In this section knowledge representation with the use of attributes is outlined. Such type of knowledge representation seems to be prevailing in knowledge engineering due to its simple, intuistic interpretation and wide applicability. Although the particular form of displaying information (notation variants) may be different, the formalism is basically the same over numerous knowledge representation formalisms. In general, it is applicable to both qualitative and quantitative data. Below a simple, generic form and its extensions are presented.

Let C denote a set of objects (elements) of interest; in case of supervision these are usually the subsequent states. These can also be physical components of the considered system or abstract concepts like performance indices characterising system behaviour. The elements of C will be further described by providing values of some attributes applicable to their characterisation. Attributes are just any preselected properties taking specific values at certain instants of time.

Let A denote a set of attributes selected to describe important features of the system under consideration, . For any attribute  let  denote a (finite) set of possible values of this attribute, or in case of real or integer numbers let it be some interval. For avoiding triviality, it is assumed that any set  contains at least two different elements. Further, functional character of the attributes is assumed, i.e. at some instant of time for any object , if  and  then .

Moreover, for the sake of simplicity it is assumed that any attribute is applicable to any object; an extension to a more general case where any object has a specific sets of attributes applicable to it is straightforward. The values of an attribute can be just listed in a set, or some order of them may be established.

A basic knowledge representation item consists of specification of some element , its attribute and a value of this attribute. Such a triplet constitutes a fact; in logical terms it constitutes an atomic formula. For intuition, the meaning of such a formula is that the value of the specified attribute for a given element is just the one provided; thus the basic relation here is equality, taken in the sense of assignment of a value to a function. Thus the basic atomic formula is always of the form:
 
 

For example, colour(light) = red denotes the fact that the value of attribute colour for light red. For attribute such as temperature and object being coolant a fact like, for example temperature(coolant) = upper_limit might denote some dangerous situation when the temperature of cooling liquid reaches some predefined upper limit. For numeric variables the representation is similar, e.g. in order to denote the fact that current speed amounts to 60 (typically written as v=60), one can construct the description value(speed)=60.

Note that when characterising one, specific object, the specification of its attributes can be even simpler, i.e. it can take the form:
 
 

If the values of two or more attributes are specified, then such a specification forms the so called conjunction of facts, a notion imported from logic. A conjunction represents a set of facts holding simultaneously. Such a conjunction of facts will be called also a simple fact formula or simple formula for short. Conjunction is denoted with the symbol . A simple formula is always of the form , where any  is a fact.

Knowledge representation with attributes is sometimes called the Object-Attribute-Value (OAV) or the Attribute-Object-Value (AOV) formalism.

Simple formulae characterise current state of a system (a more detailed discussion, definitions and a formal treatment of the problem of logical state representation has been described in the former section around in this paper). Since a simple formula constitutes in fact an abstract characterisation (usually only selected parameters/features are taken into account), as we mentioned it can in fact refer to a situation including a great number of real states.

If all the attributes apply to all objects, a complete formula characterising all the values of the attributes for all the objects displayed in form of "linear" conjunction would be clumsy and perhaps difficult to read; in such a case a tabular representation is much more transparent. A tabular form of such simple formula can look as follows:
 
 

In practice, some of the attributes may be unimportant or inapplicable to certain objects (e.g. the colour of certain elements may be not important and thus not specified (for example the weight is an attribute not applicable to an element like light). In the first case one can use the sign "_'' while in the second one the "* ". Such a weakened description forms a more general (abstract) formula.

It is often still more convenient to use a simplified tabular form similar to tables in relational databases. In such a form the columns are marked with attributes, while any row provides description of certain object. The simplified tabular form is still more readable and it allows not to repeat the names of the attributes and the names of the objects. The form is as follows:

In this subsection an extended language for knowledge representation with use of attributes is presented. The basic extension consists of admitting imprecise characterisation of the values of attributes for certain objects by allowing for the use of sets. It is no longer assumed that the value of certain attribute for a given object is given precisely. As a generalisation a set of possible values is specified rather than single value.

As in the former section let C denote a set of objects (elements) of interest. The elements of C will be described by providing values or sets of values of some attributes applicable to their characterisation. Let A denote a set of selected attributes, . For any attribute  let  denote a (finite) set of possible values of this attribute. As before, functional character of attribute values is assumed and for simplicity it is assumed that any attribute is applicable to any object.

As an extension with respect to the former subsection, a basic knowledge representation item (fact, atom) consists of any element, an attribute selected to describe it and a set of possible values of this attribute (if applicable). Such a specification of knowledge item is still called a fact or atomic formula (atom, for short). For intuition, the meaning of such a fact is that the value of the specified attribute for a given element is equal to one of the elements of the specified set. Hence the basic relation here is inclusion and the standard form of any atomic formula is always as follows:
 
 

Note that from a formal point of view facts defined as above are not a direct subclass of atoms in pure first order logic; with respect to the intended interpretation, since sets are used as arguments, specific reasoning mechanisms should be applied. For example, given two facts as and no purely logical reasoning mechanism would be able to deduce q from p, i.e. to show that pÆ q (where Æ is the symbol denoting logical consequence). In order to do this, one should transform the above atoms into "equivalent" logical formulae of the form and respectively. The meaning of an element belonging to a set of possible values is that some of the values is taken, i.e. it corresponds to logical disjunction (). Such a transformation, however, would lead normally to clumsy and long expressions, and in some cases it may be impossible (i.e. if the specified set is infinite). Thus specific inference rules should be provided.

Depending on the current needs one can admit various notation possibilities, e.g. if the set of values for attribute a is ordered, one can use typical algebraic symbols as , , , . For example, if , then the fact  can be denoted as , etc.

Note that for several facts having the same object and attribute but different sets of values, a partial order relation can be established. A more general fact will admit a wider set of possible values. The following definition introduces formally the concept of generalisation.

Definition. Consider two facts  and . Fact q is said to be more general than fact p (also fact p is said to be more specific than fact q) if and only if .

Note that if the appropriate set is not specified (i.e. an unconstrained variable is given), then one can replace it with the domain of the attribute, i.e. the set  and the generated atomic formula is equivalent to the former one. This kind of convention can be used to simplify further consideration. Note that a more general fact logically follows from a more specific one (i.e. pÆ q) provided that the specific interpretation referring to the appropriate sets of values is considered.

The above definition allows for simple check of generalisation amongst facts, provided that the appropriate sets are given explicitly. In case the sets are specified implicitly, specific, case-dependent reasoning procedures should be applied. For example, if the sets are specified as intervals, interval inclusion should be verified; this can be done by comparison of their boundary values.

As before, several single facts can be used to form simple conjunctive formulae. Similarly to the former subsection, array (tabular) representation of simple fact formulae can be admitted. An interesting problem consists in comparing such formulae with respect to which of two simple formulae is more general than the other, or to check if generalisation holds. As before, more general formula describes potentially more items, since it imposes weaker conditions. The generalisation for simple formulae will be defined as below.

Definition. Consider two simple formulae  and . Formula  is said to be more general than formula  (also formula  is said to be more specific than formula ) if and only if for any fact  of formula  there exists some fact  in formula , such that  is more general than  (Æ).

Note that generalisation defined as above is equivalent to logical entailment, i.e. a more general formula logically follows from a more specific one; thus the notation  Æ will also be used.

In the above definition it is assumed that both the above formulae are specifically reduced, i.e. no two facts in a single formula refer to the same combination of object an attribute. For example, if  then reduced (but equivalent) form of  would be . One can see that for simple formula  the generalisation was not holding according to for the non-reduced form of ; however for reduced form there is  Æ, since . From now on only reduced forms of simple formulae will be considered.

Checking if generalisation holds for simple formulae is a crucial test in situation recognition, checking if precondition of certain formula are satisfied, etc. Hence, the definitions introduced in this section constitute an important foundations for inference in knowledge based systems.