Multi-Context Systems (MCSs) were introduced in , building on the work in [12, 23], to address the need for a general framework that integrates knowledge bases expressed in heterogeneous KR formalisms. Intuitively, instead of designing a unifying language (see e.g., [13, 22], and  with its reasoner NoHR ) to which other languages could be translated, in an MCS the different formalisms and knowledge bases are considered as modules, and means are provided to model the flow of information between them (cf. [1, 17, 20] and references therein for further motivation on hybrid languages and their connection to MCSs).
More specifically, an MCS consists of a set of contexts, each of which is a knowledge base in some KR formalism, such that each context can access information from the other contexts using so-called bridge rules. Such non-monotonic bridge rules add their heads to the context’s knowledge base provided the queries (to other contexts) in their bodies are successful.
Managed Multi-Context Systems (mMCSs) were introduced in  to provide an extension of MCSs by allowing operations, other than simple addition, to be expressed in the heads of bridge rules. This allows mMCSs to properly deal with the problem of consistency management within contexts.
One recent challenge for KR languages is to shift from static application scenarios which assume a one-shot computation, usually triggered by a user query, to open and dynamic scenarios where there is a need to react and evolve in the presence of incoming information. Examples include EVOLP , Reactive ASP [11, 10], C-SPARQL , Ontology Streams  and ETALIS , to name only a few.
Whereas mMCSs are quite general and flexible to address the problem of integration of different KR formalisms, they are essentially static in the sense that the contexts do not evolve to incorporate the changes in the dynamic scenarios.
In such scenarios, new knowledge and information is dynamically produced, often from several different sources – for example a stream of raw data produced by some sensors, new ontological axioms written by some user, newly found exceptions to some general rule, etc.
To address this issue, two recent frameworks, evolving Multi-Context Systems (eMCSs)  and reactive Multi-Context Systems (rMCSs) [5, 9, 8] have been proposed sharing the broad motivation of designing general and flexible frameworks inheriting from mMCSs the ability to integrate and manage knowledge represented in heterogeneous KR formalisms, and at the same time be able to incorporate knowledge obtained from dynamic observations.
In such dynamic scenarios, where systems can have alternative evolutions, it is desirable to have some criteria of minimal change to be able to compare the possible alternatives. This problem is particularly interesting and non-trivial in dynamic frameworks based on MCSs, not only because of the heterogeneity of KR frameworks that may exist in an MCS – each of which may require different notions of minimal change –, but also because the evolution of such systems is based not only on the semantics, but also on the evolution of the knowledge base of each context.
In this paper, we study minimal change in eMCSs, by presenting some minimal change criteria to be applied to the possible evolving equilibria of an eMCS, and by discussing the relation between them.
The remainder of this paper is structured as follows. After presenting the main concepts regarding mMCSs, we introduce the framework of eMCSs. Then, we present and study some minimal change criteria in eMCSs. We conclude with discussing related work and possible future directions.
2 Multi-Context Systems
In this section, we introduce the framework of multi-context systems (MCSs). Following , a multi-context system (MCS) consists of a collection of components, each of which contains knowledge represented in some logic, defined as a triple where is the set of well-formed knowledge bases of , is the set of possible belief sets, and is a function describing the semantics of by assigning to each knowledge base a set of acceptable belief sets. We assume that each element of and is a set, and we define .
In addition to the knowledge base in each component, bridge rules are used to interconnect the components, specifying what knowledge to assert in one component given certain beliefs held in the components of the MCS. Formally, for a sequence of logics , an -bridge rule over , , is of the form
where is a set of bridge literals of the form and of the form , , with a belief formula of , and, for each , .
Bridge rules in MCSs only allow adding information to the knowledge base of their corresponding context. Note that the condition precisely guarantees that the resulting set is still a knowledge base of the context. In , an extension, called managed Multi-Context Systems (mMCSs), is introduced in order to allow other types of operations to be performed on a knowledge base. For that purpose, each context of an mMCS is associated with a management base, which is a set of operations that can be applied to the possible knowledge bases of that context. Given a management base and a logic , let be the set of operational formulas that can be built from and . Each context of an mMCS gives semantics to operations in its management base using a management function over a logic and a management base , , i.e., is the (non-empty) set of possible knowledge bases that result from applying the operations in to the knowledge base . We assume that . Now, -bridge rules for mMCSs are defined in the same way as for MCSs, except that is now an operational formula over and .
A managed Multi-Context System (mMCS) is a sequence , where each , , called a managed context, is defined as where is a logic, , is a set of -bridge rules, is a management base, is a management function over and .
Note that, for the sake of readability, we consider a slightly restricted version of mMCSs where each is still a function and not a set of functions as for logic suites .
For an mMCS , a belief state of is a sequence such that each is an element of . For a bridge literal , if and if ; for a set of bridge literals , if for every . We say that a bridge rule of a context is applicable given a belief state of if satisfies . We can then define , the set of heads of bridge rules of which are applicable in , by setting .
Equilibria are belief states that simultaneously assign an acceptable belief set to each context in the mMCS such that the applicable operational formulas in bridge rule heads are taken into account.
A belief state of an mMCS is an equilibrium of if, for every ,
3 Evolving Multi-Context Systems
In this section, we recall evolving Multi-Context Systems, which generalize mMCSs to a dynamic scenario in which contexts are enabled to react to external observations and evolve. For that purpose, we consider that some of the contexts in the MCS become so-called observation contexts whose knowledge bases will be constantly changing over time according to the observations made, similar, e.g., to streams of data from sensors.111For simplicity of presentation, we consider discrete steps in time here.
The changing observations then will also affect the other contexts by means of the bridge rules. As we will see, such effect can either be instantaneous and temporary, i.e., limited to the current time instant, similar to (static) mMCSs, where the body of a bridge rule is evaluated in a state that already includes the effects of the operation in its head, or persistent, but only affecting the next time instant. To achieve the latter, we extend the operational language with a unary meta-operation that can only be applied on top of operations.
Given a management base and a logic , we define , the evolving operational language, as
The idea of observation contexts is that each such context has a language describing the set of possible observations of that context, along with its current observation. The elements of the language of the observation contexts can then be used in the body of bridge rules to allow contexts to access the observations. Formally, an observation context is a tuple where is the observation language of and is its current observation.
We can now define evolving Multi-Context Systems.
An evolving Multi-Context Systems (eMCS) is a sequence , such that, for each , is an observation context, and, for each , is an evolving context defined as where
is a logic
is a set of bridge rules of the form
such that , and each , , is either of the form with and a belief formula of , or of the form with and
is a management base
is a management function over and .
Given an eMCS we denote by the set of knowledge base configurations for , i.e., .
Let be an eMCS. A belief state for is a sequence such that, for each , we have . We denote by the set of belief states for .
An instant observation for is a sequence such that, for each , we have that .
The notion of the set of heads of bridge rules of which are applicable in a belief state , cannot be directly transferred from mMCSs to eMCSs since bridge rule bodies can now contain atoms of the form , whose satisfaction depends on the current observation. Formally, given a belief state for and an instant observation for , we define the satisfaction of bridge literals of the form as if and if . The satisfaction of bridge literal of the form depends on the current observations, i.e., we have that if and if . As before, for a set of bridge literals, we have that if for every .
We say that a bridge rule of a context is applicable given a belief state and an instant observation if its body is satisfied by and , i.e., .
We denote by the set of heads of bridges rules of the context which are applicable given the belief state and the instant observation . In the case of eMCSs, the set may contain two types of formulas: those that contain and those that do not. As already mentioned before, the former are to be applied to the current knowledge base and not persist, whereas the latter are to be applied in the next time instant and persist.
Let be an eMCS, a belief state for , and an instant observation for . Then, for each , consider the following sets:
Note that if we want an effect to be instantaneous and persistent, then this can also be achieved using two bridge rules with identical body, one with and one without in the head.
Similar to equilibria in mMCS, the (static) equilibrium is defined to incorporate instantaneous effects based on alone.
Let be an eMCS, and an instant observation for . A belief state for is an equilibrium of given iff for each , there exists some such that .
To be able to assign meaning to an eMCS evolving over time, we introduce evolving belief states, which are sequences of belief states, each referring to a subsequent time instant.
Let be an eMCS. An evolving belief state of size for is a sequence where each , , is a belief state for .
To enable an eMCS to react to incoming observations and evolve, a sequence of observations defined in the following has to be processed. The idea is that the knowledge bases of the observation contexts change according to that sequence.
Let be an eMCS. A sequence of observations for is a sequence , such that, for each , is an instant observation for , i.e., for each .
To be able to update the knowledge bases and the sets of bridge rules of the evolving contexts, we need the following notation. Given an evolving context , and a knowledge base , we denote by the evolving context in which is replaced by , i.e., . For an observation context , given a set of observations for , we denote by the observation context in which its current observation is replaced by , i.e., . Given a knowledge base configuration for , we denote by the eMCS .
We can now define that certain evolving belief states are evolving equilibria of an given a sequence of observations for . The intuitive idea is that, given an evolving belief state for , in order to check if is an evolving equilibrium, we need to consider a sequence of eMCSs, (each with observation contexts), representing a possible evolution of according to the observations in , such that each is a (static) equilibrium of . The current observation of each observation context in is exactly its corresponding element in . For each evolving context , its knowledge base in is obtained from the one in by applying the operations in .
Let be an eMCS, an evolving belief state of size for , and an observation sequence for such that . Then, is an evolving equilibrium of size of given iff, for each , is an equilibrium of where, for each , is defined inductively as follows:
We end this section by presenting a proposition about evolving equilibria that will be useful in the next section. In Def. 10, the number of considered time instances of observations is greater or equal the size of the evolving belief state with the intuition that an equilibrium may also be defined for only a part of the observation sequence. An immediate consequence is that any subsequence of an evolving equilibrium is an evolving equilibrium.
Let be an eMCS and an observation sequence for . If is an evolving equilibrium of size of given , then, for each , and every , we have that is an evolving equilibrium of size of given the observation sequence .
4 Minimal change
In this section, we discuss some alternatives for the notion of minimal change in eMCSs. What makes this problem interesting is that in an eMCS there are different parameters that we may want to minimize in a transition from one time instant to the next one. In the following discussion we focus on two we deem most relevant: the operations that can be applied to the knowledge bases, and the distance between consecutive belief states.
We start by studying minimal change at the level of the operations. In the following discussion we consider fixed an eMCS .
Recall from the definition of evolving equilibrium that, in the transition between consecutive time instants, the knowledge base of each context of changes according to the operations in , and these depend on the belief state and the instant observation . The first idea to compare elements of this set of operations is to, for a fixed instant observation , distinguish those equilibria of which generate a minimal set of operations to be applied to the current knowledge bases to obtain the knowledge bases of the next time instant. Formally, given a knowledge base configuration and an instant observation for , we can define the set:
This first idea of comparing equilibria based on inclusion of the sets of operations can, however, be too strict in most cases. Moreover, different operations usually have different costs, and it may well be that, instead of minimizing based on set inclusion, we want to minimize the total cost of the operations to be applied. For that, we need to assume that each context has a cost function over the set of operations, i.e., , where represents the cost of performing operation .
Let be a belief state for and an instant observation for . Then, for each , we define the cost of the operations to be applied to obtain the knowledge base of the next time instant as:
Summing up the cost for all evolving contexts, we obtain the global cost of given :
Now that we have defined a cost function over belief states, we can define a minimization function over possible equilibria of eMCS for a fixed knowledge base configuration . Formally, given an instant observation for , we define the set of equilibria of given which minimize the global cost of the operations to be applied to obtain the knowledge base configuration of the next time instant as:
Note that, instead of using a global cost, we could have also considered a more fine-grained criterion by comparing costs for each context individually, and define some order based on these comparisons. Also note that the particular case of taking for every and every , captures the scenario of minimizing the total number of operations to be applied.
The function allows for the choice of those equilibria which are minimal with respect to the operations to be performed to the current knowledge base configuration in order to obtain the knowledge base configuration of the next time instant. Still, for each choice of an equilibrium , we have to deal with the existence of several alternatives in the set . Our aim now is to discuss how we can apply some notion of minimal change that allows us to compare the elements . The intuitive idea is to compare the distance between the current equilibria and the possible equilibria resulting from the elements in . Of course, given the possible heterogeneity of contexts in an eMCS, we cannot assume a global notion of distance between belief sets. Therefore, we assume that each evolving context has its own distance function between its beliefs sets. Formally, for each , we assume the existence of a distance function , i.e., satisfying for all the conditions:
There are some alternatives to extend the distance function of each context to a distance function between belief states. In the following we present two natural choices. One option is to consider the maximal distance between belief sets of each context. Another possibility is to consider the average of distances between belief sets of each context. Formally, given and two belief states of we define two functions and as follows:
We can easily prove that both and are indeed distance functions between belief states.
The functions and defined above are both distance functions, i.e., satisfy the axioms (1- 4).
We now study how we can use one of these distance functions between belief states to compare the possible alternatives in the sets , for each . Recall that the intuitive idea is to minimize the distance between the current belief state and the possible equilibria that each element of can give rise to. We explore here two options, which differ on whether the minimization is global or local. The idea of global minimization is to choose only those knowledge base configurations with , which guarantee minimal distance between the original belief state and the possible equilibria of the obtained eMCS. The idea of local minimization is to consider all possible tuples with , and only apply minimization for each such choice, i.e., for each such knowledge base configuration we only allow equilibria with minimal distance from the original belief state.
We first consider the case of pruning those tuples with which do not guarantee minimal change with respect to the original belief state. We start with an auxiliary function. Let be a belief state for , a knowledge base configuration for , and an instant observation for . Then we can define the set of knowledge base configurations that are obtained from given the belief state and the instant observation .
For each choice of a distance function between belief states, we can define the set of knowledge base configurations which give rise to equilibria of which minimize the distance to the original belief state. Let be a belief state for , a knowledge base configuration for , and and instant observations for .
|such that there is no|
Again note that applies minimization over all possible equilibria resulting from every element of . Using , we can now define a minimal change criterion to be applied to evolving equilibria of .
Let be an eMCS, an observation sequence for , and let be an evolving equilibrium of given . We assume that , with , is the sequence of knowledge base configurations associated with as in Definition 10. Then, satisfies the strong minimal change criterion for given if, for each , the following conditions are satisfied:
We call this minimal change criterion the strong minimal change criterion because it applies minimization over all possible equilibria resulting from every possible knowledge base configuration in .
The following proposition states the desirable property that the existence of an equilibrium guarantees the existence of an equilibrium satisfying the strong minimal change criterion. We should note that this is not a trivial statement since we are combining minimization of two different elements: the cost of the operations and the distance between belief states. This proposition in fact follows from their careful combination in the definition of .
Let be an observation sequence for . If has an evolving equilibrium of size given , then there is at least one evolving equilibrium of size given satisfying the strong minimal change criterion.
Note that in the definition of the strong minimal change criterion, the knowledge base configurations , for which the corresponding possible equilibria are not at a minimal distance from , are not considered. However, there could be situations in which this minimization criterion is too strong. For example, it may well be that all possible knowledge base configurations in are important, and we do not want to disregard any of them. In that case, we can relax the minimization condition by applying minimization individually for each knowledge base configuration in . The idea is that, for each fixed we choose only those equilibria of which minimize the distance to .
Formally, let be a belief state for , a knowledge base configuration for , and an instant observation for . For each distance function between belief states, we can define the following set:
Using this more relaxed notion of minimization we can define an alternative weaker criterion for minimal change to be applied to evolving equilibria of an eMCS.
Let be an eMCS, and be an observation sequence for , and be an evolving equilibrium of given . We assume that , with , is the sequence of knowledge base configurations associated with as in Definition 10. Then, satisfies the weak minimal change criterion of given , if for each the following conditions are satisfied:
In this case we can also prove that the existence of an evolving equilibrium implies the existence of an equilibrium satisfying the weak minimal change criterion. Again we note that the careful combination of the two minimizations – cost and distance – in the definition of , is fundamental to obtain the following result.
Let be an observation sequence for . If has an evolving equilibrium of size given , then has at least one evolving equilibrium of size given satisfying the weak minimal change criterion.
We can easily prove that, in fact, the strong minimal change criterion is stronger than the weak minimal change criterion.
Let be an eMCS, and be an observation sequence for , and be an evolving equilibrium of given . If satisfies the strong minimal change criterion of given , then satisfies the weak minimal change criterion of given .
We end this section by briefly discussing a global alternative to the minimization of costs of operations. Recall that both and combine minimization of distances between belief states, and minimization of costs of operations. The minimization on costs is done at each time instant. Another possibility is, instead of minimizing the cost at each time instant , to minimize the global cost of an evolving equilibrium.
We first extend the cost function over belief states to a cost function over evolving belief states. Let be an evolving belief state for , and an observation sequence for such that . Then we can define the cost of given :
Let be an observation sequence for and let . We now define the set of evolving equilibria of size of given which minimize the total cost of the operations generated by it.
Since the minimization on costs of operations is orthogonal to the minimization on distances between belief states, it would be straightforward to use to define the global cost versions of and , and use them to define the respective strong and weak minimal change criteria.
5 Related and Future Work
In this paper we have studied the notion of minimal change in the context of the dynamic framework of evolving Multi-Context Systems (eMCSs) . We have presented and discussed some alternative definitions of minimal change criteria for evolving equilibria of a eMCS.
Closely related to eMCSs is the framework of reactive Multi-Context Systems (rMCSs) [5, 9, 8] inasmuch as both aim at extending mMCSs to cope with dynamic observations. Three main differences distinguish them.
First, whereas eMCSs rely on a sequence of observations, each independent from the previous ones, rMCSs encode such sequences within the same observation contexts, with its elements being explicitly timestamped. This means that with rMCSs it is perhaps easier to write bridge rules that refer, e.g., to specific sequences of observations, which in eMCSs would require explicit timestamps and storing the observations in some context, although at the cost that rMCSs need to deal with explicit time which adds an additional overhead. Second, since in rMCSs the contexts that result from the application of the management operations are the ones that are used in the subsequent state, difficulties may arise in separating non-persistent and persistent effects, for example, allowing an observation to override some fact in some context while the observation holds, but without changing the context itself – such separation is easily encodable in eMCSs given the two kinds of bridge rules, i.e., with or without the operator. Finally, the bridge rules with the operator allow the specification of transitions based on the current state, such as for example one encoded by the rule , which do not seem possible in rMCSs. Overall, these differences seem to indicate that an interesting future direction would be to merge both approaches, exploring a combination of explicitly timestamped observations with the expressiveness of the operator.
Also interesting is to study how to perform AGM style belief revision at the (semantic) level of the equilibria, as in Wang et al , though necessarily different since knowledge is not incorporated in the contexts.
Another framework that aims at modeling the dynamics of knowledge is that of evolving logic programs EVOLP focusing on updates of generalized logic programs. Closely related to EVOLP, hence to eMCS, are the two frameworks of reactive ASP, one implemented as a solver clingo  and one described in . The system clingo extends an ASP solver for handling external modules provided at runtime by a controller. The output of these external modules can be seen as the observations of EVOLP. Unlike the observations in EVOLP, which can be rules, external modules in clingo are restricted to produce atoms so the evolving capabilities are very restricted. On the other hand, clingo permits committing to a specific answer-set at each state, a feature that is not part of EVOLP, nor of eMCSs. Reactive ASP as described in  can be seen as a more straightforward generalization of EVOLP where operations other than for self-updating a program are permitted. Whereas EVOLP employs an update predicate that is similar in spirit to the predicate of eMCSs, it does not deal with distributed heterogeneous knowledge, neither do both versions of Reactive ASP. Moreover, no notion of minimal change is studied for these frameworks.
The dynamics of eMCSs is one kind of dynamics, but surely not the only one. Studying the dynamics of the bridge rules is also a relevant, non-trivial topic, to a great extent orthogonal to the current development, which nevertheless requires investigation.
Another important issue open for future work is a more fine-grained characterization of updating bridge rules (and knowledge bases) as studied in  in light of the encountered difficulties when updating rules [24, 25, 27] and the combination of updates over various formalisms [25, 26].
Also, as already outlined in , we can consider the generalization of the notions of minimal and grounded equilibria  to eMCSs to avoid, e.g., self-supporting cycles introduced by bridge rules, or the use of preferences to deal with several evolving equilibria an eMCS can have for the same observation sequence.
We would like to thank the referees for their comments, which helped improve this paper considerably. Matthias Knorr and João Leite were partially supported by FCT under project “ERRO – Efficient Reasoning with Rules and Ontologies” (PTDC/EIA-CCO/121823/2010). Ricardo Gonçalves was supported by FCT grant SFRH/BPD/47245/2008 and Matthias Knorr was also partially supported by FCT grant SFRH/BPD/86970/2012.
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