Tài liệu Probabilistic Event Structures and Domains pdf

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A configuration x of an event structure E is a conflict-free downward closed subset
of E, i.e., a subset x of E satisfying: (1) whenever e ∈ x and e

≤ e then e

∈ x and (2)
for every e, e

∈ x, it is not the case that e # e

. Therefore, two events of a configuration
are either causally dependent or concurrent, i.e., a configuration represents a run of
an event structure where events are partially ordered. The set of configurations of E,
partially ordered by inclusion, is denoted as L(E). The set of finite configurations is
written by L
fin
(E). We denote the empty configuration by ⊥.
If x is a configurationand e is an event such that e ∈ x and x∪{e} is a configuration,
then we say that e is enabled at x. Two configurations x, x

are said to be compatible if
x ∪ x

is a configuration. For every event e of an event structure E,wedefine[e]:=↓e,
and [e):=[e]\{e}. It is easy to see that both [e] and [e) are configurations for every
event e and that therefore any event e is enabled at [e).
We say that events e
1
and e
2
are in immediate conflict, and write e
1
#
µ
e
2
when
e
1
# e
2
and both [e
1
) ∪ [e
2
] and [e
1
] ∪ [e
2
) are configurations. Note that the immediate
conflict relation is symmetric. It is also easy to see that a conflict e
1
# e
2
is immediate
if and only if there is a configuration where both e
1
and e
2
are enabled. Every conflict
is either immediate or inherited from an immediate conflict.
Lemma 2.1. In an event structure, e # e

if and only if there exist e
0
,e

0
such that e
0
≤
e, e

0
≤ e

,e
0
#
µ
e

0
.
Proof. Consider the set ([e] × [e

]) ∩ # consisting of the pairs of conflicting events,
and order it componentwise. Consider a minimal such pair (e
0
,e

0
). By minimality, any
event in [e
0
) is not in conflict with any event in [e

0
]. Since they are both lower sets
we have that [e
0
) ∪ [e

0
] is a configuration. Analogously for [e
0
] ∪ [e

0
). By definition
e
0
#
µ
e

0
. The other direction follows from the definition of #. 
2.2 Confusion-free Event Structures
The most intuitive way to add probability to an event structure is to identify “probabilis-
tic events”, such as coin flips, where probability is associated locally. A probabilistic
event can be thought of as probability distribution over a cell, that is, a set of events (the
outcomes) that are pairwise in immediate conflict and that have the same set of causal
predecessors. The latter implies that all outcomes are enabled at the same configura-
tions, which allows us to say that the probabilistic event is either enabled or not enabled
at a configuration.
Definition 2.2. A partial cell is a set c of events such that e, e

∈ c implies e #
µ
e

and
[e)=[e

). A maximal partial cell is called a cell.
We will now restrict our attention to event structures where each immediate conflict
is resolved through some probabilistic event. That is, we assume that cells are closed
under immediate conflict. This implies that cells are pairwise disjoint.
Definition 2.3. An event structure is confusion-free if its cells are closed under imme-
diate conflict.
3
Proposition 2.4. An event structure is confusion-free if and only if the reflexive closure
of immediate conflict is transitive and inside cells, the latter meaning that e #
µ
e

=⇒
[e)=[e

).
Proof. Take an event structure E. Suppose it is confusion-free. Consider three events
e, e

,e

such that e #
µ
e

and e

#
µ
e

. Consider a cell c containing e (there exists
one by Zorn’s lemma). Since c is closed under immediate conflict, it contains e

.By
definition of cell [e)=[e

). Also, since c contains e

, it must contain e

. By definition
of cell, e #
µ
e

.
For the other direction we observe that if the immediate conflict is transitive, the
reflexive closure of immediate conflict is an equivalence. If immediate conflict is inside
cells, the cells coincide with the equivalence classes. In particular they are closed under
immediate conflict. 
In a confusion-free event structure, if an evente ∈ c is enabled at a configuration x,
all the eventsof c are enabled as well. In such a case we say that the cellc is accessible at
x. The set of accessible cells at x is denoted by Acc (x). Confusion-freeevent structures
correspond to deterministic concrete data structures [NPW81,KP93] and to confusion-
free occurrence nets [NPW81].
We find it useful to define cells without directly referring to events. To this end we
introduce the notion of covering.
Definition 2.5. Given two configurations x, x

∈L(E)we say that x

covers x (written
x  x

) if there exists e ∈ x such that x

= x ∪{e}. For every finite configuration x of a
confusion-free event structure, a partial covering at x is a set of pairwise incompatible
configurations that cover x.Acovering at x is a maximal partial covering at x.
Proposition 2.6. In a confusion-free event structure if C is a covering at x,thenc=
{e|x∪{e}∈C}is a cell accessible at x.Conversely,ifcis accessible at x,then
C:= {x ∪{e}|e∈c}is a covering at x.
Proof. See Appendix B. 
In confusion-free event structures, we extend the partial order notation to cells by
writing e<c

if for some event e

∈ c

(and therefore for all such) e<e

. We write
c<c

if for some (unique) event e ∈ c, e<c

.By[c)we denote the set of events e
such that e<c.
2.3 Probabilistic Event Structures with Independence
Once an event structure is confusion-free, we can associate a probability distribution
with each cell. Intuitively it is as if we have a die local to each cell, determining the
probability with which the events at that cell occur. In this way we obtain our first
definition of a probabilistic event structure, a definition in which dice at different cells
are assumed probabilistically independent.
Definition 2.7. When f : X → [0, +∞] is a function, for every Y ⊆ X, we define
f[Y ]:=

x∈Y
f(x).Acell valuation on a confusion-free event structure E,≤,# is
a function p : E → [0, 1] such that for every cell c, we have p[c]=1.
4
Assuming probabilistic independence of all probabilistic events, every finite configura-
tion can be given a “probability” which is obtained as the product of probabilities of its
constituent events. This gives us a function L
fin
(E) → [0, 1] which we can characterise
in terms of the order-theoretic structure of L
fin
(E) by using coverings.
Proposition 2.8. Let p be a cell valuation and let v : L
fin
(E) → [0, 1] be defined by
v(x)=Π
e∈x
p(e). Then we have
(a) (Normality) v(⊥)=1;
(b) (Conservation) if C is a covering at x,thenv[C]=v(x);
(c) (Independence) if x, y are compatible, then v(x) · v(y)=v(x∪y)·v(x∩y).
Proof. Straightforward. 
Definition 2.9. A configuration valuation with independenceon a confusion-free event
structure E is a function v : L
fin
(E) → [0, 1] that satisfies normality, conservation
and independence. The configuration valuation associated with a cell valuation p as in
Prop. 2.8 is denoted by v
p
.
Lemma 2.10. If v : L
fin
(E) → [0, 1] satisfies conservation, then it is contravariant,
i.e.:
x ⊆ x

=⇒ v(x) ≥ v(x

) .
Proof. By induction on the cardinality of x

\ x.Ifx=x

then v(x)=v(x

).Take
x⊆x

and consider a maximal event e in x

\ x.Letx

:= x

\{e}. By induction
hypothesis v(x) ≥ v(x

).Letcbe the cell of e and C be the c-covering of x

.By
conservation,

y∈C
v(y)=v(x

). Since for every y ∈ C we have that v(y) ≥ 0,then
it must also be that v(y) ≤ v(x

).Butx

∈Cso that v(x

) ≤ v(x

) ≤ v(x). 
Proposition 2.11. If v is a configuration valuation with independence and p : E →
[0, 1] is a mapping such that v([e]) = p(e) · v([e)) for all e ∈ E,thenpis a cell
valuation such that v
p
= v.
Proof. See Appendix B. 
Independence is essential to prove Proposition 2.11. We will show later (Theorem
5.3) the sense in which this condition amounts to probabilistic independence.
We give an example. Take the following confusion-free event structure E
1
: E
1
=
{a, b, c, d} with the discrete causal ordering and with a #
µ
b and c #
µ
d.Werepresent
immediate conflict by a curly line.
a
/o
/o
/o
b
c
/o
/o
/o
d
We define a cell valuation on E
1
by p(a)=1/3,p(b)=2/3,p(c)=1/4,p(d)=
3/4. The corresponding configuration valuation is defined as
• v
p
(⊥)=1;
• v
p
({a})=1/3,v
p
({b})=2/3,v
p
({c})=1/4,v
p
({d})=3/4;
• v
p
({a, c})=1/12,v
p
({b, c})=1/6,v
p
({a, d})=1/4,v
p
({b, d})=1/2.
5
In the event structure above, a covering at ⊥ consists of {a}, {b}, while a covering at
{a} consists of {a, c}, {a, d}.
We conclude this section with a definition of a probabilistic event structure. Though,
as the definition indicates, we will consider a more general definition later, one in which
there can be probabilistic correlations between the choices at different cells.
Definition 2.12. A probabilistic event structure with independence consists of a confu-
sion-free event structure together with a configuration valuation with independence.
3 A Process Language
Confusion-freeness is a strong requirement. But it is still possible to give a seman-
tics to a fairly rich language for probabilistic processes in terms of probabilistic event
structures with independence. The language we sketch is a probabilistic version of
value passing CCS. Following an idea of Milner, used in the context of confluent pro-
cesses [Mil89], we restrict parallel composition so that there is no ambiguity as to which
two processes can communicate at a channel; parallel composition will then preserve
confusion-freeness.
Assume a set of channels L. For simplicity we assume that a common set of values
V may be communicated over any channel a ∈ L. The syntax of processes is given by:
P ::= 0 |

v∈V
a!(p
v
,v).P
v
| a?(x).P | P
1
P
2
| P \ A |
P [f] | if b then P
1
else P
2
| X | rec X.P
Here x ranges over value variables, X over process variables, A over subsets of chan-
nels and f over injective renaming functions on channels, b over boolean expressions
(which make use of values and value variables). The coefficients p
v
are real numbers
such that

v∈V
p
v
=1.
A closed process will denote a probabilistic event structure with independence, but
with an additional labelling function from events to output labels a!v, input labels a?v
where a is a channel and v avalue,orτ. At the cost of some informality we explain the
probabilistic semantics in terms of CCS constructions on the underlying labelled event
structures, in which we treat pairs of labels consisting of an output label a!v and input
label a?v as complementary.(See e.g.the handbookchapter [WN95] or [Win82,Win87]
for an explanation of the event structure semantics of CCS.) For simplicity we restrict
attention to the semantics of closed process terms.
The nil process 0 denotes the empty probabilistic event structure. A closed output
process

v∈V
a!(p
v
,v).P
v
can perform a synchronisation at channel a, outputting a
value v with probability p
v
, whereupon it resumes as the process P
v
. Each P
v
,for
v ∈V, will denote a labelled probabilistic event structure with underlying labelled
event structure E[[ P
v
]] . The underlying event structure of such a closed output process
is got by the juxtaposition of the family of prefixed event structures
a!v.E [[ P
v
]] ,
6
with v ∈ V , in which the additional prefixing events labelled a!v are put in (immedi-
ate) conflict; the new prefixing events labelled a!v are then assigned probabilities p
v
to
obtain the labelled probabilistic event structure.
A closed input process a?(x).P synchronises at channel a, inputting a value v and
resuming as the closed process P [v/x]. Such a process P [v/x] denotes a labelled prob-
abilistic event structure with underlying labelled event structure E[[ P [ v/x]]]. The under-
lying labelled event structure of the input process is got as the parallel juxtaposition of
the family of prefixed event structures
a?v.E [[ P [ v/x]]] ,
with v ∈ V ; the new prefixing events labelled a?v are then assigned probabilities 1.
The probabilistic parallel composition corresponds to the usual CCS parallel com-
position followed by restricting away on all channels used for communication. In order
for the parallel composition P
1
P
2
to be well formed the set of input channels of P
1
and P
2
must be disjoint, as must be their output channels. So, for instance, it is not
possible to form the parallel composition

v∈V
a!(p
v
,v).0a?(x).P
1
a?(x).P
2
.
In this way we ensure that no confusion is introduced through synchronisation.
We first describe theeffect of the parallel composition on the underlyingevent struc-
tures of the two components, assumed to be E
1
and E
2
. This is got by CCS parallel
composition followed by restricting away events in a set S:
(E
1
| E
2
) \ S
where S consists of all labels a!v, a?v for which a!v appears in E
1
and a?v in E
2
,or
vice versa. In this way any communication between E
1
and E
2
is forced when possible.
The newly introduced τ -events, corresponding to a synchronisation between an a!v-
event with probability p
v
and an a?v-event with probability 1, are assigned probability
p
v
.
A restriction P \ A has the effect of the CCS restriction
E[[ P ]] \{a!v, a?v | v ∈ V & a ∈ A}
on the underlying event structure; the probabilities of the events which remain stay the
same. A renaming P [f] has the usual effect on the underlying event structure, proba-
bilities of events being maintained. A closed conditional (if b then P
1
else P
2
) has the
denotation of P
1
when b is true and of P
2
when b is false.
The recursive definition of probabilistic event structures follows that of event struc-
tures [Win87] carrying the extra probabilities along. Though care must be taken to en-
sure that a confusion-free event structure results: one way to ensure this is to insist that
for rec X.P to be well-formed the process variable X may not occur under a parallel
composition.
7
4 Probabilistic Event Structures and Domains
The configurations L(E), ⊆ of a confusion-free event structure E, ordered by inclu-
sion, form a domain, specifically a distributive concrete domain (cf. [NPW81,KP93]).
In traditional domain theory, a probabilistic process is denoted by a continuous valu-
ation. Here we show that, as one would hope, every probabilistic event structure with
independence corresponds to a unique continuous valuation. However not all continu-
ous valuations arise in this way. Exploring why leads us to a more liberal notion of a
configuration valuation, in which there may be probabilistic correlation between cells.
This provides a representation of the normalised continuous valuations on distributive
concrete domains in terms of probabilistic event structures. (Appendix A includes a
brief survey of the domain theory we require and some of the rather involved proofs of
this section. All proofs of this section can be found in [Var03].)
4.1 Domains
The configurations of an event structure form a coherent ω-algebraic domain, whose
compact elements are the finite configurations [NPW81]. The domain of configurations
of a confusion free has an independent equivalent characterisation as distributive con-
crete domain (for a formal definition of what this means, see [KP93]).
The probabilistic powerdomain of Jones and Plotkin [JP89] consists of continuous
valuations, to be thought of as denotations of probabilistic processes. A continuous
valuation on a DCPO D is a function ν defined on the Scott open subsets of D,taking
values on [0, +∞], and satisfying:
• (Strictness) ν(∅)=0;
• (Monotonicity) U ⊆ V =⇒ ν(U) ≤ ν(V );
• (Modularity) ν(U)+ν(V)=ν(U∪V)+ν(U∩V);
• (Continuity) if J is a directed family of open sets, ν


J

=sup
U∈J
ν(U).
A continuous valuation ν is normalised if ν(D)=1.LetV
1
(D)denote the set of
normalised continuous valuations on D equipped with the pointwise order: ν ≤ ξ if for
all open sets U, ν(U) ≤ ξ(U). V
1
(D) is a DCPO [JP89,Eda95].
The open sets in the Scott topology represent observations. If D is an algebraic
domain and x ∈ D is compact, the principal set ↑ x is open. Principal open sets can be
thought of as basic observations. Indeed they form a basis of the Scott topology.
Intuitively a normalised continuous valuation ν assigns probabilities to observa-
tions. In particular we could think of the probability of a principal open set ↑ x as rep-
resenting the probability of x.
4.2 Continuous and Configuration Valuations
As can be hoped, a configurationvaluation with independenceon a confusion-freeevent
structure E corresponds to a normalised continuous valuation on the domain L(E), ⊆,
in the following sense.
8
Proposition 4.1. For every configuration valuation with independence v on E there is
a unique normalised continuous valuation ν on L(E) such that for every finite configu-
ration x, ν(↑ x)=v(x).
Proof. The claim is a special case of the subsequent Theorem 4.4. 
While a configuration valuation with independence gives rise to a continuous val-
uation, not every continuous valuation arises in this way. As an example, consider the
event structure E
1
as defined in Section 2.3. Define
• ν(↑{a})=ν(↑{b})=ν(↑{c})=ν(↑{d})=1/2;
• ν(↑{a, d})=ν(↑{b, c})=1/2;
• ν(↑{a, c})=ν(↑{b, d})=0;
and extend it to all open sets by modularity. It is easy to verify that it is indeed a con-
tinuous valuation on L(E
1
). Define a function v : L
fin
(E
1
) → [0, 1] by v(x):=ν(↑x).
This is not a configuration valuation with independence; it does not satisfy condition
(c) of Proposition 2.8. If we consider the compatible configurations x := {a},y := {c}
then v(x ∪ y) · v(x ∩ y)=0<1/4=v(x)·v(y).
Also continuous valuations “leaking” probability do not arise from probabilistic
event structures with independence.
Definition 4.2. Denote the set of maximal elements of a DCPO D by Ω(D). A nor-
malised continuous valuation ν on D is non-leaking if for every open set O ⊇ Ω(D),
we have ν(O)=1.
This definition is new, although inspired by a similar concept in [Eda95]. For the sim-
plest example of a leaking continuous valuation, consider the event structure E
2
con-
sisting of one event e only, and the valuation defined as ν(∅)=0,ν(↑⊥)=1,
ν(↑{e})=1/2. The corresponding function v : L
fin
(E
2
) → [0, 1] violates condition
(b) of Proposition 2.8. The probabilities in the cell of e do not sum up to 1.
We analyse how valuations without independence and leaking valuations can arise
in the next two sections.
4.3 Valuations Without Independence
Definition 2.12 of probabilistic event structures assumes the probabilistic independence
of choice at different cells. This is reflected by condition (c) in Proposition 2.8 on which
it depends. In the first example above, the probabilistic choices in the two cells are not
independent: once we know the outcome of one of them, we also know the outcome
of the other. This observation leads us to a more general definition of a configuration
valuation and probabilistic event structure.
Definition 4.3. A configuration valuation on a confusion-free event structure E is a
function v : L
fin
(E) → [0, 1] such that:
(a) v(⊥)=1;
(b) if C is a covering at x,thenv[C]=v(x).
9
A probabilistic event structure consists of a confusion-free event structure together with
a configuration valuation.
Now we can generalise Proposition 4.1, and provide a converse:
Theorem 4.4. For every configuration valuation v on E there is a unique normalised
continuous valuation ν on L(E) such that for every finite configuration x, ν(↑ x)=
v(x). Moreover ν is non-leaking.
Proof. See Appendix C. 
Theorem 4.5. Let ν be a non-leaking continuous valuation on L(E). The function v :
L
fin
(E) → [0, 1] defined by v(x)=ν(↑x)is a configuration valuation.
Proof. See Appendix C. 
Using this representation result, we are also able to characterise the maximal ele-
ments in V
1
(L(E)) as precisely the non-leaking valuations—a fact which is not known
for general domains.
Theorem 4.6. Let E be a confusion-free event structure and let ν ∈V
1
(L(E)).Thenν
is non-leaking if and only if it is maximal.
Proof. See [Var03], Prop. 7.6.3 and Thm. 7.6.4. 
4.4 Leaking Valuations
There remain leaking continuous valuations, as yet unrepresented by any probabilistic
event structures. At first sight it might seem that to account for leaking valuations it
would be enough to relax condition (b) of Definition 4.3 to the following
(b’) if C is a covering at x,thenv[C]≤v(x).
However, it turns out that this is not the right generalisation, as the following example
shows. Consider the event structure E
3
where E
3
= {a, b} with the flat causal ordering
and no conflict. Define a “leaking configuration valuation” on E
3
by v(⊥)=v({a})=
v({b})=1,v({a, b} )=0. The function v satisfies conditions (a) and (b’), but it cannot
be extended to a continuous valuation on the domain of configurations. However, we
can show that the leaking of probability is attributable to an “invisible” event.
Definition 4.7. Consider a confusion-free event structure E = E,≤,#. For every
cell c we consider a new “invisible” event ∂
c
such that ∂
c
∈ E and if c = c

then
∂
c
= ∂
c

.Let∂={∂
c
|cis a cell}. We define E
∂
to be E
∂
, ≤
∂
, #
∂
,where
• E
∂
=E∪∂;
•≤
∂
is ≤ extended by e ≤
∂
∂
c
if for all e

∈ c, e ≤ e

;
• #
∂
is # extended by e #
∂
∂
c
if there exists e

∈ c, e

≤ e.
So E
∂
is E extended by an extra invisible event at every cell. Invisible events can absorb
all leaking probability, as shown by Theorem 4.9 below.
10
Definition 4.8. Let E be a confusion-free event structure. A generalised configuration
valuation on E is a function v : L
fin
(E) → [0, 1] that can be extended to a configuration
valuation on E
∂
.
It is not difficult to prove that, when such an extension exists, it is unique.
Theorem 4.9. Let E be a confusion-freeevent structure. Let v : L
fin
(E) → [0, 1].There
exists a unique normalised continuous valuation ν on L(E) with v(x)=ν(↑x), if and
only if v is a generalised configuration valuation.
Proof. See [Var03], Thm. 6.5.3. 
The above theorem completely characterises the normalised continuous valuations
on distributive concrete domains in terms of probabilistic event structures.
5 Probabilistic Event Structures as Probabilistic Runs
In the rest of the paper we investigate how to adjoin probabilities to event structures
which are not confusion-free.Inorder to do so, we find it useful to introduce two notions
of probabilistic run.
Configurations represent runs (or computation paths) of an event structure. What is
a probabilistic run (or probabilistic computation path) of an event structure? One would
expect a probabilistic run to be a form of probabilistic configuration, so a probability
distribution over a suitably chosen subset of configurations. As a guideline we con-
sider the traditional model of probabilistic automata [Seg95], where probabilistic runs
are represented in essentially two ways: as a probability measure over the set of max-
imal runs [Seg95], and as a probability distribution over finite runs of the same length
[dAHJ01].
The first approach is readily available to us, and where we begin. As we will see,
according to this view probabilistic event structures over an underlying event structure
E correspond precisely to the probabilistic runs of E.
The proofs of the results in this section are to be found in the appendix.
5.1 Probabilistic Runs of an Event Structure
The first approach suggests that a probabilistic run of an event structure E be taken to
be a probability measure on the maximal configurations of L(E).
Some basic notion of measure theory can be found in Appendix A. Let D be an
algebraic domain. Recall that Ω(D) denotes the set of maximal elements of D and
that for every compact element x ∈ D the principal set ↑ x is Scott open. The set
K(x):=↑x∩Ω(D)is called the shadow of x. We shall consider the σ-algebra S on
Ω(D) generated by the shadows of the compact elements.
Definition 5.1. A probabilistic run of an event structure E is a probability measure
on Ω(L(E)), S,whereSis the σ-algebra generated by the shadows of the compact
elements.
There is a tight correspondence between non-leaking valuations and probabilistic runs.
11
Theorem 5.2. Let ν be a non-leaking normalised continuous valuation on a coherent
ω-algebraic domain D. Then there is a unique probability measure µ on S such that for
every compact element x, µ(K(x)) = ν(↑ x).
Let µ be a probability measure on S. Then the function ν defined on open sets by
ν(O)=µ(O∩Ω(D)) is a non-leaking normalised continuous valuation.
Proof. See Appendix C. 
According to the result above, probabilistic event structures over a common event
structure E correspond precisely to the probabilistic runs of E. Among these we can
characterise probabilistic event structures with independence in terms of the standard
measure-theoretic notion of independence. In fact, for such a probabilistic event struc-
ture, every two compatible configurations are probabilistically independent, given the
common past:
Proposition 5.3. Let v be a configuration valuation on a confusion-free event structure
E.Letµ
v
be the corresponding measure as of Propositions 4.1 and Theorem 5.2. Then,
v is a configuration valuation with independence iff for every two finite compatible
configurations x, y
µ
v

K(x) ∩ K(y) | K(x ∩ y)

= µ
v

K(x) | K(x ∩ y)

· µ
v

K(y) | K(x ∩ y)

.
Proof. See Appendix C. 
Note that the definition of probabilistic run of an event structure does not require
that the eventstructure is confusion-free.It thus suggests a general definition of a proba-
bilistic event structure as an event structure with a probability measure µ on its maximal
configurations, even when the event structure is not confusion-free. This definition, in
itself, is however not very informative and we look to an explanation in terms of finite
probabilistic runs.
5.2 Finite Runs
What is a finite probabilistic run? Following the analogy heading this section, we want
it to be a probability distribution over finite configurations. But which sets are suitable
to be the support of such distribution? In interleaving models, the sets of runs of the
same length do the job. For event structures this won’t do.
To see why consider the event structure with only two concurrent events a, b.The
only maximal run assigns probability 1 to the maximal configuration {a, b}. This corre-
sponds to a configuration valuation which assigns 1 to both {a} and {b}.Nowtheseare
two configurations of the same size, but their common “probability” is equal to 2! The
reason is that the two configurations are compatible: they do not represent alternative
choices. We therefore need to represent alternative choices, and we need to represent
them all. This leads us to the following definition.
Definition 5.4. Let E be an event structure. A partial test of E is a set C of pairwise
incompatible configurations of E.Atest is a maximal partial test. A test is finitary if all
its elements are finite.
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