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Precise Enterprises and Imperfect Data P. Chountas & I. Petrounias Department of Computation, UMIST, PO Box 88, Manchester M60 1QD, UK e-mail: {chountap, ilias}@sna.co.umist.ac.uk Fax: + 44 161 200 3324, Tel: + 44 161 200 3386 Abstract One of the main uses of an information system is the representation and management of large amounts of indicative information from multiple sources describing the state of some enterprise. Most current information systems model enterprises that are crisp. A crisp enterprise is defined as one that is highly quantifiable; all relationships are fixed, and all attributes are atomic valued. The premises on which this paper is based are precise enterprises, where data are imperfect. In such cases information can be certain, imprecise or uncertain, temporal, or any possible combination of each two of them, depending on the application domain. Additionally, in domains where the information is perfect, all information sources are absolutely reliable and trustworthy. In more speculative domains, like diagnosis, where, information may be asserted relatively to some time intervals in which it is possibly defined and probably believed. In such domains different sources of information may be assigned different degrees of reliability. This paper is presenting a framework for the conceptual integration and uniform treatment of all these types of information. Keywords Value imperfection, temporal imperfection, multiple information sources, consistency principle, conceptual modelling, nested relational databases. 1. Introduction Imperfect information is the partial knowledge of the true value of the real world. It is an epistemic property caused by lack of information. Elements of the enterprise ontology, involved in information imperfection are: • The might happen ability of things, or the tendency of things to occur. • The concept of time • The information source or provider A database attempts to represent an abstract version of the enterprise reality; the level of it is determined by the expected applications. Work in this area is considering imperfect information arising from elements of the enterprise ontology but always in isolation form each other. This paper is suggesting that the elements involved in information imperfection are related and affecting each other and concepts at the specification level. More than one information provider might be describing the same fragment of information, expressing logical views about the same facts that are defined over the 3-dimensional space of time, belief and reliability, named as the multidimensionality of the source. Facts are expressing associations between objects but not in isolation, always in some form of relationship with each other. The rest of the paper is organised as follows. Section 2 evaluates existing work dealing with imperfect information. Section 3 presents a framework for dealing with imprecise temporal information. Section 4 presents a formalism for capturing and representing temporal and value imperfection in a multisource environment, where each source is carrying a degree of reliability. Section 5 provides the mapping of the formalism to a nested relational database. Section 6 suggests a set of NF2 algebraic operators. Section 7 points at work in progress. 2. Related Work The main stream of researchers is dealing with either temporal or value imperfection. The difference between temporal and value imperfection can be characterised as that between “do not know what” and “do not know when” information. Approaches for representing temporal or value imperfection can be weighted or unweighted. Weights are normalised values in the range [0,1]. Weights are assigned to alternatives or possibilities of an imperfect value. For a possibility a weight deals with the might happen ability of the possibility to be the actual value. Unweighted imperfect information may be either restricted or unrestricted [1]. Other researchers are considering the existence of multiple conflicting sources accommodated in a database. All these are assuming that there are no intentional inconsistencies between different sources. It is assumed that the point of conflict is that two or more information sources provide different answers to the same query (extensional inconsistencies), provided that those sources do not have internal disharmonies (internal extensional inconsistencies). This section evaluates approaches in the literature according to the following criteria: relevance to real world representation of concepts, power offered by the proposed model, minimality of concepts (formalisms should not contain overlapping concepts), formality in the representation to avoid ambiguities. In addition, it examines whether support for temporal uncertainty is offered by the models and whether uncertainty about both temporal and value aspects is supported. Finally, models are evaluated as to whether they offer support for multiple information sources. 2.1 Possibilistic or Fuzzy Databases When considering uncertain information enterprises are considered as either precise or vague. In vague enterprises it is assumed that attribute values are not precise and are presented as linguistic terms. Early possibilistic approaches extended the relational model with the acceptance of fuzzy functional [2] and fuzzy multivalued dependencies [3]. Based on the definition of a fuzzy resemblance relation EQ(UAL) over domains of attributes, a set of inference rules for fuzzy functional and multivalued dependencies is proposed ([2], [3]). The main disadvantage of both is that they support only a limited number of possible associations between the elements of the application domain in order to keep the models in 1NF. Recently these ideas moved towards the support of similarity relations in a nested relational model for representing uncertain, complex data [4]. In the case of precise enterprises, information can be certain, uncertain or imprecise. The key notion is that while one value applies to the enterprise, the database extension may contain a set. The application of imprecise or uncertain information to the precise enterprise means that the value in the database is a possibility distribution. This is taken to show the limits of knowledge concerning the actual value and the significance of ordering. Other research has made attempts to identify whether the uncertainty property can be presented as part of ER diagrams [5]. This is influenced by approaches for treating uncertainty proposed by the AI community [6] and is trying to embody linguistic terms, as part of the ER formalism or as part of the relational theory. 2.2 Probabilistic Databases The key notion here is that while one value applies to the certain enterprise, the database extension may contain a set (probabilistic distributions). In that way value imperfection is accommodated. The field of probabilistic databases covers a wide spectrum of different approaches. Probabilistic weights are used to express that an attribute value can be a set of alternative data values ([7], [8]) or to express the likelihood that a tuple belongs to a relation. Other approaches [9] are using separate probabilistic weights to express the logic view that a tuple belongs to a relation and different probabilistic weights to express the intent that an attribute value may be a set of alternative data values. There is a debate on whether an interval of probabilities or a single probability is better for expressing the tendency of things to occur ([9], [10]). There is also concern if events should be considered dependent or independent. However, uncertainty is treated only at the database level ignoring the specification level, leading to complex probabilistic reasoning with no knowledge of the primitive notions of the model that can produce imperfect information. The main issue is if the model is in 1NF or in NF2. It is argued in this paper that imperfection should firstly be considered at the conceptual level. If imperfection is not accommodated by conceptual modelling formalisms (i.e. ER diagrams, object role modelling approaches) then it cannot appear in the resulting databases (which are after all the result of a mapping from the conceptual schema). 2.3 Temporal Probabilistic Databases While in reality a time interval applies to an event, in temporal probabilistic databases the database extension may contain a set of possible intervals. In valid time indeterminacy [1] it is known that an event did in fact occur but is not known exactly when. The model is presented as an extension of the SQL data model. If a tuple k in relation R is timestamped with the interval [t1...t2], then this is interpreted as tuple k holds at some point t in interval [t1...t2]. Query constructs are defined to specify belief (Correlation credibility) in the underlying data and their plausibility (Ordering Plausibility) in the relationships among the data. However, valid time indeterminacy is treated at the database level instead of arising from the conceptual level that states exactly which primitive notions may be involved in valid time indeterminacy. A probabilistic temporal algebra is suggested in [11] for expressing information of the following type: tuple d is in relation R at some point of time in interval [t1, t2] with probability between p1 and p2. A range of probability distributions is supported to allocate the probability measure over the set of time points of the interval. Different valid times related to a tuple may have different probability distributions in nature. The main problem in ([1], [11]) is that if the type of the probability distribution is known then it is known beforehand that some time points in an interval have greater probability, thus a subinterval of the initial time interval is more probable. Therefore, there is a fact somewhere that makes our knowledge about the real world more explicit but it is not present in the conceptual schema or in the database. Temporal probabilistic databases are a natural extension of probabilistic databases. Imperfection of the information is treated only at the database level ignoring the specification level, leading to complex probabilistic reasoning with no explicit specification of concepts. 2.4 Databases with Multiple Information Sources The key notion here is the representation of a certain enterprise, where only one value applies, while the database extension may contain a set because of different conflicting sources. The IST approach [12] is using information source vectors to accommodate multiple conflicting sources and define the conditions under which a tuple is valid. Each attribute value in a tuple is associated with an information source vector to state whether an attribute value is valid, therefore certainty about certainty can be expressed. [13] assumes that data models can be mapped, resolving only existential inconstancies. Both approaches are ignoring intentional inconsistencies between different sources since both approaches are treating conflicting values at the database level ignoring the specification level. Furthermore, it is assumed that there are no internal extensional inconsistencies. Both are modelling the certain world. Models are trying to resolve information coming from different sources, which are conflicting. 3. Considerations for a Dynamic Conceptual Model In any enterprise environment of multiple information sources it is undeniable that more than one sources describe the same portion of the enterprise world differently. The conceptual model is acting as a gateway between different sources, permitting different sources to express information in a single and highly abstract level, the level of concepts (metamodel level). The approach followed here is based on a type of object role modelling formalism [14]. A fact is a true logical proposition about the modelled world. Each fact instance is a semantically irreducible proposition in the real world about one or more entity instances. Irreducible means that the fact cannot be split into facts involving fewer entities without loss of information. A dynamic database environment is presenting certain or plausible information about the past and present of the modelled world. Therefore, there is a need to express imperfect information as a part of the fact formalism and to identify the impact of the time and belief dimension on it, before proceeding with database considerations: • If a fact is related to the belief dimension, with a degree of belief less than one it is simply declared that an association between objects possibly stands in the enterprise world (value imperfection). • If a fact is linked with the time dimension, it is simply declared that a certain association between objects is valid for a certain time period. However, if the time dimension is associated with the belief dimension it is simply declared that a certain fact is possibly defined over that period (temporal imperfection). In most of the research proposals only value or temporal imperfection can be expressed. This paper suggests that both kinds of imperfection (value and temporal) can be represented in a database environment. • If a fact is associated with the belief dimension and the time that a fact is defined over (valid time according to the temporal database literature [1]) is also allied to the belief dimension, then both value and temporal imperfection can be expressed. Expressing temporal and value imperfection simultaneously, permits the representation of statements arising from everyday enterprise activities. • In cases of either value or temporal imperfection the belief dimension is affected by the reliability of the source. The reliability of the source is expressing the humans’concern about the identity and trustiness of the source that is responsible for a particular piece of information. 4. The Temporal Multisoucre Belief Model (TMBM) The basic items that one wishes to reason about are objects in terms of the roles that they play within a domain [14]. In general a fact type is composed of the arguments shown in Figure 1, where n is the arity of the fact type. The way that one can refer to specific entities is through reference labels. If the modelled world is certain then the label value is a single value. The time interval ∆t that a fact instance is defined over has an explicit duration since both ends of the interval are unambiguously defined in the time line. If the modelled world is imperfect then a label value may be a possible multiset of values (π_Label Type) [15]. Each member (value) of the possible multiset is an alternative value with an indicative belief. The time interval that a fact is defined over, is an alternative from a multiple set of time intervals, with an indicative belief. In either the certain or imperfect modelled world the reliability of the source is forming the conclusive belief for the timestamped fact. A graphical representation of the concepts is shown in Figure 2. Fact types can be of any degree [14]. For example, (Figure 3) in a ternary fact type, there will be three entity types involved with three different roles. The relationship between two entity types of the ternary fact can be regarded as an entity type itself (objectified fact type). F = {{{<E 1 , L 1 , R 1 >,… , <E n , L n , R n > },{ ∆ t ≤ T} }, m} w here : F is a fact type consisting of k fact instances T is the time interval that an irreducible fact type is defined in the real world. E i is the ith entity type playing a role in the fact type L i is the label type (referencing Ei) R i is the ith role of the fact type ∆ t is the time interval that an irreducible fact instance is defined in the real world m is the reliability of the source that circulates a particular fact type. The reliability of the source is a domain independent variable. Figure 1: Fact Types Role-1 Entity type T Entity type Label Type Role-2 π-Label Type m Figure 2: Graphical Notation of a Fact type In Figure 3, the entity types (E) are (Supplier, Product, Location) and the corresponding reference labels are (Supplier-Name, π (Product–Name), City-Name). Supplier-Name and City- Name are deterministic label types. Product-Name is a stochastic label type (π). The meaning of a stochastic label type is that a label value can take a possible (π) set of values and each member of the set is an alternative value with an indicative belief (probability, (p)). Based on the possibility/ probability consistency principle [16] a connection between the measure of randomness (p) or observation and compatibility (π) can be achieved. In this way a fact is presenting information that is observed and testified by one or more information sources therefore a set of alternatives is defined with p > 0 and π = 1. A fact may also represent information that is compatible with its domain, p = 0 and 0 < π < 1 based on some specified or unspecified criterion. However, the information source cannot testify these values, but does not have any reason to reject them. In that way information that is more elementary and less context dependent can be represented. Any instance of the (SALE-LOCATION) in Figure 3 must exist during the period (or at the same period) that the corresponding SALE fact type exists. The following relationship between T and T1 must exist: (T1 during T) or (T1 same as T). A time period is defined as a temporal constraint over a linear hierarchy of time units, denoted Hr. H r is a finite collection of distinct time units, with linear order among those units. For instance H1 = day ⊆ month⊆ year, H2 = minute⊆ hour⊆ day⊆ month ⊆ year are all linear hierarchies of time units defined over the Gregorian Calendar. A time point in a linear hierarchy is simply an instantiation of each time unit in Hr. A calendar r consists of a linear hierarchy Hr of time units and a validity predicate that specifies a non empty set of time points. In that way an application may assume the existence of an arbitrary but fixed calendar. m SUPPLIER (S) Sells T PRODUCT (P) Supplier-Name Is related to π (Product –Name) SALE T1 Committed in LOCATION City -Name Figure 3: Uncertain Timestamped Fact In case of temporal imperfection, the time interval over which a fact instance is valid, is accompanied by an indicative belief (probability) that the relationship between T and T1 still holds. The validity lifespan that a concept is defined (e.g. SALE) over is the union of the time intervals that ‘sale’ instances are believed to be valid. If an entity type is involved in non-timestamped facts, the interval [now - t1, now] is awarded to non-timestamped fact types, where t1 is the smallest granularity of all timestamped facts that the entity type participates in. A snapshot fact keeps only current information. The reliability of a provider or source (m), (Figure 3) is acting as a creditor of trust towards facts expressed by this particular source and is associated with an instantaneous event e. The event supplies the system with the reliability of the source. The time point (te) in the time line associated with the particular event is recorded. The interval [te, now] is the valid period that the reliability measure is defined for a particular source. The ‘ now’upper bound, will updated to te1, when te1 is the time point another instantaneous event e1 is triggered and subsequently modifies the reliability measure of the source. 5. Mapping to a Nested Relational Model The 1NF relational model is simple and mathematically tractable but not rich enough to model complex objects. In order to represent complex objects hierarchical structures are used instead of flat tables [17]. A relation schema R is recursively defined as: i. If {A1 , … , An}⊂ U and A1… An are atomic valued attributes then R={A1,..., An} is a relation schema. ii. If {A1, … , An}⊂ U and A1… An are atomic valued attributes and R1,… ,Rn are relation schemas then R=( A1,… ,An,R1… .Rn ) is a relation schema . The atomic valued attributes A1,… ,An are called zero order attributes. R1,… ,Rn are called relation-valued attributes or high order attributes. m Supplies T SUPPLIER PRODUCT Supplier-Name Sale Is related to π (Product –Name) Figure 4: The Sale Fact type Consider fact type Sale in Figure 4. Supplier-Name is an atomic value attribute. π(Product – Name) declares that a single label value can be a possible (π) set of values and each member of the set is an alternative value with an indicative belief (probability, (p)). Product is a relation schema or a high order attribute. The time interval ∆t⊂ T that a fact instance is defined over may be explicitly known or may be a set of possible time intervals (π) where each interval is accompanied by an indicative belief. T is represented by a high order attribute. Another separate relation represents the 1). information source and its reliability (0< M≤ M affects only the probability measure. In Figure 5, a sample population for the fact type Sale is presented. In it two kinds of value imperfection found in 1) the real world can be modelled: the possible (π=1) and probable (0<p≤ or the possible (0<π<1) and unexpected, improbable (p=0). In Figure 6 Sale is presented as a hierarchical structure. A node can be either an atomic value attribute or a relation. Timestamped Fact Type Sale Valid Time Multisource Fact Type Sale ∆t / (p) Source Supplier-Name π(Product-Name) Name / Probability(p) Source Possibility Water / 0.5 Ivi Wine / 0.2 John 1 Oil / 0.3 Minerva [10/06/99,15/08/99] / 0.6 Ivi, John [10/07/99,15/10/99]/ 0.4 Minerva Amber Smith Cigarettes / 0 Paul 0.5 Figure 5: Multisource Timestamped Fact Type Sale 6. A Recursive NF2 Algebra A set of relational operators (Select, Project, Cartesian product and Join) is presented with the emphasis on processing queries which include join operations in the nested relational model. Operators are recursively defined so that each operator can be applied to subrelations at all levels. Selection (σ): For all nodes ∈ node S where Sa≠ Sb, if node Sa is a child of an ancestor of a node Sb, then Sa, Sb are called selection comparable nodes (Sa σ→ Sb). Timestamped Relation Sale (S) Valid Time (R1) Sale (R2) Valid ∆ t / (p) Source (SP) Supplier -Name π (Product –Name) Name / Probability (p) Source (SP) Possibility Figure 6: Nested schema tree for value and temporal imperfection For example, in figure 6 (Valid Time(R1)σ→ Supplier–Name) and (Valid Time(R1)σ→ π(Product –Name)) are selection comparable notes. Since there is a path between π(Product–Name) and Valid Time (R1) then (R1) is also comparable to Name/Probability (p). However Valid ∆t/(p), Supplier–Name are not selection comparable nodes. Selection conditions are comparisons between attributes and constants and may include also membership operators. ): Projection (π′ A projection operation is a way of accessing attribute values or relation schemas from the outermost level to the innermost level. A projection can be defined as a nesting of multiple projections in the attribute domains of a relation schema. Many project operators have been proposed in the context of nested relational models [18] but existing projection operators deal only with projection of attribute values based on a selection condition that is defined on the attribute domain (e.g. π′ (Supplier Name = ‘ John Smith’)). For all nodes ∈ node (S) if two nodes are selection comparable notes then the projection operator is defined. In figure 6 (Supplier Name π′ π(Product–Name), (π(Product– → π′ Name) → Possibility) are selection comparable notes. In this case the project operator is defined as an ordered sequence of zero level attributes and relation valued attributes (section 5). Projection operators can be either simple or complex. A simple projection involves a one level vertical or horizontal path (e.g. Supplier-Nameπ′ π(Product–Name)). In this example, for a → Supplier-Name instance the whole relation valued attribute π(Product–Name) is derived. A complex projection involves the derivation of values through paths in the tree hierarchy (e.g. Supplier Nameπ′ SP (Source Identity/reliability). Duplicates are not eliminated in the case that the → values of the timestamps are different or the conclusive beliefs are different. Cartesian Product (×ε): The idea behind the extended Cartesian product is to combine relations with common high order attributes not only at the top level but also at the subschema level. Let R be the relational relation schema and T be the schema tree of R, the path Pr = (M1...Mk) is a join-path of R if M1 is a child of root (T) and Mk is a non-leaf node of T. Path expressions describe routes along the composition hierarchy and expressions describe links between attribute domains. They flatten any nested relation structure in one way – no need to break paths in the schema into several expressions and apply a fold up operator to each one. The idea is to combine to high order relational attributes not only at the top level but also at the subschema level. The definition of the Cartesian Product does not have any major practical value, since it is clearly a mathematical operation. However, it underlines the theoretical framework for defining the P Join operator. P Join (ρ×): The same attribute names in two join relations may appear in multiple subtrees. The P join can be extended with multiple path joins, which exploit the more general situation. In Figure 5 only the information sources, are stated and not they reliability. The way that the reliability measure is changing throughout time has been discussed in section 4. Assuming that the following relation describes the information source (Figure 7), a P Join can be used to relate the reliability (m) of the source and the belief expressed for the value or temporal part of a fact instance. The source attribute presented by the Source relation (SP) in Figure 7, is evident in two relational subschemes R1, R2 of Figure 6. A join path between R1, R2 and SP can be defined. Subrelation R1 is expressing the time dimension of the fact type sale. In defining the path join between the relation SP and R1 the following relationship must exist: Valid time (SPi)∩ Valid time (Rti)≠∅ (1). If and only if (1) is true then a path join between R2 and SP can be defined. Otherwise, it is accepted that a source can be temporally imperfect. If the time that an event occurred is not known (no matter if it is known to what extent an event did occur), the provided information is still incomplete. Source (SP) Source Identity / Source Reliability Source Valid time (SP) Figure 7: Relation SP for the Information provider Maybe P Join (mρ×): A maybe P Join is defined as an extension of the P-Join and is also an extended natural join. With the maybe P Join attribute values defined using a probability can encapsulate the belief of the source and express the conclusive belief in a single complex value, thus forming higher level attributes with complex data types. The conclusive belief for a possible value is defined as the product of reliability and probability measure Cp=(p×m). The same applies when two probabilities have to be joined (p1×p2). When elements of different possibility distributions are joined then the min (π1...πn) possibility is the common one. If n sources are expressing the same belief (p) for an attribute value, having different degrees of reliability (m) then the conclusive belief (Cp) of the attribute value is defined by the following formula: Cp =min (m1× p, m2×p… mn×p) (2) The time interval that the conclusive belief is defined is the intersection of the time intervals that the sources (SP1… SPn) are defined. ∆t Cp = ∆t1SP1∩ ∆t2SP2∩ … ..∩ ∆tnSPn (3) It should be stated that (1) must be always true. Figure 8 presents relation (Smρ×SP) after applying the Maybe P Join. Relation S is from figures 5 and 6. Relation SP is from figure 7. Union (∪ ): Union compatibility in the fact formalism (section 4), means that two relations are union compatible if and only if they have the same arity or degree and their corresponding attributes are based on the same domain. Attribute names may not be the same. Attributes may be zero level or (e.g. supplier Name) or relation value attributes π(Product–Name). If two relations are not union compatible the project operator can be used to identify the union compatible attributes of the relation. The defined projection operator is a way of accessing attribute values or relation schemas from the outermost level to the innermost level, thus projecting zero or high order attributes at different levels in the nested schema tree. ((∆t / Cp1), (∆t Cp1, Source Identity)) Supplier-Name π (Product-Name) (((Name /C p),(∆tCp,Source Identity))Possibility) Figure 8: S mρ×SP Maybe P Join Example In Figure 8 projecting the source identity from the outermost level, the sources supplying the possible set of times that the fact sale occurred are known. Applying the project operator in the π(Product–Name), a relation-valued attribute, the sources giving the possibility that a fact instance is defined are also known. Applying the union operator the total population of sources involved in either value or temporal imperfection, or both are derived. Instances of a timestamped fact with the same entity instances involved are considered different if the values of the timestamps are different or the conclusive beliefs are different. Intersection (∩ ): The intersection operation is defined in analogy with the union operator. Relations must be union compatible. Similarly as in the case of the union operator the project can be applied to guarantee union compatible zero level or relation value attributes. In figure 8 projecting the source identity from the outermost level, and intersecting them with the source identity after applying the project operator in the π(Product–Name), the members from the population of sources that are involved in both temporal and value imperfection are derived. Difference (− ): The difference operation accepts as inputs two zero level or relation value attributes and returns instances that it will be members of the population of the first operand that are not members of the population of the second operand. The definition is based on the intuition that two attributes (zero level, relation valued) r, s represents information that two different actors (sources) have about the same world then r− s should represent the information about the real world that r has and s does not 7. Conclusions A single conceptual framework has been proposed for treating either value or temporal information imperfection. A conceptual model describes the real world and descriptions at the database level must be defined according to the conceptual model that states the static and dynamic elements of an application domain that may generate imperfect information. Certainty about certainty can be expressed. An algebra was presented in order to manipulate imperfect information at the database level. Extensional inconsistencies between different sources can be represented and queried. Wok is carried out in order to model and query internal extensional inconsistencies of a source. 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