Unrestricted Stone Duality for Markov Processes

Robert Furber∗, Dexter Kozen†, Kim Larsen∗, Radu Mardare∗ and Prakash Panangaden‡ ∗Aalborg University, Denmark †Cornell University ‡McGill University, Montreal, Canada

Abstract—Stone duality relates logic, in the form of Boolean probabilities of events or expected behaviour. Recent papers on algebra, to spaces. Stone-type dualities abound in computer Markovian logic [11], [17], [18] have established completeness science and have been of great use in understanding the rela- and finite model properties for such systems. tionship between computational models and the languages used to reason about them. Recent work on probabilistic processes In this paper we focus on the duality of a certain class has established a Stone-type duality for a restricted class of of models of probabilistic computation, namely Markov tran- Markov processes. The dual category was a new notion—Aumann sition systems, with a certain class of Boolean algebras with algebras—which are Boolean algebras equipped with countable operators that behave like probabilistic modalities, the Aumann family of modalities indexed by rational probabilities. In this algebras. Aumann algebra is the algebraic analogue of Marko- article we consider an alternative definition of Aumann algebra that leads to dual adjunction for Markov processes that is a vian logic; that is, it is to Markovian logic what Boolean duality for many measurable spaces occurring in practice. This algebra is to propositional logic. They were first defined in extends a duality for measurable spaces due to Sikorski. In [19], where a restricted form of the duality was established. particular, we do not require that the probabilistic modalities The duality was shown to hold only under certain (somewhat preserve a distinguished base of clopen sets, nor that morphisms artificial) assumptions, to wit: of Markov processes do so. The extra generality allows us to give a perspicuous definition of event bisimulation on Aumann • The Aumann algebra must be countable. algebras. • The Borel sets of the Markov transition system must be generated by a distinguished countable family of clopen I.INTRODUCTION sets, and morphisms must preserve the distinguished clopens. Dualities in computer science have enjoyed a recent spate These assumptions were made in order to apply the Rasiowa– of popular interest. Since Plotkin and Smyth’s discovery of a Sikorski lemma [20], a lemma of logic that is dual to the Baire Stone-type duality between the predicate-transformer seman- category theorem of topology. The RSL/BCT implies that tics of Dijkstra and state-transformer semantics [1], [2], it has certain “bad” ultrafilters (those not satisfying the countably become increasingly apparent that dualities are ubiquitous in many infinitary defining conditions of countable Aumann al- computer science, having appeared in automata and formal gebras) can be deleted from the Stone space without changing language theory, automated deduction, programming language the supported algebra of measurable sets, since the “good” semantics and verification, domain theory, and concurrency ultrafilters are topologically dense. theory [3]–[14]. Although not a perfect duality due to these restrictions, the Dualities are important because they establish canonical groundwork laid in that paper nevertheless led to significant connections between computational models and the languages advances in the completeness of Markovian logics [21], [22]. used to reason about them. A duality gives an exact character- Previously, strong completeness theorems had used a powerful ization of the power of a state transition system by showing infinitary axiom scheme called the countable additivity rule, how the system determines a corresponding logic or algebra which has uncountably many instances. Moreover, one needs in a canonical way, and vice versa. Moreover, algebra homo- to postulate Lindenbaum’s lemma (every consistent set of morphisms correspond directly to structure-preserving maps or formulas extends to a maximally consistent set), which for bisimulations of the transition system, allowing mathematical these logics is conjectured but not proven. The duality result arguments to be transferred in both directions. of [19] gives rise to a complete axiomatization that does The original duality of Stone [15] asserts that the category not involve infinitary axiom schemes with uncountably many of Boolean algebras and Boolean algebra homomorphisms is instances and that satisfies Lindenbaum’s lemma. contravariantly equivalent to the category of Stone spaces and In this paper we improve the duality of [19] to a full- continuous maps. Jonsson and Tarski [16] extended Stone’s fledged Stone-like duality between Markov transition systems result to Boolean algebras with modal operators and Stone and Aumann algebras based on Sikorski’s Stone duality for spaces with transitions. A recent surge of interest in prob- σ-perfect σ-fields [23], [24] and σ-spatial Boolean algebras. abilistic systems, due largely to impetus from the artificial A σ-Boolean algebra is σ-spatial if every element, other intelligence and machine learning communities, has led to the than ⊥, is contained in an σ-complete ultrafilter; this is the study of various logics with constructs for reasoning about algebraic analogue of Lindenbaum’s lemma. The construction 978-1-5090-3018-7/17/$31.00 c 2017 IEEE does not use the RSL/BCT, thus the restrictions mentioned above are no longer necessary. However, we need to change σ-spatial if every element is contained in a σ-ultrafilter.1 If the definition of an Aumann algebra, so this duality is not we use σ-homomorphisms as morphisms, σ-Boolean algebras strictly a generalization of [19]. In particular, the axiom AA8 form a category σ-BA, with σ-spatial algebras forming a full that we use has uncountably many instances. We say that subcategory σ-BAsp. + the duality is Stone-like because the duality is no longer Define Q0 = Q ∩ [0, 1] and R = R ∩ [0, ∞). an algebraic/topological duality in the strict sense of the word, as the “topological” side is axiomatized in terms of A. Measurable Spaces and Measures measure-theoretic properties of the state space, not topological properties as with traditional Stone-type results. This is an easy If F ⊆ PM, the σ-field generated by F is the smallest price to pay, because it brings the relevant properties of the σ-field containing F. state space needed for duality into sharp relief. In any measurable space (X, Σ), a point x ∈ X defines a Our key results are: σ-ultrafilter on Σ by • a version of the duality of [19] for all σ-spatial Aumann algebras, not just countable ones; and hxi = {S ∈ Σ | x ∈ S}. • the removal of the assumptions that Markov processes Sikorski introduced the term σ-perfect for those measurable be countably generated and that maps preserve the dis- spaces for which h-i is a bijection from X to the set of σ- tinguished clopens. ultrafilters on Σ [24, pp. 1, 98]. The paper is organized as follows. In §II, we briefly review Measurable spaces form a category Mes with measurable the necessary background material. In §III, we describe Siko- maps as morphisms, and σ-perfect measurable spaces form a rski’s Stone duality for measurable spaces. We do this so as to full subcategory PMes. fix notation and also because it will be used in two different A nonnegative real-valued set function µ is said to be finitely places later in the article. It takes the form of an adjunction additive if µ(A∪B) = µ(A)+µ(B) whenever A∩B = ∅, and that becomes a categorical duality when restricted to objects countably additive if µ(S A ) = P µ(A ) for a countable for which the unit and counit are isomorphisms. In §IV, we i i i i pairwise-disjoint family of measurable sets. A measure on a recall Halmos’s description of free σ-Boolean algebras and measurable space (X, Σ) is a countably additive set function what a presentation of a σ-Boolean algebra is. We then give µ :Σ → +. A measure is a probability measure if in addition a presentation of the Borel σ-field of [0, 1] as an abstract σ- R µ(X) = 1, and a subprobability measure if µ(X) ≤ 1. We Boolean algebra. In §V, we describe an alternative definition of use G(X, Σ) to denote the set of subprobability measures on Aumann algebras and prove a duality for Aumann algebras and (X, Σ). (discrete-time, continuous-space, time-homogeneous) Markov We can view G(X, Σ) as a measurable space by considering processes extending the duality for σ-Boolean algebras and the σ-field generated by the sets {µ ∈ G(X, Σ) | µ(S) ≥ r} measurable spaces in §III. Finally, in §VI we describe the for S ∈ Σ and r ∈ [0, 1]. This is the least σ-field on G(X, Σ) generalization of labelled Markov processes and how event such that all maps µ 7→ µ(S): G(X, Σ) → [0, 1] for S ∈ Σ bisimulation is formulated in the setting of Aumann algebras. are measurable, where the real interval [0, 1] is endowed its II.BACKGROUND Borel σ-field. In the usual way, we extend G to a functor See Johnstone [25] for a detailed introduction to Stone Mes → Mes by defining duality and its ramifications, as well as an account of several −1 other related mathematical dualities such as Priestley duality G(f)(µ)(T ) = µ(f (T )), for distributed lattices and Gelfand duality for C∗-algebras. We assume knowledge of basic notions from measure theory where f :(X, Σ) → (Y, Θ) is a measurable map, µ ∈ G(X, Σ) and topology such as field of sets, σ-field, measurable set, and T ∈ Θ. It is worth mentioning that G is the subprobabilis- measurable space, measurable function, measure, topology, tic Giry monad [28]. open and closed sets, continuous functions, and the Borel algebra of a topology (denoted by Bo(X) for X a topological B. Markov Processes space in this article). See [26], [27] for a more thorough Markov processes (MPs) are models of probabilistic systems introduction. with a continuous state space and discrete-time probabilistic Throughout, we use σ-Boolean algebra to refer to a σ- transitions [28]–[30]. complete Boolean algebra, i.e. a Boolean algebra with count- able joins and meets, and use σ-field to refer to a pair (X, F) Definition 1 (Markov process). A Markov process (MP) where X is a set, and F a family of subsets that is a σ-Boolean is a measurable space (X, Σ) equipped with a measurable algebra under set-theoretic operations. To avoid confusion we map θ :(X, Σ) → G(X, Σ). A Markov process is said to do not use the term σ-algebra. be σ-perfect iff (X, Σ) is. Maps of Markov processes are The natural notion of an ultrafilter on a σ-Boolean algebra “zig-zags”, i.e. if (X1, Σ1, θ1) and (X2, Σ2, θ2) are Markov is a σ-ultrafilter, i.e. an ultrafilter that is additionally closed under countable meets. We say that a σ-Boolean algebra is 1This terminology is inspired by the theory of locales. processes, a measurable map f :(X1, Σ1) → (X2, Σ2) is a We define the category Aumann to have σ-Aumann algebras map of Markov processes if as objects and σ-Aumann algebra morphisms as its mor-

f phisms, and AumannSp to be the full subcategory on σ- X1 / X2 spatial Aumann algebras.

θ1 θ2 The reader may verify that AA1 and AA3-AA6 are the same   as in [19, §4.1], and that AA7 and AA8 imply the AA7 of the G(X1) / G(X2) G(f) original definition. Note that AA2 originates in [22, Table 3], and is the small change needed to account for subprobability commutes. Then Markov processes and morphisms thereof distributions. Note also that AA2 with r = 0 is inconsistent form a category Markov, with σ-perfect Markov processes with AA1. Additionally, AA1 is implied by AA7 with s = 0. forming a full subcategory PMarkov. A σ-spatial Aumann algebra is defined to be a σ-Aumann In a Markov process (X, Σ, θ), and θ is called the transition algebra whose underlying σ-Boolean algebra is σ-spatial. function. For x ∈ X, θ(x):Σ → [0, 1] is a subprobability measure on the state space (X, Σ). For S ∈ Σ, the value III.DUALITYFOR MEASURABLE SPACES θ(x)(S) ∈ [0, 1] represents the probability of a transition from x to a state in S. In this section, we express Sikorski’s duality for measurable The condition that θ be a measurable function X → spaces [24, §24, §32][23, 2.1-2] as an adjunction, and describe G(X, Σ) is equivalent to the condition that for fixed S ∈ Σ, how this adjunction can be restricted to an equivalence (see the function x 7→ θ(x)(S) is a measurable function X → [0, 1] [32, Part 0, Proposition 4.2] for a proof that this is possible (see e.g. [30, Proposition 2.9]). for any adjunction). If (X, Σ) is a measurable space, then Σ is a σ-Boolean C. Aumann Algebras algebra. We can use this to define a functor F : Mes → Aumann Algebra (AA) [19] is the algebraic analogue of σ-BAop by defining F (X, Σ) = Σ, and for f :(X, Σ) → Markovian logic [11], [17], [18]. It is so named in honor of (Y, Θ) a measurable map and T ∈ Θ Robert Aumann, who has made fundamental contributions to −1 probabilistic logic [31]. F (f)(T ) = f (T ). Definition 2 (Aumann algebra). A σ-Aumann algebra is a This is easily verified to be a functor. The following is easy tuple (A, (Lr)r∈Q0 ), where A is a σ-Boolean algebra, Q0 = to verify using the fact that sets can be distinguished by their Q ∩ [0, 1] and each Lr : A → A, such that the following points. axioms hold, where a, b are arbitrary elements of A and r, s Lemma 3. For any measurable space (X, Σ), Σ is a σ-spatial are elements of Q0: σ-Boolean algebra. (AA1) > ≤ L0(a) So we can also regard F as having the type Mes → (AA2) Lr(⊥) ≤ ⊥, where r > 0 σ-BAop. The analogous functor in Stone duality takes the (AA3) L (a) ≤ ¬L (¬a) if r + s > 1 sp r s Boolean algebra of clopens of a Stone space. (AA4) Lr(a ∧ b) ∧ Ls(a ∧ ¬b) ≤ Lr+s(a) if r + s ≤ 1 The reader familiar with Stone duality will already be (AA5) ¬Lr(a ∧ b) ∧ ¬Ls(a ∧ ¬b) ≤ ¬Lr+s(a) if r + s ≤ 1 expecting ultrafilters to be involved in the definition of a (AA6) a ≤ b implies Lr(a) ≤ Lr(b) functor the other way. For A a σ-Boolean algebra, we define V σ (AA7) Lr(a) = Ls(a) U (A) to be the set of all σ-ultrafilters on A. Given an element rimage of A under the map - . L M L M Lemma 4. - is a surjective morphism of σ-Boolean algebras We define a morphism of σ-Aumann algebras f : A → F(A),L andM F(A) is a σ-field [24, §24.1]. We also have (A, (Lr)) → (B, (Mr)) to be a σ-Boolean homomorphism that A is σ-spatial iff - is an isomorphism. such that f(Lr(a)) = Mr(f(a)) for all a ∈ A, i.e. such that L M the following diagram commutes for all r ∈ Q0: The proof is omitted. op f We can now define G : σ-BA → PMes on objects A / B as G(A) = (U σ(A), F(A)), where F(A) = A . On σ- f : A → B G(f) L M L M homomorphisms , is defined for each r r σ   u ∈ U (B) as A / B. −1 f G(f)(u) = f (u). In order to prove that F a G, we define the unit and counit i.e. it is not possible to produce this set by taking powersets of the adjunction. For a measurable space (X, Σ), for each or unions of strictly smaller families of strictly smaller sets. element x ∈ X, we can define an ultrafilter hxi ∈ U σ(Σ) as This was first proven by Ulam [36, Lemma 1 and Satz 4]. It is therefore not possible to prove the existence, or even hxi = {S ∈ Σ | x ∈ S}. the consistency, of a non-σ-spatial discrete set [37, Theorem

We define the unit ηX :(X, Σ) → G(F (X, Σ)) and counit 12.12]. A : F (G(A)) → A to be IV. PRESENTATIONS OF σ-BOOLEAN ALGEBRAS

ηX (x) = hxi In this section we describe Halmos’s construction of the free σ-Boolean algebra on a set, how to give presentations of A(a) = a . L M σ-fields in terms of generators and relations, how to define op The direction of A is reversed because we use σ-BA . measurable maps in terms of presentations, and give a presen- Theorem 5. (F, G, η, ) is an adjunction making G a right tation of Bo([0, 1]). We note that this point that presentations adjoint to F . In fact, G maps into PMes and by restricting F can be defined in a more general setting of universal algebra and G to the categories where the unit and counit are isomor- for theories with operations of countable arity, and Lemma 8 op and Proposition 9 (i) could have been given as a reference to phisms, they define an adjoint equivalence σ-BAsp 'PMes. such a general theorem rather than being proved in this special In passing, the above theorem shows that PMes and setting. However, we will stick with a presentation closer to σ-BAsp are reflective [33, §IV.3] subcategories of Mes and measure theory than universal algebra. σ-BA, respectively. We have the usual forgetful functor U : σ-BA → Set. The reader might object to the definition of a σ-perfect measurable space as only attempting to “solve” a problem by Proposition 7 (Halmos). The functor U has a left adjoint H, defining it out of existence. To address this potential criticism, given on objects by we show that there are many σ-perfect measurable spaces H(X) = Ba(2X ), occurring in practice using a theorem of Hewitt. First, we where 2X is given the product topology. This is also a left recall that on a topological space X, the Baire σ-field Ba(X) adjoint to the restriction of U to σ-BA . can be defined to be the σ-field generated by the zero sets, the sp subsets of X of the form f −1(0) for some continuous map For the proof, see [38, §23, Theorem 14]. X → R. If X is metrizable, Ba(X) is the same as the Borel We have already seen the definition of a σ-ultrafilter. A σ- σ-field. We also need to refer to the concept of a realcompact ideal in a σ-Boolean algebra A is a subset I ⊆ A that is space. We omit the definition [34, §5.9], but we only need the downward closed (i.e. if b ≤ a and a ∈ I, then b ∈ I) and fact that every σ-compact Hausdorff space and every separably closed under countable joins. As any σ-ideal is an ideal, we metrizable space2 is realcompact, as is shown in [34, §8.2]. can define A/I to be the set of equivalence classes of elements of A modulo the relation a ∼ b ⇔ a4b ∈ I, where 4 is the Theorem 6 (Hewitt). For a completely regular space X, symmetric difference, as usual, and σ-Boolean operations are (X, Ba(X)) is σ-perfect iff X is realcompact. well-defined with respect to this equivalence relation. See [35, Theorem 16] for the proof3. If K is a subset of a σ-Boolean algebra A, we can define Therefore the Borel σ-field of any separable metric space, the sub-σ-Boolean algebra B generated by K to be the e.g. a Polish space or analytic space, and the Baire σ-field smallest σ-Boolean algebra in A containing K. This can of any compact Hausdorff space are σ-perfect. We warn the equivalently be defined as either the intersection of all σ- reader that it is not the case that the Baire σ-field of a Boolean subalgebras of A that contain K, or by building up locally compact space is σ-perfect, nor the Borel σ-field of elements of B as σ-Boolean combinations of elements of K an unmetrizable compact Hausdorff space, and σ-perfectness using countable ordinals, as in the Borel hierarchy [39, §II.3]. is not preserved under σ-subfields (even though σ-spatiality is If B = A, we say that A is generated by K. Note that K will preserved under subalgebras). not necessarily generate A as a Boolean algebra, in general For example, consider an uncountable set X. Take Σ to countable operations will be necessary. consist of countable sets and their complements (co-countable Recall that H : Set → σ-BA is the free (σ-spatial) σ- sets), the countable co-countable σ-field. Then we define u Boolean algebra on a set, from Proposition 7, in the following to be the set of co-countable sets. This is a non-principal σ- lemma. ultrafilter on Σ, so (X, Σ) is a measurable space that is not Lemma 8. A set K ⊆ A generates A iff the universal map σ-perfect. But note that if we take X = , this is a σ-subfield R ˜i : H(K) → A: of the Borel σ-field. ηK One thing to note is that the smallest cardinality of a set X K / U(H(K)) H(K) such that (X, P(X)) is not σ-spatial is strongly inaccessible, U˜i ˜i 2 i Not requiring completeness. #   3Hewitt uses Q-space to mean realcompact space. U(A) A is surjective, where i is the inclusion morphism. This is a countable family of closed subsets of [0, 1]. To define the relations, we write We say a σ-ideal I ⊆ A is generated by a subset R ⊆ I if I is the smallest σ-ideal containing R, equivalently if Br = ηK ([r, 1])

( ∞ ) _ Then we define the relations as I = a ∈ A | ∃(bi)i∈N .∀i ∈ N .bi ∈ R and a ≤ bi . ( ! ) i=1 ^ R = {(Br ∧ Bs)4Bs}r

(A, (Lr)r∈Q0 ) → (Mr)r∈Q0 we define G(g) exactly as for measurable maps between σ-perfect measurable spaces, we σ-Boolean algebras, i.e. G(g)(u) = g−1(u). can apply the categorical duality from Theorem 5 to deduce λ = θ ◦ G(g) Proposition 14. With the above definition, G is a functor g(a) a from the equation above, and therefore the Aumannop → PMarkov. diagram commutes.

Proof. By Proposition 13, it is defined correctly on objects. If Theorem 15. F is a left adjoint to G, and when restricted they op we have a morphism of σ-Aumann algebras (A, (Lr)r∈Q0 ) → define adjoint equivalences AumannSp ' PMarkov. (B, (Mr)r∈Q0 ), by Theorem 5, this defines a measurable map G(f): U σ(B) → U σ(A), and the identity map and Proof. Recall the natural transformations - : A → F (G(A)) L M composition are preserved. Therefore we only need to show and h-i :(X, Σ) → G(F (X, Σ)) from Theorem 5. If we show that G(f) is a map of Markov processes, i.e. that the diagram: that - is a morphism of Aumann algebras and h-i a morphism of MarkovL M processes, then the commutativity of the naturality G(g) diagrams and the triangle diagrams defining an adjunction U σ(B) / U σ(A) follows from the proofs in Theorem 5, and we have shown λ θ F is a left adjoint to G.   We first show that - is a σ-Aumann algebra morphism. G(U σ(B)) / G(U σ(A)) L M G(G(g)) That is to say, we want to show that for all a ∈ A and r ∈ Q0 that Lr(a) = Mr( a ), where (Mr)r∈Q0 is the Aumann commutes, where θ and λ are the morphisms defining the algebraL structureM on FL(GM (A)). Well, Markov processes on U σ(A) and U σ(B) respectively. σ The bottom left path is Lr( a ) = {u ∈ U (A) | θ(u)( a ) ≥ r} see (3) L M σ L M −1 = {u ∈ U (A) | θa(u) ≥ r} see (5) G(G(g))(λ(u))( a ) = λ(u)(G(g) ( a )) −1 L M L M = θa ([r, 1]) Now, = Lr(a) see (4). −1 σ −1 L M G(g) ( a ) = {u ∈ U (B) | u ∈ G(g) ( a )} We now show that h-i is a morphism of Markov processes, L M L M = {u ∈ U σ(B) | G(g)(u) ∈ a } i.e. the following diagram commutes σ L M = {u ∈ U (B) | a ∈ G(g)(u)} h-i (X, Σ) G(F (X, Σ)) = {u ∈ U σ(B) | a ∈ g−1(u)} / σ = {u ∈ U (B) | g(a) ∈ u} θ λ = {u ∈ U σ(B) | u ∈ g(a) }   L M G(X, Σ) / G(G(F (X, Σ))), = g(a) , G(h-i) L M so the bottom left path is equal to where λ is map making G(F (X, Σ)) a Markov process. In equations, what we want to show is that G(h-i) ◦ θ = λ ◦ h-i. λ(u)( g(a) ) = λg(a)(u). (6) Recall that the σ-field on G(F (X, Σ)) consists of elements L M of the form S for S ∈ Σ, so we want to show that, for all Note that by Theorem 6 this duality can be applied to any x ∈ X and SL ∈M Σ, of the Stone-Markov processes with countable base considered in [19], although the dual Aumann algebra will be the Borel G(h-i)(θ(x))( S ) = λ(hxi)( S ). (8) sets, not the base of clopens. L M L M If we expand the definition of G on the left hand side, we get VI.EVENT BISIMULATION AND DUALITYFOR LABELLED −1 MARKOV PROCESSES G(h-i)(θ(x))( S ) = θ(x)(h-i ( S )). L M L M Given a measurable space (X, Σ), a labelled Markov pro- e Now, we can simplify the argument of θ(x) as follows cess is a tuple (X, Σ, (θ )e∈E), where E is a set of labels e −1 and for each e ∈ E, θ : X → G(X, Σ) is a measurable h-i ( S ) = {x ∈ X | hxi ∈ S } e e L M L M function. If (X, Σ, (θ )e∈E) and (Y, Θ, (λ )e∈E) are labelled = {x ∈ X | S ∈ hxi} Markov processes with the same label set E, we say that = {x ∈ X | x ∈ S} = S, a measurable function f :(X, Σ) → (Y, Θ) is a morphism of labelled Markov processes if it is a morphism of Markov so, all together, the left hand side of (8) is θ(x)(S). For the processes f :(X, Σ, θe) → (Y, Θ, λe) for each e ∈ E. For right hand side, we can expand the definition any set of labels, we have a category LabMarkovE of E- labelled Markov processes and their morphisms. It should now λ(hxi)( S ) = λS(hxi) L M be obvious how to define the full subcategory of σ-perfect according to (5). We now prove that λS(hxi) = θ(x)(S) by labelled Markov processes, PLabMarkovE. showing that, for all r ∈ Q0, λS(hxi) ≥ r iff θ(x)(S) ≥ r. We can define a labelled σ-Aumann algebra to be e (A, (Lr)e∈E,r∈Q0 ), such that for each label e ∈ E, e −1 (A, (Lr)r∈Q0 ) is a σ-Aumann algebra. A morphism of E- λS(hxi) ≥ r ⇔ hxi ∈ λS ([r, 1]) e e labelled σ-Aumann algebras (A, (Lr)) → (B, (Mr )) is a σ- ⇔ hxi ∈ Lr(S) (4) Boolean homomorphism A → B that is a σ-Aumann algebra L M ⇔ Lr(S) ∈ hxi homomorphism for each e ∈ E. For each set of labels E, we have categories LabAumannE and LabAumannSp ⇔ x ∈ Lr(S) E defined in the familiar way. By working with each e ∈ E ⇔ θ(x)(S) ≥ r (3). independently, we can define F,G and a duality as in Theorem Because every real is the supremum of the rationals below it, 15. this implies that θ(x)(S) = λS(hxi), and therefore that (8) In the context of labelled Markov processes, an event e holds. bisimulation on (X, Σ, (θ )e∈E) is defined to be a sub σ- e As we explained at the start of the proof, this suffices to field Λ ⊆ Σ such that (X, Λ, (θ |Λ)e∈E) is a labelled Markov e show that F is a left adjoint to G. We can show that F and G process, where for each e ∈ E, θ |Λ : X → G(X, Λ) is the e define an adjoint equivalence PMarkov ' AumannSp by function such that for each x ∈ X, θ |Λ(x) is the restriction e showing that when - and h-i are, respectively, σ-Boolean of θ (x) to Λ. This notion was originally defined in [40, algebra isomorphismsL M and measurable isomorphisms, they Definition 4.3], as a version of the notion of probabilistic are σ-Aumann algebra isomorphisms and Markov process bisimulation [41] that is more adapted to probabilistic logics. isomorphisms. e Theorem 16. Let (X, Σ, (θ )e∈E) be a labelled Markov −1 We do so as follows. We first need to show that - is a process. A σ-field Λ ⊆ Σ is an event bisimulation iff it is a σ- σ-Aumann algebra homomorphism. First, we observeL M that the e Aumann subalgebra of F (X, Σ, (θ )e∈E), i.e. iff it is preserved equality Mr( a ) = Lr(a) implies by the Aumann algebra operations. L M L M −1 - (Mr( a )) = Lr(a) Proof. We start with the only if direction, which is to say L M L M −1 = Lr( - ( a )). we show that an event bisimulation Λ is also an Aumann L M L M subalgebra of the Aumann algebra F (X, Σ, (θe)). As every element of the algebra F (G(A)) is of the form a Since an event bisimulation is a σ-field supporting a labelled −1 for some a ∈ A, we have shown that - is an AumannL M Markov process, when F is applied to it becomes an Aumann L M e e algebra isomorphism. algebra. As θ |Λ is the restriction of θ to Λ at all points It is a generally true fact that a measurable isomorphism e x ∈ X, we get that Λ and Σ agree on the effect of the Lr that is a map of Markov processes is an isomorphism of operators for all r ∈ 0. This shows that Λ is an Aumann −1 Q Markov processes, but we give the special case that h-i is subalgebra of Σ. a morphism of Markov processes here: For the other direction, suppose A0 is an Aumann subal- gebra of F (X, Σ, (θe) ). We prove that A is an event G(h-i) ◦ θ = λ ◦ h-i ⇔ G(h-i) ◦ θ ◦ h-i−1 = λ e∈E 0 bisimulation of (X, Σ, (θe) ) as follows. We need to show −1 −1 e∈E ⇔ θ ◦ h-i = G(h-i ) ◦ λ. that for each e ∈ E, e θ |A0 :(X, A0) → G(X, A0) is measurable. This is equivalent to showing that for each a ∈ [10] M. Mislove, J. Ouaknine, D. Pavlovic, and J. Worrell, “Duality for La- belled Markov Processes,” in FOSSACS, ser. Lecture Notes In Computer A0 and each B ∈ Bo([0, 1]), Science, I. Walukiewicz, Ed., vol. 2987, 2004, pp. 393–407. (θe| )−1(p−1(B)) ∈ A . [11] R. Goldblatt, “On the role of the Baire category theorem in the A0 a 0 foundations of logic,” Journal of Symbolic logic, pp. 412–422, 1985. [12] ——, “Deduction systems for coalgebras over measurable spaces,” To do this, it is sufficient to prove that for any rational r ≤ 1, Journal of Logic and Computation, vol. 20, no. 5, pp. 1069–1100, 2010. e −1 −1 [13] M. J. Gabbay, T. Litak, and D. 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