomain coincides with standard constructions of spaces modelling probabilistic bahaviour.

In domain theory [3], one works with dcpos with the Scott topology. The probabilistic powerdomain for arbitrary dcpos was introduced by Jones and Plotkin [6,5]. For any dcpo D the set of probability valuations again V1(D)

again forms a dcpo. We refer to V1 (D) with the Scott topoplogy as the domain-theoretic probabilistic powerdomain. (In contrast to [5], we consider probability valuations rather than subprobability valuations.) It follows from results in [5] that, when D is a continuous pointed dcpo, then so is the domain-theoretic probabilistic powerdomain over D. Moreover, it follows from [4] that, under the same conditions, the weak and Scott topologies on V1(D) coincide. Hence we obtain:

Corollary 4.3 The abstract probabilistic powerdomain over a continuous pointed dcpo D carries the Scott topology, and hence coincides with the domain-theoretic probabilistic powerdomain over D.

In analysis, for any compact Hausdorff space X, one considers a space M1(X) of regular Borel probability measures (also known as Radon probability measures) endowed with the weak topology (also known as the vague topology), which is again compact Hausdorff. This situation generalizes to stably compact spaces, which are the T0 analogues of compact Hausdorff spaces, see [1]. Further, for stably compact spaces X, the stably compact space M1(X) is homeomorphic to V1(X) [1, Theorem 36]. Thus we have:

Corollary 4.4 The abstract probabilistic powerdomain over a stably compact space X is homeomorphic to the space M1(X) of regular Borel probability measures with the weak topology.

It would be interesting to generalize Corollary 4.4 to include all locally compact sober spaces, since this would then subsume both locally compact Hausdorff spaces from analysis and continuous dcpos from domain theory.

5 Other Computational Effects

The general approach we have taken to characterizing Pprob(X) has nothing to do with probability! It potentially adapts to other "computational effects", so long as these are generated by a collection of "algebraic operations" in the sense of Plotkin and Power [7].

We assume a signature £ of basic operations and a (topological) £-algebra O, acting as algebra of observations. One can now define successively:

abstract effect structure — analogously to abstract choice structure,

complete £-algebra — analogously to complete choice algebra, and

abstract effect space — analogously to abstract probabilistic powerdomain. We consider a few possible examples.

In the case of probabilistic choice, dealt with in detail in this article, £ contains just one binary operation, 0, and O is the algebra (I<, 0).

Other forms of nondeterministic choice are potentially addressed by retaining a single binary operation, but varying the observation algebra. One should obtain a "lower powerdomain" using (S, V) for observations (recall that S is Sierpinski space), an "upper powerdomain" using (S, A), and a "convex powerdomain" using ({{^}, T}, {T}}, U) for O.

Other examples require different signatures. For example, for nontermina-tion, only a single constant ± is needed, and (S, ±) is the natural observation algebra. To combine nontermination and probabilistic choice, take the signature containing one binary operation for probabilistic choice and one constant for nontermination and use (I<, 0, 0) for the observation algebra. In this case, we have calculated the associated abstract effect space in Top, and proved that it is V<i(X) of subprobability valuations on X, again with the weak topology.

Finally, we mention that none of the basic ideas above are at all dependent on working in Top as the ambient category. The notion of abstract effect space makes sense in any category with finite products. All that is needed is a chosen algebra of observations. It would be interesting to see if there are other interesting mathematical constructions that can be captured as abstract effect spaces in appropriate categories.

Postscript

While producing this extended abstract for the MFPS proceedings we learnt with sadness of the untimely death of Claire Jones in October 2005. In her PhD thesis of 1990 [5], Claire established the definition of probabilistic powerdomain for arbitrary dcpos, and proved many of the fundamental results in the area. Claire was always modest about the achievements of her PhD. The second author recalls telling her several times how much he liked her thesis, to which she would always respond: "Ah, but have you read it? There's not much in it!" Time has told a different story. Fifteen years on, Claire's thesis rightly remains the primary reference in the field.

References

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[3] Gierz, G., K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove and D. Scott, "Continuous Lattices and Domains," CUP, 2003.

[4] Heckmann, R., Spaces of valuations, Annals of the New York Academy of Sciences 806 (1996), pp. 174-200.

[5] Jones, C., "Probabilistic Non-determinism," PhD thesis, University of Edinburgh, 1990.

[6] Jones, C. and G.D. Plotkin, A probabilistic powerdomain of evaluations. In Logic in Computer Science, IEEE Computer Society Press, 1989, pp. 186-195.

[7] Plotkin, G.D. and A.J. Power, Computational effects and operations: An overview, Electr. Notes Theor. Comput. Sci. 73, 2002, pp. 149-163.

[8] Smyth, M.B., Topology, in: Handbook of Logic in Computer Science I, OUP, 1992 pp. 641-761.