A Behavioural Pseudometric for Probabilistic ... - Semantic Scholar
A Behavioural Pseudometric for Probabilistic Transition Systems Franck van Breugel a 1 and James Worrell b 2 ;
;
York University, Department of Computer Science 4700 Keele Street, Toronto, M3J 1P3, Canada b Tulane University, Department of Mathematics 6823 St Charles Avenue, New Orleans LA 70118, USA a
Abstract Boolean-valued logics and associated discrete notions of behavioural equivalence sit uneasily with semantic models featuring quantitative data, like probabilistic transition systems. In this paper we present a pseudometric on a class of probabilistic transition systems yielding a quantitative notion of behavioural equivalence. The pseudometric is de ned via the terminal coalgebra of a functor based on a metric on the space of Borel probability measures on a metric space. States of a probabilistic transition system have distance 0 if and only if they are probabilistic bisimilar. We also characterize our distance function in terms of a real-valued modal logic. Key words: probabilistic transition system, pseudometric, probabilistic bisimilarity, terminal coalgebra, real-valued modal logic
1 Introduction The majority of veri cation methods for concurrent systems only produce qualitative information. Questions like \Does the system satisfy its speci cation?" and \Do the systems behave the same?" are answered \Yes" or \No". Giacalone, Jou and Smolka [14], Huth and Kwiatkowska [19] and Desharnais, Gupta, Jagadeesan and Panangaden [12] have pointed out that such discrete Boolean-valued reasoning sits uneasily with semantic models featuring quantitative data, like probabilistic transition systems. 1 2
Supported by Natural Sciences and Engineering Research Council of Canada. Supported by the US Oce of Naval Research.
Preprint submitted to Elsevier Science
26 October 2001
Real numbers are ideal entities. Computers, and humans for that matter, only deal with approximations. If we only have an approximate description of a system, it makes no sense to ask if any two states behave exactly the same. Even if we have a precise description of a system, we may still want to express the idea that a state satis es a formula with high probability or that two states exhibit almost the same behaviour. Furthermore, the problem of automatically checking if a state satis es a formula or verifying that two states behave exactly the same will prove intractable owing to the impossibility of exact computation of reals. The basis of any computational treatment of the reals, and hence of automatic veri cation of probabilistic systems, has to involve approximation. The probabilistic modal logic of Larsen and Skou [24] adds probability thresholds to traditional modal logic. In this logic one has a formula haiq ' which is satis ed if the sum of the probabilities of transitions labelled with a to states satisfying ' exceeds q 2 [0; 1]. We might hope to be able to check that an in nite state (or even continuous) probabilistic transition system satis es such a formula by model checking some nite approximant of the system. However, as we illustrate below, the truth of a given formula is not continuous with respect to the obvious notion of approximation on transition systems. This phenomena is discussed extensively by Desharnais et al. in [12]. In a similar vein, Giacalone et al. [14] and Desharnais et al. [11] have criticized all-or-nothing notions of behavioural equivalence for probabilistic transition systems such as Larsen and Skou's probabilistic bisimulation [24]. Recall that a probabilistic bisimulation is an equivalence relation on the state space of a transition system such that related states have exactly the same probability of making a transition into any equivalence class. Thus, for instance, the states s and s of the probabilistic transition system 0
0
s
> 1 @` @ @@ ~~ ~~ ~ ~~ ~ a;1 0@ @@ @@ @ ~~ a; 21 @ @ ~~ a; 21
s
s
1
+ @ a; @@ 2 @
~~ ~~ 1 a; 2
s
0
?
2
are only probabilistic bisimilar if is 0. However, the two states behave almost the same for very small di erent from 0. In the words of [11], behavioural equivalences like probabilistic bisimilarity are not robust, since they are too sensitive to the exact probabilities of the various transitions. The lack of robustness of these logics and behavioural equivalences has a number of implications. First of all, the logics and behavioural equivalences are in general not suited for reasoning about in nite state systems in terms of 2
their nite approximants. For example, the in nite state system 1 2
s
1
s |
s
0
1 4
1 8
2
1
s
1 16
s
s : : :
4
7
1
1
s : : :
s
3
5
8
1
1
s : : :
s
6
9
1
s ::: 10
and its nite approximant 1 2
s
1
s
1 4
s |
s
2
1
4
1
s
0
1 4
s
3
5
1
s
6
are not probabilistic bisimilar. The systems also do not satisfy the same formulae. For example, assuming that all transitions are labelled with a, the formula hai 161 hai 12 hai 12 hai 12 true is satis ed by the in nite state system but not by its nite approximant. Secondly, the probabilities of the transitions are often approximations themselves. For example, may be approximated by 0.32. Since the systems 1
1 a;
s s
1
| || || | ~| |
s
0B
B B a; ?1 BB BB B
s
0:32 |
s
2
1
b;1
s
3
|| |~ |
| ||
s
0B
BB B B 0:68 BB B
s
2
1
3
are neither probabilistic bisimilar nor satisfy the same formulae, one cannot use the latter to reason about the former when using probabilistic bisimilarity or the probabilistic modal logic of Larsen and Skou. Finally, any inexactness in calculations to decide whether a formula is satis ed in a state or whether a relation on states is a probabilistic bisimulation may result in an incorrect conclusion. To the best of our knowledge, the earliest attempt to address some of the 3
problems with probabilistic bisimulation as outlined above is the paper of Giacalone et al. [14]. They de ne a pseudometric on the states of a (restricted type of) probabilistic transition system, yielding a smooth, quantitative notion of behavioural equivalence. A pseudometric di ers from an ordinary metric in that di erent elements, that is, states, can have distance 0. The distance between states, a real number between 0 and 1, can be used to express the similarity of the behaviour of those states. The smaller the distance, the more alike the behaviour is. In particular, the distance between states is 0 if they are behaviourally indistinguishable. The present paper is mostly inspired by the work of Desharnais et al. [11] and De Vink and Rutten [33]. The former introduce a pseudometric on a class of probabilistic transition systems more general than that considered by Giacalone et al. This pseudometric is de ned via a non-standard semantics for a probabilistic modal logic, where formulae get interpreted as measurable functions into the interval [0; 1] rather than as Boolean-valued functions. We shall show that what we present in this paper is essentially a coinductive account of a closely related pseudometric. We will also discuss some of the advantages conferred by such an account. De Vink and Rutten [33] showed that a restricted class of probabilistic transition systems can be viewed as coalgebras. Taking this view, one can transfer results from the general theory of coalgebras to the setting of probabilistic systems. (Many di erent kinds of transition system can be viewed as coalgebras. For an overview of the general theory of coalgebras, we refer the reader to Rutten's [30].) Using one such result De Vink and Rutten established the existence of a terminal object in their category of coalgebras. By de nition there is a unique map from an arbitrary coalgebra to the terminal coalgebra. Furthermore, De Vink and Rutten showed that the kernel of the unique map coincides with probabilistic bisimilarity. Each of these results has an analog in the non-probabilistic case. Thus the coalgebraic approach o ers a uniform conceptual framework for di erent operational models. In this paper we exploit the coalgebraic framework to de ne a notion of quantitative behavioural equivalence for probabilistic transition systems. In particular, we de ne a pseudometric on the states of a probabilistic transition system in terms of the terminal coalgebra of an endofunctor P on the category of pseudometric spaces and nonexpansive maps. The de nition of P is based on a metric on Borel probability measures. This metric is known as Hutchinson metric [18], the Kantorovich metric [20] and also as the Wasserstein metric. P -coalgebras can be seen as probabilistic transition systems with discrete or continuous state spaces. The terminal P -coalgebra provides for a notion of approximate equivalence similar to the pseudometric of Desharnais et al. mentioned above. In fact, we de ne a pseudometric on the state space of a probabilistic transition system, seen as a P -coalgebra, as the pseudometric 4
kernel of the unique map to the terminal P -coalgebra. That is, the distance between two states is the distance between their images under the unique map to the terminal coalgebra. Moreover states are at distance 0 just in case they are probabilistic bisimilar in the sense of Larsen and Skou. So far we have motivated our concern for de ning a notion of quantitative behavioural equivalence by examples featuring probabilistic transition systems with discrete state spaces. However our framework is suciently general to model probabilistic transition systems the state space of which is continuous, like [0; 1]. We refer the reader to [12] for a discussion of the importance of modelling continuous as well discrete systems. The rest of this paper is organized as follows. In Section 2 we present some minor variations on results of [1,32] which allow us to prove that a terminal P -coalgebra exists. Pseudometric kernels are introduced in Section 3. In Section 4, we present the above mentioned metric on Borel probability measures. In Section 5, we study the key ingredient of the functor P . In Section 6, we present the functor P and we show that all discrete probabilistic transition systems and a large class of continuous probabilistic transition systems can be viewed as P -coalgebras. We introduce our pseudometric in Section 7. In Section 8, we introduce a pseudometric de ned in terms of a modal logic a la Desharnais et al. The pseudometrics introduced in Section 7 and 8 are related in Section9. In the concluding section we present related and future work. The reader is assumed to be familiar with some elementary category theory, metric space theory and probability theory. For more details, we refer the reader to, for example, the texts of Mac Lane [26], Sutherland [31] and Billingley [6].
2 A Pseudometric Terminal Coalgebra Theorem In this section, we introduce coalgebras and a mild generalization of Rutten and Turi's metric terminal coalgebra theorem [32]. For more details about the theory of coalgebras we refer the reader to, for example, Rutten's [30].
De nition 1 Let C be a category. Let F : C ! C be a functor. An F -coalgebra consists of an object C in C together with an arrow f : C ! F (C ) in C .
The object C is called the carrier. The arrow f is called the structure. An F -homomorphism from an F -coalgebra hC; f i to an F -coalgebra hD; gi is an
5
arrow : C ! D in C such that F ( ) f = g .
C
/
D g
f
F (C )
F (D)
F ( )
/
The F -coalgebras and F -homomorphisms form a category. If this category has a terminal object, then this object is called the terminal F -coalgebra.
In the rest of this section, we restrict our attention to the category PMet of 1-bounded pseudometric spaces and nonexpansive functions. A pseudometric space is 1-bounded if all its distances are bounded by 1. The assumption of 1-boundedness is made once and for all here and is not always explicitly stated in our results. A function is nonexpansive if it does not increase any distances. We denote the collection of nonexpansive functions from the space X to the space Y by X !1 Y . This collection can be turned into a pseudometric space by endowing the functions with the supremum metric. 1
Let c be a constant between 0 and 1. A function is c-contractive if it decreases all distances by at least a factor c. This notion can be lifted to functors as follows.
De nition 2 A functor F : PMet ! PMet is locally c-contractive if for 1
1
all pseudometric spaces X and Y , the function
FX;Y : (X !1 Y ) ! (F (X ) !1 F (Y )) de ned by
FX;Y (f ) = F (f ) is c-contractive.
In the rest of this section, we restrict ourselves to locally contractive functors. Furthermore, we focus on functors which preserve equivalence and completeness.
De nition 3 A functor F : PMet ! PMet preserves equivalence and com1
1
pleteness if for all complete metric spaces X , F (X ) is a complete metric space. A functor which preserves equivalence and completeness can be restricted to a functor on the category of complete metric spaces and nonexpansive functions. 6
A locally contractive functor F which preserves equivalence and completeness has a xed point. That is, there exists a space, say x (F ), and an isometry from x (F ) to F ( x (F )). Recall that an isometry is a function which preserves all distances and which is onto.
Lemma 4 For each locally c-contractive functor F : PMet ! PMet which 1
1
preserves equivalence and completeness, there exists a unique complete metric space x (F ) such that there is an isometry i : x (F ) ! F ( x (F )).
PROOF. Immediate consequence of [32, Theorem 7.2]. 2 Next, we show that h x (F ); ii is a terminal F -coalgebra. For the rest of this section, we x hX; f i to be an F -coalgebra. To characterize the unique F homomorphism from the F -coalgebra hX; f i to the F -coalgebra h x (F ); ii we introduce the following function.
De nition 5 The function h
X;f
by
i
: (X !1 x (F )) ! (X !1 x (F )) is de ned
hX;f i ( ) = i?1 F ( ) f:
X
/
i?1
f
F (X )
x (O F )
F ( )
/
F ( x (F ))
Since the functor F is locally c-contractive, we have that the function hX;f i is c-contractive.
Proposition 6 The function h
X;f
i
is c-contractive.
PROOF. Similar to the proof of [32, Proposition 7.1]. 2 Because x (F ) is a complete metric space, X !1 x (F ) is a complete metric space as well. Obviously, the space X !1 x (F ) is nonempty. Since hX;f i is a contractive function from a nonempty complete metric space to itself, we can conclude from Banach's theorem that it has a unique xed point x (hX;f i ). This xed point x (hX;f i ) is the limit of a sequence ( n)n, where is an arbitrary nonexpansive function from X to x (F ) and n = ( n). 0
+1
Lemma 7 The function x (h i) is the unique F -homomorphism from the F -coalgebra hX; f i to the F -coalgebra h x (F ); ii. X;f
7
PROOF. Similar to the proof of [32, Proposition 7.1]. 2 Theorem 8 For every locally c-contractive functor F : PMet ! PMet 1
1
which preserves equivalence and completeness there exists a terminal F -coalgebra h x (F ); ii and x (F ) is a complete metric space.
PROOF. Immediate consequence of Lemma 4 and 7. 2 We will exploit both the terminal F -coalgebra h x (F ); ii and the unique F homomorphism x (hX;f i ) when de ning our pseudometric. If the functor F also preserves compactness, then we can conclude that the carrier of the terminal F -coalgebra is a compact metric space.
Theorem 9 For every locally c-contractive functor F : PMet ! PMet 1
1
which preserves equivalence and compactness there exists a terminal F -coalgebra h x (F ); ii and x (F ) is a compact metric space.
PROOF. Similar to the proof of Theorem 8 exploiting [1, Theorem 4.4] of Alesi, Baldan and Belle. 2
The only di erence between the results of [1,32] mentioned above and the results presented in this section is that former are about metric spaces whereas the latter are about pseudometric spaces.
3 Pseudometric Kernels In this section, we introduce pseudometric kernels. Our pseudometric on the states of a probabilistic transition system will be de ned as a pseudometric kernel. A function from a set S to a pseudometric space X de nes a distance function d on S . We call this distance function the pseudometric kernel induced by . The distance between s and s in S is de ned as the distance of their -images in the pseudometric space X . 1
2
De nition 10 Let : S ! X . The distance function d : S S ! [0; 1] is
de ned by
d (s ; s ) = dX ( (s ); (s )): 1
2
1
2
8
One can easily verify that the pseudometric kernel d is a pseudometric. Note that s and s have distance 0 if they are mapped by to the same element in X . For example, if is a constant function then all distances are 0. 1
2
If the functions and are similar, that is, they are close to each other with respect to the supremum metric, then the induced pseudometric kernels d 1 and d 2 are similar as well. 1
2
Proposition 11 For all ; 2 S ! X and s , s 2 S , jd (s ; s ) ? d (s ; s )j 2 d ! ( ; ): 1
1
1
2
2
1
2
1
S
2
X
1
2
2
PROOF. jd (s ; s ) ? d (s ; s )j = jdX ( (s ); (s )) ? dX ( (s ); (s ))j dX ( (s ); (s )) + dX ( (s ); (s )) [triangle inequality] 2 dX !S ( ; ): 1
1
2
1
1
1
1
2
1
1
2
2
1
1
2
2
1
1
2
2
2
2
2
2
2 If we have a sequence of functions ( n)n which converges to , then, by the above proposition, the kernels d n converge to d . This idea is the basis of an algorithm for calculating our pseudometric. More details will be provided in the concluding section. In order to exploit a pseudometric kernel to provide the set S of states of a probabilistic transition system with a pseudometric, we need to introduce the pseudometric space X and the function . The former will be (the carrier of) a terminal coalgebra and the latter will be the unique homomorphism from the probabilistic transition system viewed as a coalgebra to the terminal coalgebra. The details will be provided in the following sections.
4 A Pseudometric on Borel Probability Measures The set of Borel probability measures on a space can be turned into a pseudometric space in several ways (see, for example, Rachev's book [29]). In this section, we introduce a pseudometric on Borel probability measures which gives rise to meaningful distances on probabilistic transition systems. This (pseudo)metric is known as the Hutchinson metric [18], the Kantorovich metric [20] and also as the Wasserstein metric. 9
Let X be a (1-bounded) pseudometric space. We denote the set of Borel probability measures on X by M (X ).
De nition 12 The distance function d
de ned by
dM
M (X )
: M (X ) M (X ) ! [0; 1] is
8Z 9 Z < = ( ; ) = sup : f d ? f d j f 2 X !1 [0; 1] ; :
(X )
1
2
1
2
X
X
Before presenting an example, let us rst check that this distance function is indeed a pseudometric.
Proposition 13 The distance function d
M (X )
is a pseudometric.
PROOF. For all nonexpansive functions f : X ! [0; 1], Z
Z
Z
X
X
X
0 = 0 d f d 1 d = 1: Hence, dM
(X )
( ; ) 2 [0; 1].
Obviously, dM
1
(X )
2
(; ) = 0.
To prove symmetry, it suces to observe that for each nonexpansive function f : X ! [0; 1], the function 1 ? f : X ! [0; 1] is nonexpansive as well, and
Z
Z
(1 ? f ) d = 1 ? f d:
X
X
Since for each nonexpansive function f : X ! [0; 1],
Z
Z f d ? f d X 0Z X Z 1 0Z 1 Z = @ f d ? f d A + @ f d ? f d A 1
3
1
X
2
2
X
X
dM X ( ; ) + dM X ( ; ); (
)
1
2
the distance function dM
(
(X )
)
2
3
X
3
satis es the triangle inequality. 2 10
Example 14 Let the set fx ; x g be endowed with the discrete metric, that is, 0
1
all distances are either 0 or 1. Let be the discrete Borel probability measure determined by
(fx g) = + (fx g) = ? 0
1 2
1
1 2
The measures 0 and have distance . This is witnessed by the function mapping x0 to 0 and x1 to 1. Let the set [0; 1] be endowed with the Euclidean metric. For each 2 [0; 1], consider the Borel probability measure determined by
([x` ; xr ]) =
Z
g dx
[x` ;xr ]
where the function g is de ned by
8 > < 4x ? + 1 if x 2 [0; ] g (x) = > : ?4x + 3 + 1 if x 2 [ ; 1] 1 2
1 2
The measures 0 and 1 are f de ned by
1 24
apart. In this case, a witness is the function
8 > < x if x 2 [0; ] f (x) = > : ? x if x 2 [ ; 1] 1 2
1 2
1 2
Next, we note that M preserves equivalence.
Proposition 15 X is a metric space if and only if M (X ) is a metric space. PROOF. See, for example, Edgar's textbook [13, Proposition 2.5.14]. 2 As we will see in the proof of Theorem 31, we exploit the above result to apply the pseudometric terminal coalgebra theorem. In the rest of this paper, we focus on Borel probability measures which are completely determined by their values for the compact subsets of the space X. 11
De nition 16 A Borel probability measure on X is tight if for all > 0 there exists a compact subset K of X such that (X n K ) < .
Under quite mild conditions on the space, for example, completeness and separability, every measure is tight (see, for example, Parthasarathy's textbook [28, Theorem II.3.2]). Discrete Borel probability measures are tight. All measures presented in Example 14 are tight. We denote the set of tight Borel probability measures on X by Mt (X ). We are interested in these tight measures because of the following
Theorem 17 (1) X is complete if and only if Mt (X ) is complete. (2) X is compact if and only if Mt (X ) is compact.
PROOF. See, for example, the texts of Edgar [13, Theorem 2.5.25] and
Barnsley [5, Theorem 9.5.1]. 2
As we will see in Section 7, the fact that Mt preserves completeness will allow us to apply the pseudometric terminal coalgebra theorem (see the proof of Theorem 31). The preservation of compactness will also be exploited in Section 7.
5 The Functor M
t
We extend Mt to a functor on the category PMet of 1-bounded pseudometric spaces and nonexpansive functions. Furthermore, we show that the functor is locally nonexpansive and preserves isometric embeddings. 1
Let X and Y be pseudometric spaces. Let f : X ! Y be a nonexpansive function. To extend Mt to a functor we have to de ne a nonexpansive function Mt (f ) from tight measures on X to tight measures on Y .
De nition 18 The function M (f ) : M (X ) ! M (Y ) is de ned by M (f )() = f ? : t
t
t
1
Next, we prove that the measure Mt (f )() is tight and that the function Mt (f ) is nonexpansive.
Proposition 19 The measure M (f )() is tight. t
12
PROOF. Let > 0. Since is tight, there exists a compact subset K of X such that (X n K ) < . Because f is nonexpansive, f (K ) is a compact subset of Y . Since f ? (Y n f (K )) is a subset of X n K , we can conclude that ( f ? ) (Y n f (K )) < . Hence, f ? is tight. 2
1
1
1
Proposition 20 The function M (f ) is nonexpansive. t
PROOF. For all , 2 M (X ), 1
t
2
dMt Y (8Mt (f )( ); Mt (f )( )) 9 Z > > < dX (v; w) if v 2 X and w 2 X dX Y (v; w) = > dY (v; w) if v 2 Y and w 2 Y > > :1 otherwise. +
Mt is the functor introduced in Section 5. Act ? is the power functor. For an object X in PMet , Act X is the 1
Act -indexed product of copies of X equipped with the supremum metric.
The functor P is de ned by
R = 1+c? Q = Mt R P = (Act ?) Q 17
A P -coalgebra consists of a pseudometric space S together with a nonexpansive function t : S ! P (S ). The space S corresponds to the set of states of the probabilistic transition system. Given a state s and an action a, the tight Borel probability measure ts;a on R (S ) captures the reaction on action a of the system in state s. We use R (S ) to represent subprobabilities on S . The probability of refusal on action a in state s is given by ts;a (1). The role of c ? will be discussed later.
Proposition 29 Every discrete probabilistic transition system can be represented by a P -coalgebra. PROOF. We endow the set of states S of the system with the discrete metric. Consequently, every subset of the pseudometric space R (S ) is a Borel set. For every state s and action a, the Borel probability measure ts;a is the discrete Borel probability measure determined by
ts;a (1) = probability of refusal of action a in state s ts;a (fs0g) = probability of making an a-transition from state s to state s0 Obviously, the measure ts;a is tight. Because S is endowed with the discrete metric, the function t from S to P (S ) is nonexpansive. 2
Example 30 The continuous probabilistic transition system of Example 27 can be viewed as a P -coalgebra by endowing its state space with the Euclidean metric.
A continuous probabilistic transition system can be viewed as a P -coalgebra if its set S of states can be endowed with a pseudometric dS such that
the Borel -algebra induced by the pseudometric dS coincides with the algebra of the system,
for all states s and actions a, the system's subprobability measure ts;a is tight, and
the system's transition function is nonexpansive. We refer the reader forward to the conclusion for further discussion of these restrictions. Note that each P -coalgebra can be interpreted as a continuous probabilistic transition system, since nonexpansive functions are measurable. 18
7 A Pseudometric on Probabilistic Transition Systems We present our pseudometric on probabilistic transition systems. Furthermore, we show that states have distance 0 if and only if they are probabilistic bisimilar. Exploiting the pseudometric terminal coalgebra theorem, we prove that there exists a terminal P -coalgebra.
Theorem 31 There exists a terminal P -coalgebra h x (P ); ii. PROOF. According to America and Rutten's [2, Theorem 5.4], the functors 1 and + are locally nonexpansive and the scaling functor c is locally contractive. As we have seen in Proposition 22, the functor Mt is locally nonexpansive. As a consequence, the functor P is locally contractive. According to Proposition 15 and Theorem 17, the functor Mt , and hence the functor P , preserves equivalence and completeness. Therefore, we can conclude from Theorem 8 that there exists a terminal P -coalgebra h x (P ); ii. 2
The carrier of the terminal P -coalgebra is a compact metric space. We will exploit this property in Section 9.
Proposition 32 x (P ) is a compact metric space. PROOF. By Proposition 15 and Theorem 17, the functor M , and hence the t
functor P , preserves equivalence and compactness. Hence, we can conclude from Theorem 9 that the metric space x (P ) is compact. 2
As a consequence of Theorem 31, from each P -coalgebra hS; ti|a P -coalgebra represents a probabilistic transition system as we have seen in the previous section|there exists a unique P -homomorphism to the terminal P coalgebra.
S
/
x (P )
t
i
P (S )
P ( x (P ))
P ( )
/
The pseudometric kernel induced by is a pseudometric on the set underlying the carrier S of the P -coalgebra, that is, on the set of states of the probabilistic 19
transition system. Instead of d we will often write dC to stress its coalgebraic nature. Since the identity map on x (P ) is the unique P -homomorphism from the terminal P -coalgebra to itself, we can conclude that the coalgebraic pseudometric dC on the set underlying the carrier of the terminal P -coalgebra coincides with the metric d x P on the carrier of the terminal P -coalgebra. (
)
To compute some coalgebraic distances, we present a characterization of the pseudometric on Q (S ).
Proposition 33 For all , 2 Q (S ), 1
2
8Z 9 Z < = ( ; ) = sup : fd ? fd j f 2 c S !1 [0; 1] ; + ( (1) ? (1)):
dQ S
1
( )
2
1
S
1
2
2
S
PROOF. dQ S (8 ; ) 9 > > Z < Z = = sup > fd ? fd j f 2 R (S ) !1 [0; 1] > :R S ; R S 80 9 1 0 1 Z Z < = = sup : @r (1) + fd A ? @r (1) + fd A j r 2 [0; 1] ^ f 2 c S !1 [0; 1] ; S S 8 0Z 9 1 Z < = = sup : @ fd ? fd A + r ( (1) ? (1)) j r 2 [0; 1] ^ f 2 c S !1 [0; 1] ; S 8Z S 9 Z < = = sup : fd ? fd j f 2 c S !1 [0; 1] ; + ( (1) ? (1)): ( )
1
2
1
2
( )
( )
1
1
1
2
2
1
1
2
1
2
S
2
2
S
2
Example 34 Consider the discrete probabilistic transition system introduced
in Example 24. Let be the unique P -homomorphism from the P -coalgebra representing this system to the terminal P -coalgebra. Then
dC (s ; s ) = d x P ( (s ); (s )) = dP x P (i ( (s )); i ( (s ))) [i is an isometry] = dP x P (P ( ) (t (s )); P ( ) (t (s ))) [ is a P -homomorphism] 2
3
(
)
2
(
(
))
(
(
))
3
2
3
2
3
20
8 9 > > Z < Z = = sup > (f ) dts2;a ? (f ) dts3;a j f 2 c x (P ) !1 [0; 1] > : x P ; x P +(tsn2;a (1) ? ts3;a (1)) [Proposition 33] o = sup f ( (s )) j f 2 c x (P ) !1 [0; 1] + (0 ? 1) = 1: (
)
(
)
3
The other distances can be computed similarly and are collected in the following table.
s c
s
0
s s s s s
s
1
1
s s 2
3
2
0
c2 +2c
1 1
s
0
4
c2 +2c 4
+
2
c
3
1
+ c2 c + ((1 + )c ? c) c c2 +2c 4
?
c2 2c
c2 +2
2
4
2
2
c
1
c
c
2
2
1
+ c 1 1
The distance between states is a trade-o between the depth of observations needed to distinguish the states and the amount each observation di erentiates the states. The relative weight given to these two factors is determined by c lying between 0 and 1: the smaller the value of c the greater the discount on observations made at greater depth. This is re ected by the fact that dC (s ; s ) = c dC (s ; s ) in the above example. 0
1
0
1
Example 35 For the continuous probabilistic transition system of Example 27 we have that dC (0; 1) = 24c .
We conclude this section by showing that our pseudometric contains probabilistic bisimilarity.
Proposition 36 Let hS; ti be a P -coalgebra representing a probabilistic tran-
sition system. Let S be an analytic space. States have distance 0 if and only if they probabilistic bisimilar.
PROOF. For all s , s 2 S , 1
2
dC (s ; s ) = 0 i dL (s ; s ) = 0 [Theorem 46] i s and s are probabilistic bisimilar [see [10, Corollary 6.1.6 and Theorem 6.1.10]] 1
2
1
1
2
2
2 21
8 A Real-Valued Modal Logic We present a real-valued modal logic. This logic is closely related to the probabilistic modal logic of Larsen and Skou [24] and to a real-valued modal logic introduced by Desharnais, Gupta, Jagadeesan and Panangaden [11]. Along the lines of the latter paper, we de ne a pseudometric in terms of the logic. In the next section, we show that this pseudometric is closely related to the one we introduced in Section 7. Desharnais et al. de ned a pseudometric in terms of a real-valued modal logic. Their work builds on ideas of Kozen [21] to generalize logic to handle probabilistic phenomena. A minor variation on their logic is introduced in the following de nition.
De nition 37 The logic L is de ned by ' ::= 1 j hai ' j min ('; ') j 1 ? ' j ' ? q where a is an action and q is a rational in [0; 1].
Informally, there is the following correspondence between formulae in L and formulae in the probabilistic modal logic of Larsen and Skou. True is represented by 1, conjunction is represented by min, negation by 1 ? , and the modal connective haiq decomposes as hai and ? q. In analogy to one of De Morgan's laws, max can be expressed in the logic in terms of min and 1 ? as follows: max ('; ) = 1 ? min (1 ? '; 1 ? ): Given a probabilistic transition system represented by the P -coalgebra hS; ti, each formula ' can be interpreted as a function 'hS;ti from S to [0; 1] as follows.
De nition 38 For each ' 2 L, the function 'h i : S ! [0; 1] is de ned by S;t
1hS;ti (s) = 1 R (hai ')hS;ti (s) = c S 'hS;ti dts;a (min ('; ))hS;ti (s) = min ('hS;ti (s); hS;ti (s)) (1 ? ')hS;ti (s) = 1 ? 'hS;ti (s) (' ? q)hS;ti (s) = 'hS;ti (s) ? q 22
where
8 > < r ? q if r q r? q = > : 0 otherwise.
Next, we verify that for each formula ', the function 'hS;ti is c-contractive and hence measurable.
Proposition 39 For all ' 2 L, the function 'h i is c-contractive. S;t
PROOF. By structural induction on '. We only consider the most interesting case.
(hai ')hS;ti (s ) ? (hai ')hS;ti (s ) Z Z = c 'hS;ti dts1;a ? c 'hS;ti dts2;a S S Z Z = c 'hS;ti dts1;a ? 'hS;ti dts2;a S S 8Z 9 Z Z Z < = = c max : 'hS;ti dts1;a ? 'hS;ti dts2;a; (1 ? ')hS;ti dts1;a ? (1 ? ')hS;ti dts2;a; S S S S c dQ S (ts1;a ; ts2;a) [Proposition 33 and 'hS;ti and (1 ? ')hS;ti are c-contractive by induction] c dP S (ts1 ; ts2 ) c dS (s ; s ) [t is nonexpansive] 1
2
( )
( )
1
2
2 The logic L induces a pseudometric as follows.
De nition 40 The distance function d : S S ! [0; 1] is de ned by d (s ; s ) = sup 'h i (s ) ? 'h i (s ): L
L
1
2
2L
'
S;t
1
S;t
2
Clearly, the above introduced distance function is a pseudometric. Our logic di ers from the one presented by Desharnais et al. [11]. Instead of ' ? q they write b'cq . Furthermore, they introduce d'eq . In the presence of 23
negation, d'eq is redundant as it is equivalent to min ('; 1 ? b1cq ). Finally, they introduce a countable supremum over formulae. The logic considered by Desharnais in [10] lacks negation, but does include d'eq and max. The presence of negation in our logic has an impact on the distances as is shown in
Example 41 Consider the following probabilistic transition system. sQ
0
a; 12
a;1
s
3
|| || || | |~
s
1B
B B a; 1 BB 2 BB B
a; 21
sQ
s
4
5
|| || || | ~|
sQ
2
a; 12
a;1
The system in state s0 terminates with probability 0, in state s1 with probability 12 and in state s2 with probability 1. The expected number of transitions to termination starting in state s0 , s1 and s2 is 1, 1, and 2, respectively. Based on these kind of observations, one may infer that state s0 behaves more like state s1 than2 state s2 . This is re ected by the pseudometric dL. The states s0 and s1 are c2 apart, witnessed by hai hai 1. The states s0 and s2 are at distance c2 which is witnessed by the formulae 'n de ned by 2?c
8 > 0 and f 2 hS; d i ! [0; 1]. It suces to show that there exists a formula ' in L such that f and 'h i are at most apart. Since for all s , s 2 S , L
1
S;t
1
2
dL (s ; s ) = sup 'hS;ti (s ) ? 'hS;ti (s ) '2L c dS (s ; s ) [Proposition 39] 1
2
1
1
2
2
and the space hS; dS i is compact, we can conclude that hS; dLi is a compact pseudometric space. According to Proposition 42, the set (2) is a subset of hS; dLi !1 [0; 1]. Obviously, (2) is closed under min and max. Let s , s 2 S . Hence, according to Ash's [3, Lemma A.7.2], it suces to show that there exists a formula ' in L such that f (si) and 'hS;ti (si) are at most apart. 1
2
Without loss of generality, assume that f (s ) f (s ). Since 1
2
= f (s ) ? f (s ) dL (s ; s ) [f is nonexpansive] = sup 'hS;ti (s ) ? 'hS;ti (s ) 1
2
1
2L
'
2
1
2
there exists a formula ' such that ? 'hS;ti (s ) ? 'hS;ti (s ). Let p, q and r be rationals in [0; 1] such that 1
25
2
p 2 ['hS;ti (s ) ? ; 'hS;ti (s )] q 2 [ ? ; ] r 2 [f (s ); f (s ) + ] 2
2
2
2
We leave it to the reader to verify that the formula 1 ? ((1 ? min (' ? p; 1 ? (1 ? q))) ? r) has the desired property. 2 Note that 1, min, max, 1 ? and ? q all play a role in the above proof. We will exploit this result in the next section to relate the coalgebraic and logical distances. Interpreting a formula with respect to di erent P -coalgebras may give rise to di erent functions. These functions are related as follows.
Proposition 44 Let be a P -homomorphism from a P -coalgebra hS; ti to a P -coalgebra hS 0; t0i. Then for all formulae ', 'h 0 0 i = 'h i: S ;t
S;t
PROOF. By structural induction on '. We only present the most interesting case. For all s 2 S , (hai 'Z)hS0;t0i ( (s)) = c 'hS0 ;t0i dt0 s ;a =c =c
S0
Z
S0
Z
S0
Z
( )
'hS0 ;t0i d(P ( )(t))s;a [t0 = P ( ) t] 'hS0 ;t0i d(ts;a ? ) 1
= c ('hS0;t0 i ) dts;a =c
ZS
'hS;ti dts;a [induction]
S
= (hai ')hS;ti (s):
2 26
Also this result will be used in the next section where we relate the coalgebraic and logical pseudometric.
9 Relating the Coalgebraic and Logical Distances For a large class of probabilistic transition systems we have introduced a coalgebraic distance function dC and a logical distance function dL. In this section we relate the two pseudometrics. Before considering the general case, we rst relate the two distance functions on the set underlying the carrier of the terminal P -coalgebra. Recall that the coalgebraic pseudometric dC on the set underlying the carrier of the terminal P -coalgebra coincides with the metric d x P on the carrier of the terminal P -coalgebra. (
)
Proposition 45 For all x , x 2 x (P ), 1
2
dL (x ; x ) = d (x ; x ): C c 1
2
1
2
PROOF. Consider the function which maps each x 2 x (P ) to itself. For all x , x 2 x (P ), 1
2
dL ( (x ); (x )) c = dL (x ; x ) c sup'2L 'h x P ;ii (x ) ? 'h x P ;ii (x ) = c dC (x ; x ) ['h x P ;ii is c-contractive by Proposition 39] 1
1
2
2
(
1
1
)
2
(
(
)
2
)
Consequently, is a nonexpansive function from the space h x (P ); dC i to the space hh x (P ); dcL i. Next, we introduce a structure t such that hh x (P ); dcL i; ti is a P -coalgebra. Because is nonexpansive, each Borel set of R h x (P ); dcL i is also a Borel set of R h x (P ); dC i. Therefore, we can de ne the function t for x 2 x (P ), a 2 Act and Borel set B of R h x (P ); dcL i by
tx;a (B ) = ix;a (B ): Since the function is nonexpansive and the measure ix;a is tight, we can conclude that the measure tx;a is tight as well (cf. Proposition 19). 27
To conclude that t is the structure of a P -coalgebra with carrier h x (P ); dcL i, we have left to show that t is nonexpansive. Let x , x 2 x (P ). Then 1
dP h x
(P );
dL c i
(tx1 ; tx2 ) = sup dQ h x a2Act
(P );
dL c i
2
(tx1 ;a; tx2;a ):
Let a 2 Act. Without loss of generality, assume that tx1;a (1) tx2;a (1). Then, dL (tx1 ;a ; tx2 ;a ) 8ci 9 > > Z < Z = = sup > f dtx1 ;a ? f dtx2;a j f 2 c h x (P ); dcL i !1 [0; 1] > [Proposition 33] : x P ; x P Z Z sup 'h x P ;ii dtx1;a ? 'h x P ;ii dtx2 ;a [Proposition 32 and 43]
dQ h x
(P );
(
)
2L x (P )
'
=
(
(
)
sup'2L (hai ')h x
(
x (P )
)
i (x1 ) ? (hai ')h x (P );ii (x2 )
(P );i
dL (xc ; x ) : 1
)
c
2
From the de nition of t and we can easily derive that is a P -homomorphism from the P -coalgebra hh x (P ); dC i; ii to the P -coalgebra hh x (P ); dcL i; ti. We denote the unique P -homomorphism from the P -coalgebra hh x (P ); dcL i; ti to the terminal P -coalgebra hh x (P ); dC i; ii by .
h x (P ); dcL i l t
P h x (P ); dcL il
h x (P ); dC i +
P ( )
i
P h x (P ); dC i +
P ()
Obviously, the identity map on x (P ) is the unique P -homomorphism from the terminal P -coalgebra hh x (P ); dC i; ii to itself. Since is also such a P -homomorphism, we can conclude that equals the identity map on x (P ). Therefore, both and are isometries. This observation completes the proof. 2 The above proof makes use of Proposition 43. In the proof of that result 1, min, max, 1 ? and ? q all play a role. In the above proof, also the modality hai is used. 28
Next, we consider the general case. Consider a probabilistic transition system represented by the P -coalgebra hS; ti. We relate its coalgebraic pseudometric dC and its logical pseudometric dL in
Theorem 46 For all s , s 2 S , 1
2
dL (s ; s ) = d (s ; s ): C c 1
2
1
2
PROOF. We denote the unique P -homomorphism from the P -coalgebra hS; ti to the terminal P -coalgebra h x (P ); ii by . For all s , s 2 S , 1
dL (s ; s ) c d L ( (s ); (s )) [Proposition 44] = c = dC ( (s ); (s )) [Proposition 45] = dC (s ; s ): 1
2
1
1
2
1
2
2
2
(3) (4) (5) (6)
Note that (3) and (4) refer to di erent logical pseudometrics: the one on S and the one on x (P ), respectively. Also notice that (5) and (6) refer to di erent coalgebraic pseudometrics: the one on x (P ) and the one on S , respectively. 2 In [9], we studied a minor variation on the functor P . In that paper, we considered the functor
P 0 = c Act Mt (1 + ?): This functor is also locally contractive and preserves equivalence and completeness and, therefore, has a terminal coalgebra. The carriers of the terminal P -coalgebra and the terminal P 0-coalgebra are related as follows.
Proposition 47 x (P 0) = c x (P ). PROOF. According to Lemma 4, there exists an isometry i from x (P ) to P ( x (P )). Clearly, i is also an isometry from c x (P ) to cP ( x (P )) = P 0 (c x (P )). Using Lemma 4 again, we can conclude that x (P 0) = c x (P ). 2 Consequently, the coalgebraic pseudometrics induced by the functor P 0 coincide with the logical pseudometrics. 29
10 Conclusion 10.1 Related Work
As we have already seen in Section 8 and 9, our coalgebraic pseudometric is closely related to the logical pseudometric of Desharnais et al. [10,11]. In [11], they also introduce a probabilistic process algebra. A number of combinators of the process algebra, including probabilistic choice, are shown to be nonexpansive. This is a quantitative analogue of probabilistic bisimulation being a congruence. It allows for compositional veri cation of probabilistic transition systems. Since our coalgebraic pseudometric is related to their logical pseudometric, we can conclude that those combinators are also nonexpansive with respect to our pseudometric. Furthermore, Desharnais et al. present a decision procedure for their pseudometric. That is, they provide an algorithm to approximate the logical distances to a prescribed degree of accuracy. The algorithm involves the generation of a representative set of formulae of their real-valued modal logic. They only consider formulae with a restricted number of nested occurrences of the modal connective. Their algorithm approximates the distances in exponential time. In [8], we present an algorithm to approximate our coalgebraic distances. The problem of approximating such distances can be reduced to a particular linear programming problem: the transshipment problem. Since the latter problem can be solved in polynomial time, we obtain a polynomial time decision procedure for our distances. We see this practical algorithm as one of the advantages of our coalgebraic approach over the logical approach of Desharnais et al. Another advantage of our approach is that we work within a uniform framework, the theory of coalgebras. We do not know whether there exists a terminal coalgebra of our functor for c equals 1, and hence we cannot use our framework to de ne a pseudometric when c equals 1. However, the logical approach of Desharnais et al. also works in that case. Furthermore, Desharnais et al. consider a larger class of continuous probabilistic transition systems than we do in this paper. However, we are con dent that we can extend our results as we will discuss below. In conclusion, we believe that both approaches have their merits and demerits. The results in Section 9 are very valuable as they allows us to transfer results from the one setting to the other. As far as we know, [14] by Giacalone et al. is the rst paper to advocate the use of pseudometric spaces to provide a robust and quantitative notion of behavioural equivalence. They stress the importance of combinators being nonexpansive with respect to the pseudometric, making compositional veri cation possible. The class of discrete probabilistic transition systems they consider is rather restricted. A decade later, we are able to deal with all discrete probabilistic transition systems and a large class of continuous probabilistic 30
transition systems. De Vink and Rutten [33] show that discrete probabilistic transition systems and some continuous probabilistic transition systems can be viewed as coalgebras. Their main contribution is the proof that the kernel of the homomorphism from a coalgebra, representing a probabilistic transition system, to the terminal coalgebra coincides with probabilistic bisimilarity. They only exploit metrics to represent continuous systems as coalgebras. Their metric on the terminal coalgebra only provides qualitative information. For example, in De Vink and Rutten's setting the states s and s of the system presented in the introduction are c apart if di ers from 0. More generally, the distance between two states in their setting is cn where n is the depth of a probabilistic bisimulation between them. De Vink and Rutten consider the endofunctor 0
0
Act (1 + Mc (c ?))
on the category of complete ultrametric spaces and nonexpansive functions. Mc denotes the Borel probability measures with compact support. The main di erences between our functor and their functor are the following. First of all, they consider a distance function on Borel probability measures [33, De nition 5.3] di erent from the one presented in De nition 12. Their distance function only captures qualitative information as the above example illustrates. Secondly, they consider the category of complete ultrametric spaces and nonexpansive functions whereas we consider the considerably larger category pseudometric spaces and nonexpansive functions. This allows us to captures many more interesting continuous probabilistic transition systems as coalgebras, including systems where the state space is the real interval [0; 1] endowed with the Euclidean metric. Furthermore, they consider Borel probability measures with compact support whereas we consider the more general tight Borel probability measures. Again this allows us to represent more systems as coalgebras. Finally, their model only allows states to refuse transitions with probability 0 or 1. In conclusion, our functor allows to model many more interesting continuous systems, and all the results for their functor in [33, Section 5] can be generalized to our setting. 3
Baier and Kwiatkowska [4] study a functor which is closely related to the one of De Vink and Rutten. Our work can be compared to theirs in the same way it is compared to the work of De Vink and Rutten in the paragraph above. In his thesis [17], Den Hartog exploits ultrametric spaces very similar to the terminal coalgebra of De Vink and Rutten. The metric structure is only used to model in nite behaviour. As a consequence, qualitative information suces. The proof of [33, Theorem 5.8] is incomplete. We also have no proof for this result in our setting.
3
31
We believe that metrics closely related to the one we present in this paper may be used in his setting as well, possibly providing additional quantitative information about his models. Kwiatkowska and Norman [22,23,27] present a number of closely related metrics. Like Den Hartog, they use their metric as a means to model recursion. However, their metric is not an ultrametric and contains quantitative information. Let us compare the metric introduced by Norman in [27, Section 6.1] with our pseudometric. Consider the following probabilistic transition system.
s
0
1 2
s
| || || | ~| 3B BB BB B 1 B
s |
1B
1
BB 1 BB2 BB
1 2
s B
B
4
s
5
1
/
s
| || || | |~
2
1 2
1
6
s |
z
Clearly, the states s and s are not probabilistic bisimilar. In Norman's setting the states have distance 0. In our pseudometric, states only have distance 0 if they are probabilistic bisimilar. In our setting the states are c2 c apart. This example shows that his distance function gives rise to a topology di erent from ours. The main di erences between his and our approach are the following. First of all, he uses a linear-time model whereas we consider a branching-time model. Secondly, he only handles discrete systems whereas we also consider continuous ones. Finally, we use the usual categorical machinery and various standard constructions whereas his de nitions are more ad-hoc. We believe however that his metric can also be characterized by means of a terminal coalgebra. 0
1
+2 4
Results similar to the ones in this paper have been presented by the second author in his thesis [34, Chapter 4] in the setting of bimodules and generalized metric spaces. As we have seen, the functor Mt , and hence the functor P , preserves isometric embeddings. As a consequence, the coalgebraic distance of states s and s can be characterized as the smallest R (s ; s ) where R is a bimodule satisfying certain conditions (see [34, Theorem 4.5.12] for the details). This is the quantitative analogue of the characterization of probabilistic bisimilarity as the largest probabilistic bisimulation. 1
2
1
2
10.2 Future Work
Let us isolate two distinct consequences of our use of the pseudometric presented in Section 4. First of all, we can talk about approximate equivalence 32
of states. Secondly, we can model a large class of continuous probabilistic transition systems as coalgebras. An apparent restriction with regard to the latter point is the requirement that the structure of a P -coalgebra, that is, the system's transition function, be nonexpansive. Properly speaking, continuous probabilistic transition systems as formulated in De nition 26 are coalgebras of (a variant of) the Giry monad on the category of measurable spaces and measurable functions [15]. However, we conjecture that the terminal P -coalgebra h x (P ); ii is also terminal when seen as a coalgebra of the Giry functor, and that our results can be extended to continuous probabilistic transition systems in general. In Proposition 32 we have shown that the carrier of our terminal coalgebra is compact and hence separable. Furthermore, we conjecture that the unique homomorphism from the initial algebra of a nitary version of P |this nitary version represents nite discrete probabilistic transition systems with rational probabilities|to the terminal P -coalgebra is a dense embedding. Hence, every continuous system can be approximated by a nite one (see also [12]).
Acknowledgements The authors would like to thank the Amsterdam Coordination Group, Josee Desharnais, Abbas Edalat, Joel Ouaknine, Prakash Panangaden, Jan Rutten and Erik de Vink for discussion. The rst author is thankful to Stephen Watson for the joint study of metrics on probability measures.
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[6] P. Billingsley. Probability and Measure. Series in Probability and Mathematical Statistics. Wiley, New York, third edition, 1995. [7] R. Blute, J. Desharnais, A. Edalat, and P. Panangaden. Bisimulation for labelled Markov processes. In Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science, pages 149{158, Warsaw, June/July 1997. IEEE. [8] F. van Breugel and J. Worrell. An algorithm for quantitative veri cation of probabilistic systems. In K.G. Larsen and M. Nielsen, editors, Proceedings of the 12th International Conference on Concurrency Theory, volume 2154 of Lecture Notes in Computer Science, pages 336{350, Aalborg, August 2001. SpringerVerlag. [9] F. van Breugel and J. Worrell. Towards quantitative veri cation of probabilistic systems. In F. Orejas, P.G. Spirakis, and J. van Leeuwen, editors, Proceedings of 28th International Colloquium on Automata, Languages and Programming, volume 2076 of Lecture Notes in Computer Science, pages 421{432, Crete, July 2001. Springer-Verlag. [10] J. Desharnais. Labelled Markov Processes. PhD thesis, McGill University, Montreal, November 1999. [11] J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labeled Markov systems. In J.C.M. Baeten and S. Mauw, editors, Proceedings of the 10th International Conference on Concurrency Theory, volume 1664 of Lecture Notes in Computer Science, pages 258{273, Eindhoven, August 1999. SpringerVerlag. [12] J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Approximating labeled Markov processes. In Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science, pages 95{106, Santa Barbara, June 2000. IEEE. [13] G.A. Edgar. Integral, Probability, and Fractal Measures. Springer-Verlag, Berlin, 1998. [14] A. Giacalone, C.-C. Jou, and S.A. Smolka. Algebraic reasoning for probabilistic concurrent systems. In Proceedings of the IFIP WG 2.2/2.3 Working Conference on Programming Concepts and Methods, pages 443{458, Sea of Gallilee, April 1990. North-Holland. [15] M. Giry. A categorical approach to probability theory. In B. Banaschewski, editor, Proceedings of the International Conference on Categorical Aspects of Topology and Analysis, volume 915 of Lecture Notes in Mathematics, pages 68{ 85, Ottawa, August 1981. Springer-Verlag. [16] R.J. van Glabbeek, S.A. Smolka, and B. Ste en. Reactive, generative and strati ed models of probabilistic processes. Information and Computation, 121(1):59{80, August 1995. [17] J.I. den Hartog. Extending Semantical Models with Probabilistic Notions. PhD thesis, Free University, Amsterdam. In preparation.
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36
;
York University, Department of Computer Science 4700 Keele Street, Toronto, M3J 1P3, Canada b Tulane University, Department of Mathematics 6823 St Charles Avenue, New Orleans LA 70118, USA a
Abstract Boolean-valued logics and associated discrete notions of behavioural equivalence sit uneasily with semantic models featuring quantitative data, like probabilistic transition systems. In this paper we present a pseudometric on a class of probabilistic transition systems yielding a quantitative notion of behavioural equivalence. The pseudometric is de ned via the terminal coalgebra of a functor based on a metric on the space of Borel probability measures on a metric space. States of a probabilistic transition system have distance 0 if and only if they are probabilistic bisimilar. We also characterize our distance function in terms of a real-valued modal logic. Key words: probabilistic transition system, pseudometric, probabilistic bisimilarity, terminal coalgebra, real-valued modal logic
1 Introduction The majority of veri cation methods for concurrent systems only produce qualitative information. Questions like \Does the system satisfy its speci cation?" and \Do the systems behave the same?" are answered \Yes" or \No". Giacalone, Jou and Smolka [14], Huth and Kwiatkowska [19] and Desharnais, Gupta, Jagadeesan and Panangaden [12] have pointed out that such discrete Boolean-valued reasoning sits uneasily with semantic models featuring quantitative data, like probabilistic transition systems. 1 2
Supported by Natural Sciences and Engineering Research Council of Canada. Supported by the US Oce of Naval Research.
Preprint submitted to Elsevier Science
26 October 2001
Real numbers are ideal entities. Computers, and humans for that matter, only deal with approximations. If we only have an approximate description of a system, it makes no sense to ask if any two states behave exactly the same. Even if we have a precise description of a system, we may still want to express the idea that a state satis es a formula with high probability or that two states exhibit almost the same behaviour. Furthermore, the problem of automatically checking if a state satis es a formula or verifying that two states behave exactly the same will prove intractable owing to the impossibility of exact computation of reals. The basis of any computational treatment of the reals, and hence of automatic veri cation of probabilistic systems, has to involve approximation. The probabilistic modal logic of Larsen and Skou [24] adds probability thresholds to traditional modal logic. In this logic one has a formula haiq ' which is satis ed if the sum of the probabilities of transitions labelled with a to states satisfying ' exceeds q 2 [0; 1]. We might hope to be able to check that an in nite state (or even continuous) probabilistic transition system satis es such a formula by model checking some nite approximant of the system. However, as we illustrate below, the truth of a given formula is not continuous with respect to the obvious notion of approximation on transition systems. This phenomena is discussed extensively by Desharnais et al. in [12]. In a similar vein, Giacalone et al. [14] and Desharnais et al. [11] have criticized all-or-nothing notions of behavioural equivalence for probabilistic transition systems such as Larsen and Skou's probabilistic bisimulation [24]. Recall that a probabilistic bisimulation is an equivalence relation on the state space of a transition system such that related states have exactly the same probability of making a transition into any equivalence class. Thus, for instance, the states s and s of the probabilistic transition system 0
0
s
> 1 @` @ @@ ~~ ~~ ~ ~~ ~ a;1 0@ @@ @@ @ ~~ a; 21 @ @ ~~ a; 21
s
s
1
+ @ a; @@ 2 @
~~ ~~ 1 a; 2
s
0
?
2
are only probabilistic bisimilar if is 0. However, the two states behave almost the same for very small di erent from 0. In the words of [11], behavioural equivalences like probabilistic bisimilarity are not robust, since they are too sensitive to the exact probabilities of the various transitions. The lack of robustness of these logics and behavioural equivalences has a number of implications. First of all, the logics and behavioural equivalences are in general not suited for reasoning about in nite state systems in terms of 2
their nite approximants. For example, the in nite state system 1 2
s
1
s |
s
0
1 4
1 8
2
1
s
1 16
s
s : : :
4
7
1
1
s : : :
s
3
5
8
1
1
s : : :
s
6
9
1
s ::: 10
and its nite approximant 1 2
s
1
s
1 4
s |
s
2
1
4
1
s
0
1 4
s
3
5
1
s
6
are not probabilistic bisimilar. The systems also do not satisfy the same formulae. For example, assuming that all transitions are labelled with a, the formula hai 161 hai 12 hai 12 hai 12 true is satis ed by the in nite state system but not by its nite approximant. Secondly, the probabilities of the transitions are often approximations themselves. For example, may be approximated by 0.32. Since the systems 1
1 a;
s s
1
| || || | ~| |
s
0B
B B a; ?1 BB BB B
s
0:32 |
s
2
1
b;1
s
3
|| |~ |
| ||
s
0B
BB B B 0:68 BB B
s
2
1
3
are neither probabilistic bisimilar nor satisfy the same formulae, one cannot use the latter to reason about the former when using probabilistic bisimilarity or the probabilistic modal logic of Larsen and Skou. Finally, any inexactness in calculations to decide whether a formula is satis ed in a state or whether a relation on states is a probabilistic bisimulation may result in an incorrect conclusion. To the best of our knowledge, the earliest attempt to address some of the 3
problems with probabilistic bisimulation as outlined above is the paper of Giacalone et al. [14]. They de ne a pseudometric on the states of a (restricted type of) probabilistic transition system, yielding a smooth, quantitative notion of behavioural equivalence. A pseudometric di ers from an ordinary metric in that di erent elements, that is, states, can have distance 0. The distance between states, a real number between 0 and 1, can be used to express the similarity of the behaviour of those states. The smaller the distance, the more alike the behaviour is. In particular, the distance between states is 0 if they are behaviourally indistinguishable. The present paper is mostly inspired by the work of Desharnais et al. [11] and De Vink and Rutten [33]. The former introduce a pseudometric on a class of probabilistic transition systems more general than that considered by Giacalone et al. This pseudometric is de ned via a non-standard semantics for a probabilistic modal logic, where formulae get interpreted as measurable functions into the interval [0; 1] rather than as Boolean-valued functions. We shall show that what we present in this paper is essentially a coinductive account of a closely related pseudometric. We will also discuss some of the advantages conferred by such an account. De Vink and Rutten [33] showed that a restricted class of probabilistic transition systems can be viewed as coalgebras. Taking this view, one can transfer results from the general theory of coalgebras to the setting of probabilistic systems. (Many di erent kinds of transition system can be viewed as coalgebras. For an overview of the general theory of coalgebras, we refer the reader to Rutten's [30].) Using one such result De Vink and Rutten established the existence of a terminal object in their category of coalgebras. By de nition there is a unique map from an arbitrary coalgebra to the terminal coalgebra. Furthermore, De Vink and Rutten showed that the kernel of the unique map coincides with probabilistic bisimilarity. Each of these results has an analog in the non-probabilistic case. Thus the coalgebraic approach o ers a uniform conceptual framework for di erent operational models. In this paper we exploit the coalgebraic framework to de ne a notion of quantitative behavioural equivalence for probabilistic transition systems. In particular, we de ne a pseudometric on the states of a probabilistic transition system in terms of the terminal coalgebra of an endofunctor P on the category of pseudometric spaces and nonexpansive maps. The de nition of P is based on a metric on Borel probability measures. This metric is known as Hutchinson metric [18], the Kantorovich metric [20] and also as the Wasserstein metric. P -coalgebras can be seen as probabilistic transition systems with discrete or continuous state spaces. The terminal P -coalgebra provides for a notion of approximate equivalence similar to the pseudometric of Desharnais et al. mentioned above. In fact, we de ne a pseudometric on the state space of a probabilistic transition system, seen as a P -coalgebra, as the pseudometric 4
kernel of the unique map to the terminal P -coalgebra. That is, the distance between two states is the distance between their images under the unique map to the terminal coalgebra. Moreover states are at distance 0 just in case they are probabilistic bisimilar in the sense of Larsen and Skou. So far we have motivated our concern for de ning a notion of quantitative behavioural equivalence by examples featuring probabilistic transition systems with discrete state spaces. However our framework is suciently general to model probabilistic transition systems the state space of which is continuous, like [0; 1]. We refer the reader to [12] for a discussion of the importance of modelling continuous as well discrete systems. The rest of this paper is organized as follows. In Section 2 we present some minor variations on results of [1,32] which allow us to prove that a terminal P -coalgebra exists. Pseudometric kernels are introduced in Section 3. In Section 4, we present the above mentioned metric on Borel probability measures. In Section 5, we study the key ingredient of the functor P . In Section 6, we present the functor P and we show that all discrete probabilistic transition systems and a large class of continuous probabilistic transition systems can be viewed as P -coalgebras. We introduce our pseudometric in Section 7. In Section 8, we introduce a pseudometric de ned in terms of a modal logic a la Desharnais et al. The pseudometrics introduced in Section 7 and 8 are related in Section9. In the concluding section we present related and future work. The reader is assumed to be familiar with some elementary category theory, metric space theory and probability theory. For more details, we refer the reader to, for example, the texts of Mac Lane [26], Sutherland [31] and Billingley [6].
2 A Pseudometric Terminal Coalgebra Theorem In this section, we introduce coalgebras and a mild generalization of Rutten and Turi's metric terminal coalgebra theorem [32]. For more details about the theory of coalgebras we refer the reader to, for example, Rutten's [30].
De nition 1 Let C be a category. Let F : C ! C be a functor. An F -coalgebra consists of an object C in C together with an arrow f : C ! F (C ) in C .
The object C is called the carrier. The arrow f is called the structure. An F -homomorphism from an F -coalgebra hC; f i to an F -coalgebra hD; gi is an
5
arrow : C ! D in C such that F ( ) f = g .
C
/
D g
f
F (C )
F (D)
F ( )
/
The F -coalgebras and F -homomorphisms form a category. If this category has a terminal object, then this object is called the terminal F -coalgebra.
In the rest of this section, we restrict our attention to the category PMet of 1-bounded pseudometric spaces and nonexpansive functions. A pseudometric space is 1-bounded if all its distances are bounded by 1. The assumption of 1-boundedness is made once and for all here and is not always explicitly stated in our results. A function is nonexpansive if it does not increase any distances. We denote the collection of nonexpansive functions from the space X to the space Y by X !1 Y . This collection can be turned into a pseudometric space by endowing the functions with the supremum metric. 1
Let c be a constant between 0 and 1. A function is c-contractive if it decreases all distances by at least a factor c. This notion can be lifted to functors as follows.
De nition 2 A functor F : PMet ! PMet is locally c-contractive if for 1
1
all pseudometric spaces X and Y , the function
FX;Y : (X !1 Y ) ! (F (X ) !1 F (Y )) de ned by
FX;Y (f ) = F (f ) is c-contractive.
In the rest of this section, we restrict ourselves to locally contractive functors. Furthermore, we focus on functors which preserve equivalence and completeness.
De nition 3 A functor F : PMet ! PMet preserves equivalence and com1
1
pleteness if for all complete metric spaces X , F (X ) is a complete metric space. A functor which preserves equivalence and completeness can be restricted to a functor on the category of complete metric spaces and nonexpansive functions. 6
A locally contractive functor F which preserves equivalence and completeness has a xed point. That is, there exists a space, say x (F ), and an isometry from x (F ) to F ( x (F )). Recall that an isometry is a function which preserves all distances and which is onto.
Lemma 4 For each locally c-contractive functor F : PMet ! PMet which 1
1
preserves equivalence and completeness, there exists a unique complete metric space x (F ) such that there is an isometry i : x (F ) ! F ( x (F )).
PROOF. Immediate consequence of [32, Theorem 7.2]. 2 Next, we show that h x (F ); ii is a terminal F -coalgebra. For the rest of this section, we x hX; f i to be an F -coalgebra. To characterize the unique F homomorphism from the F -coalgebra hX; f i to the F -coalgebra h x (F ); ii we introduce the following function.
De nition 5 The function h
X;f
by
i
: (X !1 x (F )) ! (X !1 x (F )) is de ned
hX;f i ( ) = i?1 F ( ) f:
X
/
i?1
f
F (X )
x (O F )
F ( )
/
F ( x (F ))
Since the functor F is locally c-contractive, we have that the function hX;f i is c-contractive.
Proposition 6 The function h
X;f
i
is c-contractive.
PROOF. Similar to the proof of [32, Proposition 7.1]. 2 Because x (F ) is a complete metric space, X !1 x (F ) is a complete metric space as well. Obviously, the space X !1 x (F ) is nonempty. Since hX;f i is a contractive function from a nonempty complete metric space to itself, we can conclude from Banach's theorem that it has a unique xed point x (hX;f i ). This xed point x (hX;f i ) is the limit of a sequence ( n)n, where is an arbitrary nonexpansive function from X to x (F ) and n = ( n). 0
+1
Lemma 7 The function x (h i) is the unique F -homomorphism from the F -coalgebra hX; f i to the F -coalgebra h x (F ); ii. X;f
7
PROOF. Similar to the proof of [32, Proposition 7.1]. 2 Theorem 8 For every locally c-contractive functor F : PMet ! PMet 1
1
which preserves equivalence and completeness there exists a terminal F -coalgebra h x (F ); ii and x (F ) is a complete metric space.
PROOF. Immediate consequence of Lemma 4 and 7. 2 We will exploit both the terminal F -coalgebra h x (F ); ii and the unique F homomorphism x (hX;f i ) when de ning our pseudometric. If the functor F also preserves compactness, then we can conclude that the carrier of the terminal F -coalgebra is a compact metric space.
Theorem 9 For every locally c-contractive functor F : PMet ! PMet 1
1
which preserves equivalence and compactness there exists a terminal F -coalgebra h x (F ); ii and x (F ) is a compact metric space.
PROOF. Similar to the proof of Theorem 8 exploiting [1, Theorem 4.4] of Alesi, Baldan and Belle. 2
The only di erence between the results of [1,32] mentioned above and the results presented in this section is that former are about metric spaces whereas the latter are about pseudometric spaces.
3 Pseudometric Kernels In this section, we introduce pseudometric kernels. Our pseudometric on the states of a probabilistic transition system will be de ned as a pseudometric kernel. A function from a set S to a pseudometric space X de nes a distance function d on S . We call this distance function the pseudometric kernel induced by . The distance between s and s in S is de ned as the distance of their -images in the pseudometric space X . 1
2
De nition 10 Let : S ! X . The distance function d : S S ! [0; 1] is
de ned by
d (s ; s ) = dX ( (s ); (s )): 1
2
1
2
8
One can easily verify that the pseudometric kernel d is a pseudometric. Note that s and s have distance 0 if they are mapped by to the same element in X . For example, if is a constant function then all distances are 0. 1
2
If the functions and are similar, that is, they are close to each other with respect to the supremum metric, then the induced pseudometric kernels d 1 and d 2 are similar as well. 1
2
Proposition 11 For all ; 2 S ! X and s , s 2 S , jd (s ; s ) ? d (s ; s )j 2 d ! ( ; ): 1
1
1
2
2
1
2
1
S
2
X
1
2
2
PROOF. jd (s ; s ) ? d (s ; s )j = jdX ( (s ); (s )) ? dX ( (s ); (s ))j dX ( (s ); (s )) + dX ( (s ); (s )) [triangle inequality] 2 dX !S ( ; ): 1
1
2
1
1
1
1
2
1
1
2
2
1
1
2
2
1
1
2
2
2
2
2
2
2 If we have a sequence of functions ( n)n which converges to , then, by the above proposition, the kernels d n converge to d . This idea is the basis of an algorithm for calculating our pseudometric. More details will be provided in the concluding section. In order to exploit a pseudometric kernel to provide the set S of states of a probabilistic transition system with a pseudometric, we need to introduce the pseudometric space X and the function . The former will be (the carrier of) a terminal coalgebra and the latter will be the unique homomorphism from the probabilistic transition system viewed as a coalgebra to the terminal coalgebra. The details will be provided in the following sections.
4 A Pseudometric on Borel Probability Measures The set of Borel probability measures on a space can be turned into a pseudometric space in several ways (see, for example, Rachev's book [29]). In this section, we introduce a pseudometric on Borel probability measures which gives rise to meaningful distances on probabilistic transition systems. This (pseudo)metric is known as the Hutchinson metric [18], the Kantorovich metric [20] and also as the Wasserstein metric. 9
Let X be a (1-bounded) pseudometric space. We denote the set of Borel probability measures on X by M (X ).
De nition 12 The distance function d
de ned by
dM
M (X )
: M (X ) M (X ) ! [0; 1] is
8Z 9 Z < = ( ; ) = sup : f d ? f d j f 2 X !1 [0; 1] ; :
(X )
1
2
1
2
X
X
Before presenting an example, let us rst check that this distance function is indeed a pseudometric.
Proposition 13 The distance function d
M (X )
is a pseudometric.
PROOF. For all nonexpansive functions f : X ! [0; 1], Z
Z
Z
X
X
X
0 = 0 d f d 1 d = 1: Hence, dM
(X )
( ; ) 2 [0; 1].
Obviously, dM
1
(X )
2
(; ) = 0.
To prove symmetry, it suces to observe that for each nonexpansive function f : X ! [0; 1], the function 1 ? f : X ! [0; 1] is nonexpansive as well, and
Z
Z
(1 ? f ) d = 1 ? f d:
X
X
Since for each nonexpansive function f : X ! [0; 1],
Z
Z f d ? f d X 0Z X Z 1 0Z 1 Z = @ f d ? f d A + @ f d ? f d A 1
3
1
X
2
2
X
X
dM X ( ; ) + dM X ( ; ); (
)
1
2
the distance function dM
(
(X )
)
2
3
X
3
satis es the triangle inequality. 2 10
Example 14 Let the set fx ; x g be endowed with the discrete metric, that is, 0
1
all distances are either 0 or 1. Let be the discrete Borel probability measure determined by
(fx g) = + (fx g) = ? 0
1 2
1
1 2
The measures 0 and have distance . This is witnessed by the function mapping x0 to 0 and x1 to 1. Let the set [0; 1] be endowed with the Euclidean metric. For each 2 [0; 1], consider the Borel probability measure determined by
([x` ; xr ]) =
Z
g dx
[x` ;xr ]
where the function g is de ned by
8 > < 4x ? + 1 if x 2 [0; ] g (x) = > : ?4x + 3 + 1 if x 2 [ ; 1] 1 2
1 2
The measures 0 and 1 are f de ned by
1 24
apart. In this case, a witness is the function
8 > < x if x 2 [0; ] f (x) = > : ? x if x 2 [ ; 1] 1 2
1 2
1 2
Next, we note that M preserves equivalence.
Proposition 15 X is a metric space if and only if M (X ) is a metric space. PROOF. See, for example, Edgar's textbook [13, Proposition 2.5.14]. 2 As we will see in the proof of Theorem 31, we exploit the above result to apply the pseudometric terminal coalgebra theorem. In the rest of this paper, we focus on Borel probability measures which are completely determined by their values for the compact subsets of the space X. 11
De nition 16 A Borel probability measure on X is tight if for all > 0 there exists a compact subset K of X such that (X n K ) < .
Under quite mild conditions on the space, for example, completeness and separability, every measure is tight (see, for example, Parthasarathy's textbook [28, Theorem II.3.2]). Discrete Borel probability measures are tight. All measures presented in Example 14 are tight. We denote the set of tight Borel probability measures on X by Mt (X ). We are interested in these tight measures because of the following
Theorem 17 (1) X is complete if and only if Mt (X ) is complete. (2) X is compact if and only if Mt (X ) is compact.
PROOF. See, for example, the texts of Edgar [13, Theorem 2.5.25] and
Barnsley [5, Theorem 9.5.1]. 2
As we will see in Section 7, the fact that Mt preserves completeness will allow us to apply the pseudometric terminal coalgebra theorem (see the proof of Theorem 31). The preservation of compactness will also be exploited in Section 7.
5 The Functor M
t
We extend Mt to a functor on the category PMet of 1-bounded pseudometric spaces and nonexpansive functions. Furthermore, we show that the functor is locally nonexpansive and preserves isometric embeddings. 1
Let X and Y be pseudometric spaces. Let f : X ! Y be a nonexpansive function. To extend Mt to a functor we have to de ne a nonexpansive function Mt (f ) from tight measures on X to tight measures on Y .
De nition 18 The function M (f ) : M (X ) ! M (Y ) is de ned by M (f )() = f ? : t
t
t
1
Next, we prove that the measure Mt (f )() is tight and that the function Mt (f ) is nonexpansive.
Proposition 19 The measure M (f )() is tight. t
12
PROOF. Let > 0. Since is tight, there exists a compact subset K of X such that (X n K ) < . Because f is nonexpansive, f (K ) is a compact subset of Y . Since f ? (Y n f (K )) is a subset of X n K , we can conclude that ( f ? ) (Y n f (K )) < . Hence, f ? is tight. 2
1
1
1
Proposition 20 The function M (f ) is nonexpansive. t
PROOF. For all , 2 M (X ), 1
t
2
dMt Y (8Mt (f )( ); Mt (f )( )) 9 Z > > < dX (v; w) if v 2 X and w 2 X dX Y (v; w) = > dY (v; w) if v 2 Y and w 2 Y > > :1 otherwise. +
Mt is the functor introduced in Section 5. Act ? is the power functor. For an object X in PMet , Act X is the 1
Act -indexed product of copies of X equipped with the supremum metric.
The functor P is de ned by
R = 1+c? Q = Mt R P = (Act ?) Q 17
A P -coalgebra consists of a pseudometric space S together with a nonexpansive function t : S ! P (S ). The space S corresponds to the set of states of the probabilistic transition system. Given a state s and an action a, the tight Borel probability measure ts;a on R (S ) captures the reaction on action a of the system in state s. We use R (S ) to represent subprobabilities on S . The probability of refusal on action a in state s is given by ts;a (1). The role of c ? will be discussed later.
Proposition 29 Every discrete probabilistic transition system can be represented by a P -coalgebra. PROOF. We endow the set of states S of the system with the discrete metric. Consequently, every subset of the pseudometric space R (S ) is a Borel set. For every state s and action a, the Borel probability measure ts;a is the discrete Borel probability measure determined by
ts;a (1) = probability of refusal of action a in state s ts;a (fs0g) = probability of making an a-transition from state s to state s0 Obviously, the measure ts;a is tight. Because S is endowed with the discrete metric, the function t from S to P (S ) is nonexpansive. 2
Example 30 The continuous probabilistic transition system of Example 27 can be viewed as a P -coalgebra by endowing its state space with the Euclidean metric.
A continuous probabilistic transition system can be viewed as a P -coalgebra if its set S of states can be endowed with a pseudometric dS such that
the Borel -algebra induced by the pseudometric dS coincides with the algebra of the system,
for all states s and actions a, the system's subprobability measure ts;a is tight, and
the system's transition function is nonexpansive. We refer the reader forward to the conclusion for further discussion of these restrictions. Note that each P -coalgebra can be interpreted as a continuous probabilistic transition system, since nonexpansive functions are measurable. 18
7 A Pseudometric on Probabilistic Transition Systems We present our pseudometric on probabilistic transition systems. Furthermore, we show that states have distance 0 if and only if they are probabilistic bisimilar. Exploiting the pseudometric terminal coalgebra theorem, we prove that there exists a terminal P -coalgebra.
Theorem 31 There exists a terminal P -coalgebra h x (P ); ii. PROOF. According to America and Rutten's [2, Theorem 5.4], the functors 1 and + are locally nonexpansive and the scaling functor c is locally contractive. As we have seen in Proposition 22, the functor Mt is locally nonexpansive. As a consequence, the functor P is locally contractive. According to Proposition 15 and Theorem 17, the functor Mt , and hence the functor P , preserves equivalence and completeness. Therefore, we can conclude from Theorem 8 that there exists a terminal P -coalgebra h x (P ); ii. 2
The carrier of the terminal P -coalgebra is a compact metric space. We will exploit this property in Section 9.
Proposition 32 x (P ) is a compact metric space. PROOF. By Proposition 15 and Theorem 17, the functor M , and hence the t
functor P , preserves equivalence and compactness. Hence, we can conclude from Theorem 9 that the metric space x (P ) is compact. 2
As a consequence of Theorem 31, from each P -coalgebra hS; ti|a P -coalgebra represents a probabilistic transition system as we have seen in the previous section|there exists a unique P -homomorphism to the terminal P coalgebra.
S
/
x (P )
t
i
P (S )
P ( x (P ))
P ( )
/
The pseudometric kernel induced by is a pseudometric on the set underlying the carrier S of the P -coalgebra, that is, on the set of states of the probabilistic 19
transition system. Instead of d we will often write dC to stress its coalgebraic nature. Since the identity map on x (P ) is the unique P -homomorphism from the terminal P -coalgebra to itself, we can conclude that the coalgebraic pseudometric dC on the set underlying the carrier of the terminal P -coalgebra coincides with the metric d x P on the carrier of the terminal P -coalgebra. (
)
To compute some coalgebraic distances, we present a characterization of the pseudometric on Q (S ).
Proposition 33 For all , 2 Q (S ), 1
2
8Z 9 Z < = ( ; ) = sup : fd ? fd j f 2 c S !1 [0; 1] ; + ( (1) ? (1)):
dQ S
1
( )
2
1
S
1
2
2
S
PROOF. dQ S (8 ; ) 9 > > Z < Z = = sup > fd ? fd j f 2 R (S ) !1 [0; 1] > :R S ; R S 80 9 1 0 1 Z Z < = = sup : @r (1) + fd A ? @r (1) + fd A j r 2 [0; 1] ^ f 2 c S !1 [0; 1] ; S S 8 0Z 9 1 Z < = = sup : @ fd ? fd A + r ( (1) ? (1)) j r 2 [0; 1] ^ f 2 c S !1 [0; 1] ; S 8Z S 9 Z < = = sup : fd ? fd j f 2 c S !1 [0; 1] ; + ( (1) ? (1)): ( )
1
2
1
2
( )
( )
1
1
1
2
2
1
1
2
1
2
S
2
2
S
2
Example 34 Consider the discrete probabilistic transition system introduced
in Example 24. Let be the unique P -homomorphism from the P -coalgebra representing this system to the terminal P -coalgebra. Then
dC (s ; s ) = d x P ( (s ); (s )) = dP x P (i ( (s )); i ( (s ))) [i is an isometry] = dP x P (P ( ) (t (s )); P ( ) (t (s ))) [ is a P -homomorphism] 2
3
(
)
2
(
(
))
(
(
))
3
2
3
2
3
20
8 9 > > Z < Z = = sup > (f ) dts2;a ? (f ) dts3;a j f 2 c x (P ) !1 [0; 1] > : x P ; x P +(tsn2;a (1) ? ts3;a (1)) [Proposition 33] o = sup f ( (s )) j f 2 c x (P ) !1 [0; 1] + (0 ? 1) = 1: (
)
(
)
3
The other distances can be computed similarly and are collected in the following table.
s c
s
0
s s s s s
s
1
1
s s 2
3
2
0
c2 +2c
1 1
s
0
4
c2 +2c 4
+
2
c
3
1
+ c2 c + ((1 + )c ? c) c c2 +2c 4
?
c2 2c
c2 +2
2
4
2
2
c
1
c
c
2
2
1
+ c 1 1
The distance between states is a trade-o between the depth of observations needed to distinguish the states and the amount each observation di erentiates the states. The relative weight given to these two factors is determined by c lying between 0 and 1: the smaller the value of c the greater the discount on observations made at greater depth. This is re ected by the fact that dC (s ; s ) = c dC (s ; s ) in the above example. 0
1
0
1
Example 35 For the continuous probabilistic transition system of Example 27 we have that dC (0; 1) = 24c .
We conclude this section by showing that our pseudometric contains probabilistic bisimilarity.
Proposition 36 Let hS; ti be a P -coalgebra representing a probabilistic tran-
sition system. Let S be an analytic space. States have distance 0 if and only if they probabilistic bisimilar.
PROOF. For all s , s 2 S , 1
2
dC (s ; s ) = 0 i dL (s ; s ) = 0 [Theorem 46] i s and s are probabilistic bisimilar [see [10, Corollary 6.1.6 and Theorem 6.1.10]] 1
2
1
1
2
2
2 21
8 A Real-Valued Modal Logic We present a real-valued modal logic. This logic is closely related to the probabilistic modal logic of Larsen and Skou [24] and to a real-valued modal logic introduced by Desharnais, Gupta, Jagadeesan and Panangaden [11]. Along the lines of the latter paper, we de ne a pseudometric in terms of the logic. In the next section, we show that this pseudometric is closely related to the one we introduced in Section 7. Desharnais et al. de ned a pseudometric in terms of a real-valued modal logic. Their work builds on ideas of Kozen [21] to generalize logic to handle probabilistic phenomena. A minor variation on their logic is introduced in the following de nition.
De nition 37 The logic L is de ned by ' ::= 1 j hai ' j min ('; ') j 1 ? ' j ' ? q where a is an action and q is a rational in [0; 1].
Informally, there is the following correspondence between formulae in L and formulae in the probabilistic modal logic of Larsen and Skou. True is represented by 1, conjunction is represented by min, negation by 1 ? , and the modal connective haiq decomposes as hai and ? q. In analogy to one of De Morgan's laws, max can be expressed in the logic in terms of min and 1 ? as follows: max ('; ) = 1 ? min (1 ? '; 1 ? ): Given a probabilistic transition system represented by the P -coalgebra hS; ti, each formula ' can be interpreted as a function 'hS;ti from S to [0; 1] as follows.
De nition 38 For each ' 2 L, the function 'h i : S ! [0; 1] is de ned by S;t
1hS;ti (s) = 1 R (hai ')hS;ti (s) = c S 'hS;ti dts;a (min ('; ))hS;ti (s) = min ('hS;ti (s); hS;ti (s)) (1 ? ')hS;ti (s) = 1 ? 'hS;ti (s) (' ? q)hS;ti (s) = 'hS;ti (s) ? q 22
where
8 > < r ? q if r q r? q = > : 0 otherwise.
Next, we verify that for each formula ', the function 'hS;ti is c-contractive and hence measurable.
Proposition 39 For all ' 2 L, the function 'h i is c-contractive. S;t
PROOF. By structural induction on '. We only consider the most interesting case.
(hai ')hS;ti (s ) ? (hai ')hS;ti (s ) Z Z = c 'hS;ti dts1;a ? c 'hS;ti dts2;a S S Z Z = c 'hS;ti dts1;a ? 'hS;ti dts2;a S S 8Z 9 Z Z Z < = = c max : 'hS;ti dts1;a ? 'hS;ti dts2;a; (1 ? ')hS;ti dts1;a ? (1 ? ')hS;ti dts2;a; S S S S c dQ S (ts1;a ; ts2;a) [Proposition 33 and 'hS;ti and (1 ? ')hS;ti are c-contractive by induction] c dP S (ts1 ; ts2 ) c dS (s ; s ) [t is nonexpansive] 1
2
( )
( )
1
2
2 The logic L induces a pseudometric as follows.
De nition 40 The distance function d : S S ! [0; 1] is de ned by d (s ; s ) = sup 'h i (s ) ? 'h i (s ): L
L
1
2
2L
'
S;t
1
S;t
2
Clearly, the above introduced distance function is a pseudometric. Our logic di ers from the one presented by Desharnais et al. [11]. Instead of ' ? q they write b'cq . Furthermore, they introduce d'eq . In the presence of 23
negation, d'eq is redundant as it is equivalent to min ('; 1 ? b1cq ). Finally, they introduce a countable supremum over formulae. The logic considered by Desharnais in [10] lacks negation, but does include d'eq and max. The presence of negation in our logic has an impact on the distances as is shown in
Example 41 Consider the following probabilistic transition system. sQ
0
a; 12
a;1
s
3
|| || || | |~
s
1B
B B a; 1 BB 2 BB B
a; 21
sQ
s
4
5
|| || || | ~|
sQ
2
a; 12
a;1
The system in state s0 terminates with probability 0, in state s1 with probability 12 and in state s2 with probability 1. The expected number of transitions to termination starting in state s0 , s1 and s2 is 1, 1, and 2, respectively. Based on these kind of observations, one may infer that state s0 behaves more like state s1 than2 state s2 . This is re ected by the pseudometric dL. The states s0 and s1 are c2 apart, witnessed by hai hai 1. The states s0 and s2 are at distance c2 which is witnessed by the formulae 'n de ned by 2?c
8 > 0 and f 2 hS; d i ! [0; 1]. It suces to show that there exists a formula ' in L such that f and 'h i are at most apart. Since for all s , s 2 S , L
1
S;t
1
2
dL (s ; s ) = sup 'hS;ti (s ) ? 'hS;ti (s ) '2L c dS (s ; s ) [Proposition 39] 1
2
1
1
2
2
and the space hS; dS i is compact, we can conclude that hS; dLi is a compact pseudometric space. According to Proposition 42, the set (2) is a subset of hS; dLi !1 [0; 1]. Obviously, (2) is closed under min and max. Let s , s 2 S . Hence, according to Ash's [3, Lemma A.7.2], it suces to show that there exists a formula ' in L such that f (si) and 'hS;ti (si) are at most apart. 1
2
Without loss of generality, assume that f (s ) f (s ). Since 1
2
= f (s ) ? f (s ) dL (s ; s ) [f is nonexpansive] = sup 'hS;ti (s ) ? 'hS;ti (s ) 1
2
1
2L
'
2
1
2
there exists a formula ' such that ? 'hS;ti (s ) ? 'hS;ti (s ). Let p, q and r be rationals in [0; 1] such that 1
25
2
p 2 ['hS;ti (s ) ? ; 'hS;ti (s )] q 2 [ ? ; ] r 2 [f (s ); f (s ) + ] 2
2
2
2
We leave it to the reader to verify that the formula 1 ? ((1 ? min (' ? p; 1 ? (1 ? q))) ? r) has the desired property. 2 Note that 1, min, max, 1 ? and ? q all play a role in the above proof. We will exploit this result in the next section to relate the coalgebraic and logical distances. Interpreting a formula with respect to di erent P -coalgebras may give rise to di erent functions. These functions are related as follows.
Proposition 44 Let be a P -homomorphism from a P -coalgebra hS; ti to a P -coalgebra hS 0; t0i. Then for all formulae ', 'h 0 0 i = 'h i: S ;t
S;t
PROOF. By structural induction on '. We only present the most interesting case. For all s 2 S , (hai 'Z)hS0;t0i ( (s)) = c 'hS0 ;t0i dt0 s ;a =c =c
S0
Z
S0
Z
S0
Z
( )
'hS0 ;t0i d(P ( )(t))s;a [t0 = P ( ) t] 'hS0 ;t0i d(ts;a ? ) 1
= c ('hS0;t0 i ) dts;a =c
ZS
'hS;ti dts;a [induction]
S
= (hai ')hS;ti (s):
2 26
Also this result will be used in the next section where we relate the coalgebraic and logical pseudometric.
9 Relating the Coalgebraic and Logical Distances For a large class of probabilistic transition systems we have introduced a coalgebraic distance function dC and a logical distance function dL. In this section we relate the two pseudometrics. Before considering the general case, we rst relate the two distance functions on the set underlying the carrier of the terminal P -coalgebra. Recall that the coalgebraic pseudometric dC on the set underlying the carrier of the terminal P -coalgebra coincides with the metric d x P on the carrier of the terminal P -coalgebra. (
)
Proposition 45 For all x , x 2 x (P ), 1
2
dL (x ; x ) = d (x ; x ): C c 1
2
1
2
PROOF. Consider the function which maps each x 2 x (P ) to itself. For all x , x 2 x (P ), 1
2
dL ( (x ); (x )) c = dL (x ; x ) c sup'2L 'h x P ;ii (x ) ? 'h x P ;ii (x ) = c dC (x ; x ) ['h x P ;ii is c-contractive by Proposition 39] 1
1
2
2
(
1
1
)
2
(
(
)
2
)
Consequently, is a nonexpansive function from the space h x (P ); dC i to the space hh x (P ); dcL i. Next, we introduce a structure t such that hh x (P ); dcL i; ti is a P -coalgebra. Because is nonexpansive, each Borel set of R h x (P ); dcL i is also a Borel set of R h x (P ); dC i. Therefore, we can de ne the function t for x 2 x (P ), a 2 Act and Borel set B of R h x (P ); dcL i by
tx;a (B ) = ix;a (B ): Since the function is nonexpansive and the measure ix;a is tight, we can conclude that the measure tx;a is tight as well (cf. Proposition 19). 27
To conclude that t is the structure of a P -coalgebra with carrier h x (P ); dcL i, we have left to show that t is nonexpansive. Let x , x 2 x (P ). Then 1
dP h x
(P );
dL c i
(tx1 ; tx2 ) = sup dQ h x a2Act
(P );
dL c i
2
(tx1 ;a; tx2;a ):
Let a 2 Act. Without loss of generality, assume that tx1;a (1) tx2;a (1). Then, dL (tx1 ;a ; tx2 ;a ) 8ci 9 > > Z < Z = = sup > f dtx1 ;a ? f dtx2;a j f 2 c h x (P ); dcL i !1 [0; 1] > [Proposition 33] : x P ; x P Z Z sup 'h x P ;ii dtx1;a ? 'h x P ;ii dtx2 ;a [Proposition 32 and 43]
dQ h x
(P );
(
)
2L x (P )
'
=
(
(
)
sup'2L (hai ')h x
(
x (P )
)
i (x1 ) ? (hai ')h x (P );ii (x2 )
(P );i
dL (xc ; x ) : 1
)
c
2
From the de nition of t and we can easily derive that is a P -homomorphism from the P -coalgebra hh x (P ); dC i; ii to the P -coalgebra hh x (P ); dcL i; ti. We denote the unique P -homomorphism from the P -coalgebra hh x (P ); dcL i; ti to the terminal P -coalgebra hh x (P ); dC i; ii by .
h x (P ); dcL i l t
P h x (P ); dcL il
h x (P ); dC i +
P ( )
i
P h x (P ); dC i +
P ()
Obviously, the identity map on x (P ) is the unique P -homomorphism from the terminal P -coalgebra hh x (P ); dC i; ii to itself. Since is also such a P -homomorphism, we can conclude that equals the identity map on x (P ). Therefore, both and are isometries. This observation completes the proof. 2 The above proof makes use of Proposition 43. In the proof of that result 1, min, max, 1 ? and ? q all play a role. In the above proof, also the modality hai is used. 28
Next, we consider the general case. Consider a probabilistic transition system represented by the P -coalgebra hS; ti. We relate its coalgebraic pseudometric dC and its logical pseudometric dL in
Theorem 46 For all s , s 2 S , 1
2
dL (s ; s ) = d (s ; s ): C c 1
2
1
2
PROOF. We denote the unique P -homomorphism from the P -coalgebra hS; ti to the terminal P -coalgebra h x (P ); ii by . For all s , s 2 S , 1
dL (s ; s ) c d L ( (s ); (s )) [Proposition 44] = c = dC ( (s ); (s )) [Proposition 45] = dC (s ; s ): 1
2
1
1
2
1
2
2
2
(3) (4) (5) (6)
Note that (3) and (4) refer to di erent logical pseudometrics: the one on S and the one on x (P ), respectively. Also notice that (5) and (6) refer to di erent coalgebraic pseudometrics: the one on x (P ) and the one on S , respectively. 2 In [9], we studied a minor variation on the functor P . In that paper, we considered the functor
P 0 = c Act Mt (1 + ?): This functor is also locally contractive and preserves equivalence and completeness and, therefore, has a terminal coalgebra. The carriers of the terminal P -coalgebra and the terminal P 0-coalgebra are related as follows.
Proposition 47 x (P 0) = c x (P ). PROOF. According to Lemma 4, there exists an isometry i from x (P ) to P ( x (P )). Clearly, i is also an isometry from c x (P ) to cP ( x (P )) = P 0 (c x (P )). Using Lemma 4 again, we can conclude that x (P 0) = c x (P ). 2 Consequently, the coalgebraic pseudometrics induced by the functor P 0 coincide with the logical pseudometrics. 29
10 Conclusion 10.1 Related Work
As we have already seen in Section 8 and 9, our coalgebraic pseudometric is closely related to the logical pseudometric of Desharnais et al. [10,11]. In [11], they also introduce a probabilistic process algebra. A number of combinators of the process algebra, including probabilistic choice, are shown to be nonexpansive. This is a quantitative analogue of probabilistic bisimulation being a congruence. It allows for compositional veri cation of probabilistic transition systems. Since our coalgebraic pseudometric is related to their logical pseudometric, we can conclude that those combinators are also nonexpansive with respect to our pseudometric. Furthermore, Desharnais et al. present a decision procedure for their pseudometric. That is, they provide an algorithm to approximate the logical distances to a prescribed degree of accuracy. The algorithm involves the generation of a representative set of formulae of their real-valued modal logic. They only consider formulae with a restricted number of nested occurrences of the modal connective. Their algorithm approximates the distances in exponential time. In [8], we present an algorithm to approximate our coalgebraic distances. The problem of approximating such distances can be reduced to a particular linear programming problem: the transshipment problem. Since the latter problem can be solved in polynomial time, we obtain a polynomial time decision procedure for our distances. We see this practical algorithm as one of the advantages of our coalgebraic approach over the logical approach of Desharnais et al. Another advantage of our approach is that we work within a uniform framework, the theory of coalgebras. We do not know whether there exists a terminal coalgebra of our functor for c equals 1, and hence we cannot use our framework to de ne a pseudometric when c equals 1. However, the logical approach of Desharnais et al. also works in that case. Furthermore, Desharnais et al. consider a larger class of continuous probabilistic transition systems than we do in this paper. However, we are con dent that we can extend our results as we will discuss below. In conclusion, we believe that both approaches have their merits and demerits. The results in Section 9 are very valuable as they allows us to transfer results from the one setting to the other. As far as we know, [14] by Giacalone et al. is the rst paper to advocate the use of pseudometric spaces to provide a robust and quantitative notion of behavioural equivalence. They stress the importance of combinators being nonexpansive with respect to the pseudometric, making compositional veri cation possible. The class of discrete probabilistic transition systems they consider is rather restricted. A decade later, we are able to deal with all discrete probabilistic transition systems and a large class of continuous probabilistic 30
transition systems. De Vink and Rutten [33] show that discrete probabilistic transition systems and some continuous probabilistic transition systems can be viewed as coalgebras. Their main contribution is the proof that the kernel of the homomorphism from a coalgebra, representing a probabilistic transition system, to the terminal coalgebra coincides with probabilistic bisimilarity. They only exploit metrics to represent continuous systems as coalgebras. Their metric on the terminal coalgebra only provides qualitative information. For example, in De Vink and Rutten's setting the states s and s of the system presented in the introduction are c apart if di ers from 0. More generally, the distance between two states in their setting is cn where n is the depth of a probabilistic bisimulation between them. De Vink and Rutten consider the endofunctor 0
0
Act (1 + Mc (c ?))
on the category of complete ultrametric spaces and nonexpansive functions. Mc denotes the Borel probability measures with compact support. The main di erences between our functor and their functor are the following. First of all, they consider a distance function on Borel probability measures [33, De nition 5.3] di erent from the one presented in De nition 12. Their distance function only captures qualitative information as the above example illustrates. Secondly, they consider the category of complete ultrametric spaces and nonexpansive functions whereas we consider the considerably larger category pseudometric spaces and nonexpansive functions. This allows us to captures many more interesting continuous probabilistic transition systems as coalgebras, including systems where the state space is the real interval [0; 1] endowed with the Euclidean metric. Furthermore, they consider Borel probability measures with compact support whereas we consider the more general tight Borel probability measures. Again this allows us to represent more systems as coalgebras. Finally, their model only allows states to refuse transitions with probability 0 or 1. In conclusion, our functor allows to model many more interesting continuous systems, and all the results for their functor in [33, Section 5] can be generalized to our setting. 3
Baier and Kwiatkowska [4] study a functor which is closely related to the one of De Vink and Rutten. Our work can be compared to theirs in the same way it is compared to the work of De Vink and Rutten in the paragraph above. In his thesis [17], Den Hartog exploits ultrametric spaces very similar to the terminal coalgebra of De Vink and Rutten. The metric structure is only used to model in nite behaviour. As a consequence, qualitative information suces. The proof of [33, Theorem 5.8] is incomplete. We also have no proof for this result in our setting.
3
31
We believe that metrics closely related to the one we present in this paper may be used in his setting as well, possibly providing additional quantitative information about his models. Kwiatkowska and Norman [22,23,27] present a number of closely related metrics. Like Den Hartog, they use their metric as a means to model recursion. However, their metric is not an ultrametric and contains quantitative information. Let us compare the metric introduced by Norman in [27, Section 6.1] with our pseudometric. Consider the following probabilistic transition system.
s
0
1 2
s
| || || | ~| 3B BB BB B 1 B
s |
1B
1
BB 1 BB2 BB
1 2
s B
B
4
s
5
1
/
s
| || || | |~
2
1 2
1
6
s |
z
Clearly, the states s and s are not probabilistic bisimilar. In Norman's setting the states have distance 0. In our pseudometric, states only have distance 0 if they are probabilistic bisimilar. In our setting the states are c2 c apart. This example shows that his distance function gives rise to a topology di erent from ours. The main di erences between his and our approach are the following. First of all, he uses a linear-time model whereas we consider a branching-time model. Secondly, he only handles discrete systems whereas we also consider continuous ones. Finally, we use the usual categorical machinery and various standard constructions whereas his de nitions are more ad-hoc. We believe however that his metric can also be characterized by means of a terminal coalgebra. 0
1
+2 4
Results similar to the ones in this paper have been presented by the second author in his thesis [34, Chapter 4] in the setting of bimodules and generalized metric spaces. As we have seen, the functor Mt , and hence the functor P , preserves isometric embeddings. As a consequence, the coalgebraic distance of states s and s can be characterized as the smallest R (s ; s ) where R is a bimodule satisfying certain conditions (see [34, Theorem 4.5.12] for the details). This is the quantitative analogue of the characterization of probabilistic bisimilarity as the largest probabilistic bisimulation. 1
2
1
2
10.2 Future Work
Let us isolate two distinct consequences of our use of the pseudometric presented in Section 4. First of all, we can talk about approximate equivalence 32
of states. Secondly, we can model a large class of continuous probabilistic transition systems as coalgebras. An apparent restriction with regard to the latter point is the requirement that the structure of a P -coalgebra, that is, the system's transition function, be nonexpansive. Properly speaking, continuous probabilistic transition systems as formulated in De nition 26 are coalgebras of (a variant of) the Giry monad on the category of measurable spaces and measurable functions [15]. However, we conjecture that the terminal P -coalgebra h x (P ); ii is also terminal when seen as a coalgebra of the Giry functor, and that our results can be extended to continuous probabilistic transition systems in general. In Proposition 32 we have shown that the carrier of our terminal coalgebra is compact and hence separable. Furthermore, we conjecture that the unique homomorphism from the initial algebra of a nitary version of P |this nitary version represents nite discrete probabilistic transition systems with rational probabilities|to the terminal P -coalgebra is a dense embedding. Hence, every continuous system can be approximated by a nite one (see also [12]).
Acknowledgements The authors would like to thank the Amsterdam Coordination Group, Josee Desharnais, Abbas Edalat, Joel Ouaknine, Prakash Panangaden, Jan Rutten and Erik de Vink for discussion. The rst author is thankful to Stephen Watson for the joint study of metrics on probability measures.
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