Approximate Databases: A Support Tool for
Approximate Reasoning
Patrick Doherty* , Martin Magnusson* , Andrzej Szałas*,*** Department of Computer and Information Science,
University of Linköping, SE-581 83 Linköping, Sweden,
email: {patdo,marma,andsz}@ida.liu.se** The University of Economics and Computer Science
Wyzwolenia 30, 10-106 Olsztyn, Poland
ABSTRACT. This paper describes an experimental platform for approximate knowledge databases

called the Approximate Knowledge Database (A KDB), based on a semantics inspired by roughsets. The implementation is based upon the use of a standard S QL database to store logicalfacts, augmented with several query interface layers implemented in JAVA through which extensional, intensional and local closed world nonmonotonic queries in the form of crisp or
approximate logical formulas can be evaluated tractably. A graphical database design userinterface is also provided which simplifies the design of databases, the entering of data andthe construction of queries. The theory and semantics for A KDBs is presented in addition toapplication examples and details concerning the database implementation.
KEYWORDS: approximate reasoning, approximate databases, knowledge representation, secondorder quantifier elimination

1. Introduction
An essential component in many AI agent-based architectures is the agent’s knowledge representation component which includes a variety of knowledge and data repositories with associated inference mechanisms. In the case of robotic systems, theknowledge representation component is often intended to provide models of aspectsof the robot’s embedding environment and its own and other agent capabilities. Designing, specifying and implementing KR components in a robotic system is particularly challenging due to soft and hard real-time constraints and the fact that knowledge
structures are often derived from sensory input and fusion processes.
It is becoming increasingly important to take seriously the gap between access to
low-level sensory data and its fusion and integration with more qualitative knowledge
structures. These signal-to-symbol transformations should be viewed as an on-going
Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

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Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

process with a great deal of feedback between the levels of processing. In addition, because the embedding environments are often as complex and dynamic as those facedby humans, the knowledge representations which are used as models of the environment must necessarily be partial, elaboration tolerant and approximate in nature.
This type of requirement rules out the use of standard relational database technologies as commonly understood, where crisp relations are stored as tables and a standard
closed-world assumption is implicit in the query mechanism associated with a relational database. The focus of our research is to relax many of these standard assumptions in order to meet the requirements associated with the use of knowledge
representation components in robotic systems. For example, approximate relations
will be used instead of crisp relations, an open-world assumption will be used insteadof the closed-world assumption and our query language will support the specificationof non-monotonic queries. In order to leverage the great amount of work already donewith traditional relational database technologies, we will also show how it is possible
to develop a theory of approximate databases and queries, but yet be able to compile such representations into standard SQL queries. In this manner, tractability is
guaranteed through the use of smart encoding techniques.
A concrete domain, where such knowledge representation components are not only
desirable, but necessary, involves the use of an unmanned aerial vehicle (UAV) operating over a road and traffic environment. In this case, the UAV system must be able todynamically construct representations of objects it observes in the world and integratethese representations with other static representations such as concepts and taxonomichierarchies about spatial or temporal relations, normative behavior of vehicles, traffic
representations, etc.
In this case, the meaning of concepts such as fast or slow, small or large vehicle,
near, far, or between, have a meaning different from that in applications with othertemporal and spatial constraints. Assuming these primitive concepts as given and thatthey are continually re-grounded in changes in an operational environment via machine learning or sensor fusion, we would then like to use these primitive concepts
in our knowledge representation structures. Since they are inherently approximate,
any knowledge structure containing these concepts should also inherit or be influenced by these characteristics. For the purposes of this paper, we will assume that theapproximate concepts have been generated in some manner. We will focus on theirrepresentation, storage and use, in what we call approximate databases.
The research described in this paper was initiated within the framework of theW ITAS UAV Project.1 This was a larger project involving the design and use of autonomous unmanned aerial vehicles and the development of an integrated softwarearchitecture for low- and high-end autonomous functionality [DOH 00a, DOH 04a].
1. W ITAS is an acronym for the Wallenberg Information Technology and Autonomous Systems
Laboratory at Linköping University, Sweden. UAV is an acronym for Unmanned Aerial Vehicles. The WITAS UAV Project ended officially in 2004. See http://www.ida.liu.se/~patdo/auttek/ for continued activities in this area.

Approximate databases

3

The research with approximate databases has been greatly influenced by the fact thatthe system had to be integrated and used in a complex UAV system which has been
deployed in a number of complex scenarios involving photogrammetry, surveying,vehicle tracking and multi-platform scanning.
The UAV experimental platform offers an ideal environment for experimentation
with the knowledge representation framework we propose, because the system architecture is rich with different types of knowledge representation structures, the operational environment is quite complex and dynamic, and signal-to-symbol transformations of data are an integral part of the architecture. In addition, much of the knowledge acquired by the UAV is necessarily approximate in nature. In several of thesections in this paper, we will use examples from this application domain to describe
and motivate some of our techniques.
In this paper, we focus on a generalization of deductive database technology suitable for the characteristics of robotic domains mentioned previously. A standard deductive database (see, e.g., [ABI 95, PRZ 90]) can store ground atomic formulas torepresent factual knowledge and use, for example, Horn-clause logic formulas as deductive rules to infer additional facts. Such databases often make assumptions, suchas the assumption that the stored knowledge is precise or complete, that render themless suitable for the contexts we are interested in. Approximate databases, considered
in a number of our publications (see, e.g., [DOH 99, DOH 03b, DOH 04b, DOH 04d,
DOH 04e]) and summarized in book form in [DOH 06], relax many of these assumptions and become applicable in robotics contexts where standard techniques do not.
An important part of our investigation involved the design, development and implementation of a database engine [MAG 05a], called the Approximate Knowledge
Database (A KDB), that serves as an experimental platform for our research. Since thetechnology is built on top of traditional S QL relational database technology, its potential applicability goes well beyond the application domain described here although wehave not yet pursued its use with other domains.
The paper is structured as follows. Section 2 introduces and motivates the concepts
that are used in the remaining parts of the paper. Section 3 provides an overview
of the architecture of approximate knowledge databases. In Section 4 we present
syntax and semantics of approximate knowledge databases. Section 5 discusses therelation of our solutions to other work. In Section 6 the concepts introduced earlier
are illustrated on a surveillance mission case study. Finally, Section 7 concludes thepaper and Appendix A mentions some additional functionality.
An online interface for experimenting with A KDB s is available via
http://www.ida.liu.se/~marma/akdb/

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Semi-Horn Theory,
Minimization Policy,
Crisp Formula Query

Horn-form Deductive Rules,
Approximate Fixpoint QueryApproximate Fixpoint Query

CCQ Database

Circumscription
DLS*

Intensional Database

Inference Mechanism

Extensional Database

Query Compiler

SQL

SQL
Database

Figure 1. An overview of the A KDB architecture.
2. Overview and Preliminaries
2.1. Approximate Knowledge Databases
The approximate knowledge database augments a standard S QL database with several extension layers, each providing an extended query language building upon thelower layers. As shown in the overview in Figure 1, there are three main layers. The
extensional database implements the concept of approximate relations based on thethree-valued logic of Kleene under an open world assumption by using a compilation mechanism to regular database relations. The intensional database uses rules andan inference mechanism to deduce both new positive and negative facts. Finally, theC CQ database provides mechanisms to apply local closed world assumptions while
evaluating a query through the use of second-order circumscription that is reduced
to first-order logic using the D LS ∗ second-order quantifier elimination algorithm. A
more detailed description of the A KDB system is found in Section 3.

2.2. Approximate Relations
Approximate relations are introduced as basic elements to be stored in approximate databases. These are intended to naturally express knowledge with imprecise
concepts and should be contrasted with the crisp (precise) relations and formulas usedin standard deductive databases. Approximate relations are based on ideas from roughset theory.

Approximate databases

5

In rough set theory, a rough set is characterized by two classical sets, one representing a lower approximation to the set and one representing an upper approximation
to the set. Both lower and upper approximations to the set are defined in terms ofequivalence classes of individuals in the domain of discourse where the partitions are
generated relative to a set of attributes associated with individuals. The net result isa partitioning of individuals relative to the lower and upper approximations and thedifference between the two, the boundary region of the rough set.
For approximate sets (relations), the basic semantics is generalized in a number ofways to create a semantics for approximate databases. Most importantly, rather thanbuilding up lower and upper approximations in terms of equivalence classes on individuals, one defines different types of neighborhood functions on individuals, where
each function may be specified using different sets of constraints. Rough sets thenbecome a special case of approximate sets with a particular set of constraints.
R⊕

crisp relation (set) R
R−
R

R+

R±

Figure 2. Approximations of a relation (set).
Any approximate relation R has (see Figure 2), a positive part, denoted by R + ,containing objects known to satisfy R, a negative part, denoted by R − , containing objects known not to satisfy R, a boundary part, denoted by R ± , containing objects thatare neither known to satisfy R nor known not to, a positive-boundary part, denotedby R⊕ , containing objects in the positive or boundary part, a negative-boundary part,
denoted by R  , containing objects in the negative or boundary part.
Approximations of a set (relation) are usually given by means of a family of neighborhoods of domain elements, as defined below. Note that such approximations not
only provide us with techniques for dealing with vague concepts, but also are important in verifying whether in a given application domain one actually deals with roughconcepts, tolerance-based concepts or maybe concepts reflecting yet other similarity
constraints. Certain properties of similarities are also used to ensure the validity ofcertain assumptions as, e.g., those discussed in Section 2.5.5.
D EFINITION 1. — Let D OM be a given domain of elements. Then any
n : D OM −→ 2DOM , such that for any u ∈ D OM, n(u) = ∅

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is called a neighborhood function.
Sets n(u), for u ∈ D OM are interpreted as direct neighborhoods of elements. The
neighborhoods are usually generated by some similarity relation σ, by assuming thatdefn(x) = {y | σ(x, y)}(see, e.g., [DOH 04f, DOH 03a]). For example, when σ is an equivalence relation,
one has rough set approximations (cf. [PAW 91, PAL 04]), when σ is reflexive andsymmetric, one has tolerance-based approximations (cf. [DOH 03a]).
In some applications it is also reasonable to consider even weaker similarity relations (cf. [DOH 04f, DOH 05]).
The lower and upper approximation of R wrt n is defined as:
Rn+
Rn⊕

def

=

{u ∈ D OM : n(u) ⊆ R}

def{u ∈ D OM : n(u) ∩ R = ∅}.

=Observe that the requirement that n(u) = ∅ in Definition 1 ensures that R n+ ⊆ Rn⊕ .
def

def

In the presence of the neighborhood function n, R + = Rn+ and R⊕ = Rn⊕ .
As shown in [DOH 04f], one can obtain correspondences between similarities andapproximations using second-order quantifier elimination techniques. The techniques
described are similar to those used in modal correspondence theory. Such correspondences are crucial when one wants to make sure that the underlying reasoning mechanism used with approximate relations reflects the properties of, say, rough sets ortolerance spaces.
For example, in order to satisfy constraints associated with tolerance spaces, one +
requires that R + ⊆ R (reflexivity of σ) and R ⊆ R⊕ (symmetry of σ). In order tosatisfyconstraints associated with rough sets, one has to additionally make sure thatR+ ⊆ R+ )+ (transitivity of σ).
One of the weakest requirements on approximations is that R + ⊆ R⊕ . This requirement corresponds to the seriality of the underlying similarity relation, i.e., tothe requirement that ∀x∃yσ(x, y). This observation is crucial for the soundness andcompleteness of the Feferman-Gilmore translation discussed in Section 2.5.5, sinceR+ ⊆ R⊕ is equivalent to R + ∩ R− = ∅, required there.
R EMARK 2. — As indicated in [DOH 04f], such correspondences can also be obtained based on modal correspondence theory. 2 This is done by translating R + into
2R and R⊕ into 3R. Then the required properties of the similarity relation are exactly those of the accessibility relation in Kripke structures for modal logics. Some2. For a survey of modal correspondence theory see [BEN 84]. For automated techniques forcorrespondence theory see [GAB 92, SZA 93, NON 98, NON 99].

Approximate databases

7times one can then apply known facts from the area of modal logics, e.g. the fact thataxiom 5: +
3R → 23R or, in the terminology of approximations, R ⊕ ⊆ R⊕
can replace both properties of reflexivity and symmetry, since logic KT 5 is S5.

2

Approximate relations can be constructed in a number of ways (see, e.g., [DOH 06,
DOH 04c, DOH 00b]), which include supervised machine learning, employing expert
knowledge, and approximating logical theories.

2.3. Open World Assumption
Traditionally, classical reasoning and planning techniques have been developed forenvironments in which the reasoning agent is assumed to have complete information
about the world in which it is embedded and the only changes to the world are theeffects which result from the agent’s invocation of actions. Under this assumption, an
efficient means of representing negative information about the world in each planningor reasoning state is to apply the Closed World Assumption (C WA) [ABI 95, REI 78].
In this case, information about the world, absent in a state, is assumed to be false.
In many realistic application domains, such as robotics, the assumption of complete information is not feasible. Therefore, the C WA can not be used. The embedding
environment is simply too dynamic and information about it too sparse. For example, a UAV flying over a region cannot have a complete model of that region. New
objects are continually sensed or encountered and agents other than the UAV agent
cause change in the region. In applications such as this, an Open World Assumption(OWA), where information not known by the agent is assumed to be unknown, is the
ontologically right choice to make, but complicates both the representational and implementational aspects associated with inference mechanisms and the use of negative
information.
The C WA and the OWA represent two extremes. Quite often, a reasoning agent has
information which permits the application of the C WA locally (see [ETZ 97, DOH 00c,
DOH 03b]). For example, if an agent has a camera sensor, the agent can assume
complete information about objects in the focus of attention (FOA) of the camera;e.g., the only objects in the FOA are those identified by the image processing module.
This method of reasoning is called the Local Closed World Assumption (abbreviatedas L CWA).
The OWA is used with approximate databases. In this case, both positive andnegative information must be stored in the database. Positive and negative information about relations is given via extensional and intensional database layers. Positive and negative facts that do not follow from these layers are assumed to be unknown. We also provide a machinery of contextually closed queries (C CQ, introduced
in [DOH 03b]) for reasoning based on L CWA, which will be described more precisely
in Section 3.4.

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2.4. Second-order Quantifier Elimination
Many concepts associated with reasoning in the context of incomplete information,such as the local closed-world assumption, approximate representations of logical theories, or circumscription axioms, can often be specified in a succinct and efficientmanner using second-order logic. It is also sometimes the case that such secondorder formulas can be reduced to logically equivalent first-order or fixpoint formulas using second-order quantifier elimination techniques, such as the D LS [DOH 97]
and D LS ∗ [DOH 96a] algorithms. The D LS algorithm is an extension of the algorithm in [SZA 93] and is based on Ackermann’s lemma [ACK 35]. The D LS ∗ algorithm makes use of a fixpoint elimination theorem in [NON 98] (Theorem 4 below) and strengthens the D LS algorithm. For an early implementation of the D LS
algorithm see [GUS 96] and for newer generation implementations of the D LS andD LS ∗ algorithms, see [MAG 05b].
Although it is clear that there is no algorithm for reducing any arbitrary secondorder formula to a logically equivalent first-order formula, when the application of
one of the algorithms described above is successful, the result may be a formula offirst-order logic, validity of which (over finite databases) is in PT IME and L OG S PACE
(here we apply the D LS algorithm) or a formula of fixpoint logic, validity of which(over finite databases) is in PT IME. 3 In the latter case we apply the D LS ∗ algorithm.
In designing our solutions we always restrict the expressiveness of our knowledge representation fragments to those where second-order quantifier elimination is
guaranteed to be in PT IME. Consequently, the methods we propose in this context
are guaranteed to be tractable, but still the resulting techniques cover all PT IME computable queries on ordered databases due to the well-known expressiveness of fixpoint
queries (see, e.g., [ABI 95]).
Relations generated using fixpoint computations are one important aspect of approximate database use. In order to formulate the fixpoint theorem of [NON 98] (Theorem 4 below), which is used in our semantic theory for approximate databases, we
require the following definitions.
D EFINITION 3. — A relation symbol R is said to occur positively (respectively negatively) in a formula A if it appears under an even (respectively odd) number of negations.4 A formula A is positive w.r.t. relation symbol R iff all occurrences of R in A
are positive. A formula A is negative w.r.t. relation symbol R iff all occurrences of R
in A are negative. If B(X) is a second-order formula, where X is a k-argument relational variable and C(x̄) is a first-order formula with free variables x̄ = x 1 , . . . , xk .
Then by B[X(t̄) := C(x̄)] we mean the formula obtained from B(X) by substituting
3. Recall that fixpoint logic captures all problems solvable in deterministic polynomial time,
provided that the underlying domain is linearly ordered — see, e.g., [ABI 95, IMM 98,
EBB 95].
4. It is assumed here that all implications of the form p → q are substituted by ¬p ∨ q and allequivalences of the form p ≡ q are substituted by (¬p ∨ q) ∧ (¬q ∨ p).

Approximate databases

9each occurrence of X of the form X( t̄) in B(X) by C(t̄), renaming any newly boundvariables in C(x̄) with fresh variables.
The fixpoint theorem of [NON 98] is formulated below.
T HEOREM 4. — Assume that formula A is a first-order formula positive w.r.t. X.
– If B is a first-order formula negative w.r.t. X then
 




∃X ∀ȳ A(X) → X(ȳ) ∧ B(X) ≡ B X(t̄) := lfpX(ȳ) [A(X)[t̄]] .

(1)– If B is a first-order formula positive w.r.t. X then

 



∃X ∀ȳ X(ȳ) → A(X) ∧ B(X) ≡ B X(t̄) := gfpX(ȳ) [A(X)[t̄]] .

(2)

A survey of various approaches to second-order quantifier elimination is given
in [NON 99].

2.5. Three-valued Logic of Kleene
The semantics of approximate databases is based on the strong three-valued logicof Kleene, denoted by K 3 .5 In K3 we use three logical values{T RUE, FALSE, U NKNOWN},
with the following logical truth ordering: FALSE ≤ U NKNOWN ≤ T RUE.
We define the set {U NKNOWN, T RUE} to be the set of designated truth values. 6
Below we shall briefly discuss propositional, first-order, fixpoint and second-orderthree-valued logic, reflecting the languages used in A KDBs.
2.5.1. The Propositional CaseIn the propositional version of K 3 we build formulas based on a set P 3 of threevalued propositional variables using the standard recursive applications of the propositional connectives ∨, ∧, →.
Any valuation v : P 3 −→ {T RUE, FALSE, U NKNOWN} is extended to the set of
all formulas as follows:⎧when v(A) = FALSE⎨ T RUE
def
FALSE
when v(A) = T RUE– v(¬A) =⎩U NKNOWN when v(A) = U NKNOWN
5. This makes our solution, among others, compatible with standard S QL, where K3 is used todefine the semantics of queries when null values can appear in database relations.
6. Recall that in the area of many-valued logics, members of the set of designated truth values
act as being true and are used in the definition of entailment.

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Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.
def– v(A ∨ B) = max{v(A), v(B)}def– v(A ∧ B) = min{v(A), v(B)}⎧when v(A) = FALSE or v(B) = T RUE⎨ T RUEdef– v(A → B) =FALSE
when v(A) = T RUE and v(B) = FALSE⎩U NKNOWN otherwise.
In K3 , as in classical propositional logic, it is the case that:
v A → (B ∨ C) = v (A ∧ ¬B) → Cv (A ∧ B) → C= v A → (¬B ∨ C) .

(3)
(4)

This is crucial in the definition of expansion (see Definition 13).
The choice of K 3 is also crucial in guaranteeing that the Feferman-Gilmore translation (see Section 2.5.5) is sound and complete wrt the semantics accepted for A KDB s.
R EMARK 5. — In the literature, the semantics of deductive databases is sometimes
specified using the logic proposed in [FIT 85] (see also [PRZ 90]) as a basis, where¬, ∨, ∧ are defined as in K 3 and → is defined bydefv(A → B) = v(A) ≤ v(B)).

(5)

From our perspective, definition (5) is doubtful. First of all, equations (3) and(4), crucial to our understanding of I DB rules (defined later) are no longer valid
within this semantics. Secondly, U NKNOWN intuitively denotes a value which is not
yet determined by a robotic agent, but may be determined at a future time. ThusU NKNOWN → U NKNOWN, which is T RUE according to (5), should be U NKNOWN,
since such an implication, after determining truth values of assumption and conclusion, may equally well appear FALSE or T RUE . Moreover, one sometimes obtains
results which we find counterintuitive. For example, consider the formula:
f ourW heels(a) ∧ onRoad(a) → car(a) ∧ ¬car(a).
(6)
Assume that an agent has been able to determine that v car(a) = FALSE andhas
been unable to determine whether a has four wheels,
i.e., v f ourW heels(x) =
U NKNOWN, or whether a is on the road, i.e., v onRoad(x) = U NKNOWN. The
definition provided by formula (5) would evaluate formula (6) to be FALSE. On theother hand, intuitively (6) should not be inconsistent. Note that in K 3 , formula (6) isevaluated in this situation to be U NKNOWN.
2
2.5.2. The First-Order Case
For the first-order language with quantifiers ∃, ∀, we define v on atomic formulas
and then extend it to deal with quantifiers:

 def


– v ∃x A(x) = max {v A(x := a) }a∈ D OM


 def


– v ∀x A(x) = min {v A(x := a) }.
a∈ D OM

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2.5.3. The Second-Order Case
For the second-order language with second-order quantifiers ∃, ∀ ranging over relations, we first define v on first-order formulas and then extend it to deal with secondorder quantifiers (below R is a three-valued relation over D OM with the same arity as
relational variable X):



 def
– v ∃X A(X) = max{v A(X := R) }
R

 def


– v ∀X A(X) = min{v A(X := R) }.
R

2.5.4. Fixpoint Formulas
For formulas A(X(x̄)) positive wrt X one can define operators
lfpX(x̄) [A(X(x̄))] and gfpX(x̄) [A(X(x̄))]denoting the least and the greatest fixpoint of A(X(x̄)) (see, e.g., [ABI 95, EBB 95,
IMM 98]). Of course, the terms “least” and “greatest” are to be understood wrt aparticular ordering on relations. In A KDB we assume the truth ordering  defined asfollows (where t̄ can contain constants only): 7
def

RS ≡

{t̄ | R(t̄) ≡ T RUE} ⊆ {t̄ | S(t̄) ≡ T RUE} and{t̄ | R(t̄) ≡ FALSE} ⊇ {t̄ | S(t̄) ≡ FALSE}.

(7)

In the classical two-valued case  coincides with the standard inclusion ⊆.
Assume the (three-valued) valuation v on first-order formulas is given. It is extended to deal with fixpoint operators as follows: def – v lfpX(x̄) [A(X(x̄))] = v A(X := R) , where R is the least (wrt ) relation such that A(X := R) = R def – v gfpX(x̄) [A(X(x̄))] = v A(X := R) , where R is the greatest (wrt )
relation such that A(X := R) = R.
2.5.5. The Feferman-Gilmore Translation
In order to provide a semantics for approximate databases, we shall require thedefinition of the Feferman-Gilmore translation (see [FEF 84, GIL 74]), used in translating three-valued logic formulas into the classical two-valued logic.
D EFINITION 6. — By a Feferman-Gilmore translation of a three-valued formula A,
denoted by F G(A), we shall mean the formula obtained from A by replacing all positive literals of the form R(ȳ) by R + (ȳ) and all negative literals of the form ¬R(ȳ) by
R− (ȳ).
7. In the literature one can also find knowledge ordering, defined by replacing ⊇ in the secondline of (7) by ⊆. Observe that this ordering appears naturally in A KDBs due to the applicationof the translation defined in Section 2.5.5.

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In A KDBs, instead of dealing with a single relation symbol representing a classicalrelation, say R, we introduce two relation symbols R + and R− , the first one for representing positive facts known about R and the second one for representing negative
facts known about R.
The Feferman-Gilmore translation is sound and complete for K 3 in the sense that
formula α entails formula β in K 3 iff in the classical logic the following formula isvalid:¬(R1+ ∧ R1− ) ∧ . . . ∧ ¬(Rk+ ∧ Rk− ) → (α → β),where R1 , . . . , Rk are all atoms in α and β (see, e.g., [BUS 96]).
Observe that whenever the underlying similarity relation is at least serial then,as indicated in Section 2.2, for any relation R approximated by means of similarity
neighborhoods we have that ¬(R + ∧ R− ) holds.

2.6. Inconsistencies
Since we envision using approximate databases in a distributed manner acrossagent systems and often assume several approximate databases being used in a single
agent system, fusing information from various sources will be common place. Consequently, local inconsistencies may arise in A KDBs. Many approaches have been
proposed for dealing with such inconsistencies (see, e.g., [BEL 77]). For the purposesof this paper, we do not deal with techniques for resolving inconsistencies, but instead
assume this is left to the user of the A KDB, who can provide I DB rules for resolving
inconsistencies. This should be considered a feature since there are many strategies
for resolving inconsistencies and their choice is often dependent on a particular context. For example one can assume that whenever a tuple, say t̄, satisfies both a relation
R and its negation ¬R, then the result of query R( t̄) as well as ¬R(t̄) is U NKNOWN.
Another approach might depend on prioritizing data sources, prefer answers given
by more trustable sources, and rely on less trustable sources only when more trustableprovide answer U NKNOWN. Yet another solution might be based on providing weights
for data sources and consider the “weighted sum” of answers with thresholds allowing to determine what logical value is actually represented by the computed answer
or to apply the majority voting principle. For a discussion of possible strategies see,e.g., [DOH 02, DOH 06].

2.7. Circumscription
Circumscription is a powerful non-monotonic formalism introduced in [MCC 80](for a survey see [LIF 91]). Although circumscription is a second-order formalism,
for a rich class of formulas it can be reduced to first-order logic (see [DOH 97]) or to
the fixpoint logic using the D LS ∗ algorithm.

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If U and V are relation expressions of the same arity, then U ≤ V stands for∀x̄ (U (x̄) → V (x̄)).8 Similarly, if Ū = U1 , . . . , Un and V̄ = V1 , . . . , Vn are
similar tuples of relation expressions, i.e., for 1 ≤ i ≤ n, U i and Vi are of the samenarity, then Ū ≤ V̄ is an abbreviation for[Ui ≤ Vi ].
i=1

We write Ū = V̄ for (Ū ≤ V̄ ) ∧ (V̄ ≤ Ū ), and Ū < V̄ for (Ū ≤ V̄ ) ∧ ¬(V̄ ≤ Ū ).
D EFINITION 7. — Let P̄ = P1 . . . , Pn be a tuple of distinct relation symbols,S̄ = S1 , . . . , Sm be a tuple of distinct relation symbols disjoint with P̄ , and let
T (P̄ , S̄) be a theory. The circumscription of P̄ in T (P̄ , S̄) with varied S̄, written
C IRC (T ; P̄ ; S̄), is the sentence
T (P̄ , S̄) ∧ ∀X̄∀ȲT (X̄, Ȳ ) ∧ [X̄ ≤ P̄ ] → [P̄ ≤ X̄](8)
where X̄ = X1 . . . , Xn and Ȳ = Y1 , . . . , Ym are tuples of relation variables
similar to P̄ and S̄, respectively.

3. An Overview of the Architecture of Approximate Knowledge Databases
It is important to facilitate experimentation to provide a better feel for the utility and applicability of the ideas and techniques behind approximate databases, andsuch experience will likely only come from a real system. We have consequently
spent some considerable effort on an actual implementation called the Approximate
Knowledge Database (A KDB). Grounded in a standard S QL database, several extension layers each provide an extended query language building upon its lower layers.
Figure 1 displays an architectural overview of the system that is described in moredetail below, starting from the bottom abstraction layer, moving upwards.
It is important to emphasize that the intended user language is the language ofthe classical crisp first-order or fixpoint logic. All facts, intensional rules and queries
are then translated using the Feferman-Gilmore translation and the results are thentranslated back according to the semantics provided in Section 4. However, the user
can also access the database using the database syntax directly via a graphical userinterface (see Appendix A.1) or text files.

3.1. SQL Database
Even though the S QL database forms the fundamental layer of the system, it caneasily be replaced, choosing from a long list of S QL databases such as P OSTGRE SQL,M Y SQL, O RACLE , S YBASE, I NFORMIX , D B 2, M S S QL S ERVER, etc. Whatever
database is chosen, it will use regular, crisp, database relations to store approximate
8. Note that U ≤ V means that the extension of U is a subset of the extension of V .

14Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

relations, but need not know the details of the representation. Deciding the exact
format is the concern of the next abstraction layer, which still leaves the S QL database
the task of optimizing and executing S QL queries passed down from higher layers.
Note that for some applications a standard S QL database might not be the most efficient means of storing and retrieving data. In a robotic system a secondary memory
footprint may need to be minimized or a high frequency of low complexity queries
might call for an implementation where data is stored in primary memory, in whichcase such an implementation can be plugged into the system, bypassing the S QL interface.

3.2. Extensional Database
The Extensional Database layer (E DB) provides a mapping from approximate relations to a specific representation scheme based on regular database relations. In
particular, the positive (R + ) and negative (R − ) parts of relations are stored explicitly
in tables while the boundary, positive boundary, and negative boundary are only stored
implicitly but can still be generated through more complex queries to the database.
When all relation arguments are assumed to have finite domains, all parts of relations consist of a finite number of tuples, and the division between explicit and implicit
storage becomes a potential implementation choice point to which the best answer will
depend on the final application. We believe that, in the general case, it is profitable tosave space by storing known information explicitly and unknown information implicitly since the amount of unknown information about world state is often vastly larger
than that which is known.
As mentioned previously, logic is consistently used as the query language in allparts of the A KDB . Both the fact that the logical query language may refer explicitly
to the boundary parts that are not explicitly stored and the fact that any, arbitrarily
complex, logical formula may be used as a query contribute to the necessity of somekind of evaluation mechanism. To make this feasible, we have developed a querycompilation mechanism that will recursively transform any logical formula query intoa, sometimes very elaborate, S QL query, the only exception being fixpoint formulas,which need to be iteratively evaluated until a fixpoint is reached before the result canbe returned. By delaying the actual evaluation of any part of the query until it reaches
the S QL database we can benefit from the potential performance increase resulting
from S QL query optimization techniques employed by a particular database of choice.
In order to illustrate the use of the extensional database, let us assume that we havea qualitative approximate relation specifying the colors of cars:– Color(x, y) (the color of car x is y)
The E DB is populated with some facts, expressed using the Color relation, about
three cars, C1 , C2 , and C3 , and two colors, Black and Red, as in (9).
Color(C1 , Black) ∧ ¬Color(C1 , Red) ∧ Color(C2 , Red)

(9)

Approximate databases

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A simple query asking for the colors of cars, Color(x, y), results in the tuples
C1 , Black and C2 , Red , while ¬Color(x, y) would return C 1 , Red . The colorof C3 is not known and an explicit query about it, e.g., Color(C 3 , Black), results inthe value U NKNOWN. This is so because the OWA is being used and no additional
constraints restricting a car to have only one color are part of the theory.
More subtle queries can be formed by using the language of approximate relations
directly. Asking for cars that might not be red, Color  (x, Red), returns the cars C1
and C3 , since C1 is known not to be red and C 3 is possibly not. Any complex
formulas can be evaluated, e.g., a query asking if there is a color that all cars might
have and that some car is known to have can be expressed as ∀x[Color ⊕ (x, y)] ∧∃x[Color+ (x, y)], and returns the tuple Black .

3.3. Intensional Database
The Intensional Database layer (I DB) uses stored rules to infer additional information from the facts in the E DB. The intensional rules are approximate implications
with a single literal head, but differ from Horn-clause type rules in that the head literals
can contain negations. To deduce new facts, the rules are translated into approximate
formula fixpoint queries that must then be checked for consistency since both new
positive and new negative information might be produced.
Performing the necessary inferences for a specific query is the task of the inference
mechanism, which currently uses the naive method of repeatedly applying all I DB
rules until no new facts can be inferred, evaluating the query in this new context, andfinally withdrawing the generated facts to restore the original database state. Thismethod can clearly be improved upon using the rich set of techniques developed for
efficiently evaluating intensional database queries (see, e.g., Chapter 13 of [ABI 95]).
Continuing the previous example, we might use the I DB to add a rule that enablesus, whenever we detect the color of a car, to conclude that it has only that color andnot any other color, expressed in (10).
∀x, y1 , y2 [Color(x, y1 ) ∧ y1 = y2 → ¬Color(x, y2 )](10)

If we then evaluate the query, ¬Color(x, y), the result will be both C 1 , Red andC2 , Black , which includes the new consequence that C 2 can not be Black since it
was known to be Red.

3.4. Contextually Closed Query Database
The Contextually Closed Query database layer (C CQ) provides the functionalitythat renders locally closed world reasoning feasible. This is accomplished through theuse of circumscription in the context of a closure policy provided by the user.

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Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

A contextually closed query, as introduced in [DOH 03b], consists of a logicalformula query together with a crisp logical theory, describing the relations in the query,and a closure policy defining what relations are affected by the closure in one of thefollowing ways:– A relation may be fixed, in which case it will not be modified by the closure.
– A relation may be minimized, in which case tuples may be moved from theboundary part into the negative part of the relation in order to minimize its extension.
– A relation may be varied, in which case tuples may be moved from the boundary
part into either the positive or negative part as an effect of the minimization of otherrelations.
Closure policies reflect the ideas behind circumscription but with local theories orcontexts being attached directly to queries to contextualize them (see, e.g., [MCC 80,
LIF 91]). Of course, circumscription-based reasoning is not tractable. However, theD LS ∗ algorithm can reduce any second-order formula from the class of semi-Hornformulas into an equivalent first-order formula in polynomial time [DOH 98], whereby a semi-Horn formula we understand any formula of the form appearing inside ofbrackets {. . .} in (1) and (2). Consequently, second-order circumscription theories
can often be reduced to logically equivalent first-order theories and standard querying
techniques can then be used.
Another, important class of closure policies, the universal closure policies, computable in polynomial time, is defined in [DOH 03b]).
The major difference between circumscription and closure policies lays in the underlying methodology. Namely, contextually closed queries focus on preserving theindicated integrity constraints, expressed by means of a first-order theory, that are
to be preserved while a given query is being evaluated. The set of chosen integrity
constraints can be made dependent on a particular context of a query.
Contextually closed queries are frequently used to model the L CWA (For the connection between L CWAs and circumscription see also [DOH 00c]).
Adding another, undoubtedly approximate, relation to the previous example will
help illustrate a C CQ query:– Sporty(x) (car x is kind of sporty)
Assuming that we want to express the opinion that red cars are sporty, we formulate
the simple integrity constraint in (11).
∀x[Color(x, Red) → Sporty(x)](11)

If we apply the closure policy of minimizing the new Sporty relation while varying the Color relation, a query such as, Sporty(x), will return the tuple C 2 , since
C2 was known to be Red and therefore has to be sporty, while the query ¬Sporty(x)returns the tuples C1 and C3 since these cars can be assumed not to be Red inorder to minimize Sporty.

Approximate databases

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Running the D LS ∗ algorithm on a circumscribed logical theory can have undesirable effects on its complexity. Although encouraging results in [DOH 99] show
that, at least in the semi-Horn case, the size of the syntactic characterizations of varied and minimized relations are linear in relation to the size of the C CQ query, inpractice there is often both a need and opportunity for simplifications. A number of
equivalence-preserving simplifications are applied to each contextually closed query,
e.g. taking advantage of the unique names assumption in the A KDB .

3.5. Logic Parser
A parser for database input was generated using the JAVACC parser generator.
Our syntax supports the representation of approximate formulas in addition to regular
first-order and fixpoint formulas, but also knowledge base-specific constructs such as
relation definitions, extensional facts, intensional rules, and logical theories together
with closure policies, making it possible to specify entire use scenarios in a single text
file. This latter functionality is complementary to the user of the A KDB as a service
in a larger system, in which case one would make direct use of the A KDB interface togain access to methods for, among others, adding and retracting facts or rules.

4. Syntax and Semantics of Approximate Knowledge Databases
4.1. Syntax of Formulas
The syntax for crisp first-order formulas is defined in the standard way, assuming that “->", “|", “&", “-", “forall", “exists" denote implication, disjunction,
conjunction, negation, universal quantification and existential quantification, respectively.
E XAMPLE 8. — Examples of crisp first-order formulas:Edge(A,B)
forall x,y [exists z [Edge(x,z) & Path(z,y)] -> Path(x,y)]2

Approximate first-order formulas use the same syntax except that atomic formulas
have approximation symbols. We will refer to such formulas, assuming that “+",“-", “+-", “++" and “--" denote the parts of approximate relations + ,− ,± ,⊕ and
, respectively.
Quantifier-free (crisp and approximate) formulas without variables are called groundformulas.
E XAMPLE 9. — Examples of approximate first-order formulas:
Edge+(A,B)
forall x,y [exists z [Edge+(x,z) & Path+(z,y)] -> Path+(x,y)]2

Fixpoint formulas are like crisp first-order formulas except that there are two additional fixpoint operators, “lfp" and “gfp", denoting the least and the greatest fixpoint

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Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

respectively. Combining the approximate first-order formula and fixpoint formulagrammar extensions yields the class of approximate fixpoint formulas.
E XAMPLE 10. — Examples of fixpoint and approximate fixpoint formulas:
gfp Safe(x) [Bad(x) & forall y [-Edge(x,y) | Safe(y)]]gfp Safe(x) [Bad--(x) & forall y [Edge--(x,y) | Safe+(y)]]lfp Tc(x,y) [Edge+(x,y) | exists z [Edge+(x,z) | Tc+(z,y)]]2∗Second-order formulas are used to directly access the D LS and D LS algorithms.
They have the same syntax as crisp first-order formulas except for two additional
second-order quantifiers, “forallrelations" and “existsrelation", denoting theuniversal and existential second-order quantifier respectively.
E XAMPLE 11. — Example of a second-order formula:forallrelations P [forallrelations Q [forall x [-forall y [R(x,y) -> (P(y) | Q(y))] |forall z [R(x,z) -> P(z)] |forall v [R(x,v) -> Q(v)]
]
]
]

2

4.2. The Extensional Database Layer
4.2.1. The Language of the Extensional Database Layer
The extensional database consists of sets of positive and negative facts. We thus
assume that the language of the extensional database is a set of ground literals, i.e.,
formulas of the form R(c̄) or ¬R(c̄), where R is a relation symbol and c̄ is a tupleof constant symbols. In order to store such facts in the E DB we apply the FefermanGilmore translation (see Definition 2.5.5). Thus, in the database syntax, these facts
are expressed by R+(c̄) and R-(c̄), respectively.
4.2.2. Semantics of the Extensional Database Layer
We give the semantics of relations of the E DB layer. The semantics of formulas
can then be extended according to the three-valued logic of Kleene, as defined in
Sections 2.5.1-2.5.4.
Consider an atomic ground formula R( t̄). Let E DB c denote the set of facts representing (positive and negative) database facts of a given database instance. Then, inthat instance, if R(t̄) is an atomic ground formula, thendef

– R+(t̄) ≡ T RUE iff R+(t̄) ∈ E DBc
def– R-(t̄) ≡ T RUE iff R-(t̄) ∈ E DBc .

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Based on the above definitions one can define other approximation operators as wellas the semantics of the crisp relation:def– R++(t̄) ≡ T RUE iff R-(t̄) ≡ T RUE
def– R--(t̄) ≡ T RUE iff R+(t̄) ≡ T RUE
def– R+-(t̄) ≡ T RUE iff R+(t̄) ≡ T RUE and R-(t̄) ≡ T RUE⎧
when R+(t̄) ≡ T RUE⎨ T RUE
def
FALSE
when R-(t̄) ≡ T RUE– R(t̄) ≡⎩U NKNOWN otherwise.
Observe that a literal of E DB can be both T RUE and FALSE, resulting in inconsistencies to be resolved as discussed in Section 2.6.

4.3. The Intensional Database Layer
4.3.1. The Language of the Intensional Database Layer
The intensional database is intended to infer new facts, both positive and negative
via application of intensional rules to the E DB. The rules are of the form
±P (x̄) ← ±P1 (x̄1 ), . . . ± Pk (x̄k )where ± is either the empty string or the negation symbol ¬.
The above rule, is translated into the database syntax by applying the FefermanGilmore translation. For example, ¬P (x) ← Q(x) ∧ ¬R(x, y) is translated into
Q+(x) & R-(x,y) -> P-(x).
R EMARK 12. — Note that contrary to DATALOG, we do not require the safety condition, i.e., that a variable occurring in the rule’s head has to occur in the rule’s body.
In cases violating the safety condition, we assume that the total relation referring tovariables that are in the rule’s head and not in its body are added to the rule’s body.
More precisely, consider the ruleR(x̄) ← R1 (x̄1 ), . . . , Rk (x̄k ).

(12)

def

Let z̄ = x̄ − (x̄1 ∪ . . . ∪ x̄k ) be nonempty. 9 Let T (z̄) be a new relation symbol,
representing the total relation, i.e., for all z̄, T (z̄) holds. Rule
R(x̄) ← R1 (x̄1 ), . . . , Rk (x̄k ), T (z̄),
equivalent to (12), is safe. Note that T (z̄) is finite, since the database domain is finite.
The semantics for unsafe rules in A KDBs, is then provided by assuming that the
total relation binding the “unsafe” variables is implicitly added to each unsafe rule. 2
9. By x̄ ∪ ȳ we shall mean all variables that are in x̄ or ȳ (after removing duplicates), and byx̄ − ȳ we shall always mean all variables that are in x̄ and not in ȳ.

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Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

4.3.2. Semantics of the Intensional Database Layer
Observe that, after the translation, all rules in I DB become Horn rules, as in thecase of standard DATALOG. We then use the standard semantics (see, e.g., [ABI 95]),
according to which minimal relations satisfying the rules are computed.
The semantics of I DB is now defined similarly as in the case of E DB. Namely,
let I DBc denote the set of facts representing (positive and negative) minimal relations
computed according to DATALOG semantics over a given database instance and letR(t̄) be an atomic ground formula. Then, in that instance,def– R+(t̄) ≡ T RUE iff R+(t̄) ∈ I DBc
def– R-(t̄) ≡ T RUE iff R-(t̄) ∈ I DBc .
The semantics of approximation operators R++( t̄), R--(t̄), R+-(t̄) as well as thesemantics of the crisp relation R( t̄) and arbitrary formulas can now be defined byanalogy with Section 4.2.2.
Observe that a literal of I DB can be both T RUE and FALSE, resulting in inconsistencies to be resolved as discussed in Section 2.6.

4.4. Contextually Closed Queries
4.4.1. The Language of Contextually Closed Queries
Contextually closed queries use local contextual closure policies, LCC s, which are
expressions of the form
LCC [L1 , . . . , Lp ; K1 , . . . , Kr ] : I,

(13)

where L1 , . . . , Lp are (positive or negative) literals, K 1 , . . . , Kr are relation symbols
not appearing in L i ’s and I is a set of integrity constraints. Literals L 1 , . . . , Lp areminimized assuming that integrity constraints I are preserved and relations K 1 , . . . , Kr
can vary. Thus LCC corresponds to circumscription of I ∪ EDB ∪ IDB with relations L1 , . . . , Lp minimized (positive literals) or maximized (negative literals) andK1 , . . . , Kr allowed to vary. By an LCC assumption we mean a minimization or maximization of a single literal from L 1 , . . . , Lp in (13).
A contextually closed query consists of the query itself, which can be any fixpoint
or first-order formula together with the local closure policy representing the closure
context.
4.4.2. Semantics of Contextually Closed Queries
Let the E DB and I DB be defined as before, let I denote a finite set of integrity
constraints, and let LCC [L̄; K̄]:I be a specific LCC policy. Then, if R( t̄) is an atomic
ground formula, then under the given policy,

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21

def

– R+(t̄) ≡ T RUE iff C IRC (I ∪ I DB ∪ E DB ; L̄; K̄) |= R(c̄)
def– R-(t̄) ≡ T RUE iff C IRC (I ∪ I DB ∪ E DB ; L̄; K̄) |= ¬R(c̄).
The semantics of approximation operators R++( t̄), R--(t̄), R+-(t̄) as well as thesemantics of the crisp relation R( t̄) and arbitrary formulas can again be defined byanalogy with Section 4.2.2.
4.4.3. Implementation Issues
In general, the problem of querying an approximate database containing LCC policies defined using unrestricted logical theories is CO -NPT IME - COMPLETE. When one
restricts theories to semi-Horn, the situation becomes tractable, since for such theories
D LS ∗ succeeds (see, [DOH 96b]).
However, for the case of universal theories a more efficient computation mechanism, not involving D LS ∗ can be provided (see, [DOH 03b]). By a universal theory
we understand any set of formulas of the form,∀y[(±R1 (x1 ) ∧ · · · ∧ ±Rk (xk )) → ±R(x)](14)

where ± denotes an optional negation sign, R 1 , . . . , Rk , R are relation symbols, y is
the vector of all variables occurring in x 1 , . . . , xk , x and x ⊆ x1 ∪ . . . ∪ xk .
To compute syntactic definitions of the minimized and varied relations we first use
the equivalences reflecting laws (3) and (4):∀x[A(R) → (B(R) ∨ M (y))] ≡ ∀x[(A(R) ∧ ¬M (y)) → B(R)]∀x[(A(R) ∧ M (y)) → B(R)] ≡ ∀x[A(R) → (B(R) ∨ ¬M (y))](15)

to generate the expansion of each formula in the logical theory, where the notion of
expansion is defined below.
D EFINITION 13. — The expansion of a universal formula F , wrt a closure policy,
is defined as the least set of formulas, obtained by applying tautologies (15) to F ,such that any occurrence of a minimized or varied relation is a consequent of one
implication.
To minimize a relation we collect, from the expanded theory, all the antecedents
for which the relation must be true and make the relation false in all other cases. Thisis accomplished by first applying the Feferman-Gilmore translation to the expanded
theory and then forming a disjunction of the implications’ antecedents where the consequent is the positive part of the relation to be minimized. This serves directly asa definition of the positive part of the new minimized relation, while the negative partis defined by the rough negation of the positive part, where the rough negation of R + isR if R belongs to the set of relations to be minimized or varied given the closure policy being applied and R − otherwise (and similarly, the rough negation of R − is R⊕ or
R+ ). Definitions of the positive and negative parts of varied relations are constructedby collecting antecedents from the implications with corresponding consequents, combining them in disjunctions and finally substituting occurrences of minimized relations
by their new definitions obtained in the previous step (see [DOH 03b]).

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5. Relation to other Work
In comparison to variants of DATALOG (see [ABI 95, PRZ 90]) we are able to naturally represent approximated concepts and relations. Instead of a single relation, say
R, we consider two relations, one responsible for positive information (R + ) and theother (R − ) for negative information; then negative literals are substituted by R − using
the Feferman-Gilmore translation. We obtain negation-free I DB rules and apply thestandard DATALOG semantics for computing R + and R− (with the C WA). However,
the relation R itself is represented by its positive and negative parts, which reflects the
OWA and can result both in inconsistencies and the lack of full knowledge about R.
Moreover, we permit the use of much more complicated I DB rules than those offered by DATALOG. As mentioned before, we do not require the safety condition
that a variable occurring in the rule’s head has to occur in the rule’s body. Secondly,
users may supply I DB rules that contain negations in the head literal as well as thebody, which are then translated as noted above. Finally, the use of quantifiers is morerelaxed as existential quantifiers may be used freely in the rule body.
An approach to rough databases is provided in [VIT 03a, VIT 03b], where a logic
programming based implementation of rough databases is given. The strength of thesesolutions is in applying numerical measures and quantitative reasoning not directly
present in A KDB s. On the other hand our approach offers a much greater flexibility
with our ability to deal not only with rough set-based reasoning but also with the
combination of rough and crisp logic.
Unlike both DATALOG and the rough database approach above, the A KDB introduces the use of restricted circumscriptive policies in the form of contextually closed
queries, which allows for highly expressive nonmonotonic queries. We also provide
an algorithm for second-order quantifier elimination (D LS ∗ ) that has many important
applications, e.g. the calculation of weakest sufficient and strongest necessary conditions discussed in Appendix A.3, and that is guaranteed to succeed for the well-definedclass of semi-Horn formulas.

6. A Surveillance Mission Case Study
Consider a scenario involving a UAV that makes use of contextually closed queries
during a surveillance mission. A black car has been reported stolen and the task of
the UAV is to locate the car by investigating areas in which the car is suspected to belocated. To represent this scenario we make use of the relations:– In(x, y) (car x is in region y)– Color(x, z) (the color of car x is z)– SuspectIn(y) (the stolen car is suspected to be in region y)– Investigate(x, y) (the UAV should search for car x in region y).

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Using these relations we construct a crisp logical theory (16) expressing the behavior
we wish the UAV to exhibit. All black cars that are in a suspect region should be
investigated. If a car is known to have some color other than black it is not necessary
to look for it in any region. Finally, when we know that the searched car is not ina region, there is no point going there looking for it.
∀x, y In(x, y) ∧ SuspectIn(y) ∧ Color(x, Black)→ Investigate(x, y) ∧


(16)

∀x, y, z Color(x, z) ∧ z = Black → ¬Investigate(x, y) ∧
∀x, y ¬In(x, y) → ¬Investigate(x, y)
In A KDB (16) is translated and represented as
forall x,y [In+(x,y) & SuspectIn+(y) & Color+(x,Black)-> Investigate+(x,y)] &forall x,y,z [Color+(x,z) & z!=Black -> Investigate-(x,y)] &forall x,y [In-(x,y) -> Investigate-(x,y)]

Additionally an intensional rule (17) is added to the I DB expressing the fact that if weknow the region a car is in, it can not simultaneously be in some other region.
∀x, y1 , y2 [¬In(x, y2 ) ← In(x, y1 ) ∧ y1 = y2 ](17)

In the database syntax (17) is expressed as
forall x,y1,y2 [In+(x,y1) & y1!=y2 -> In-(x,y2)]Continuing the example, we construct a specific scenario by adding facts to the approximate knowledge base. Given three cars, C 1 , C2 and C3 , three regions, R 1 , R2
and R3 , and two colors, Black and Red, we add the facts expressed in (18). A black
car C1 is known to be in region R 1 , the car C2 is red but we do not know in which
region it is, and nothing is known about the third car C 3 . Furthermore, the stolen car
is believed to be located somewhere in region R 1 or R2 .
In+(C1,R1) &Color+(C1,Black) & Color+(C2,Red) &SuspectIn+(R1) & SuspectIn+(R2)(18)

The current knowledge base does not contain any information about which cars
and what regions are interesting for the UAV, but this is information that would be invaluable when determining appropriate strategies to search regions for target vehicles.
To acquire such information, a contextually closed query can be formulated whichtakes account of current context. In this case, new information specific to regions ofinterest can be generated nonmonotonically.

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Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

To do this, the closure policy associated with the contextually closed query will
minimize the number of suspected regions in order to avoid searching regions that
we have no specific reason to believe the stolen car to be in, while varying what carsand regions the UAV should investigate to obtain information about possible actions totake. Consequently we construct the policy of minimizing SuspectIn while varying
Investigate and fixing the remaining relations In and Color.
In order to ask contextually closed queries we first compute the expansion of (16)(recall that expansion is defined in Definition 13), where the only formula containing
an occurrence of a minimized or varied relation that is not already in the consequentis the first one. Thus, in this case, the expansion of (16) is obtained by replacing thefirst conjunct by∀x, y[In(x, y) ∧ SuspectIn(y) ∧ Color(x, Black)→ Investigate(x, y)]∀x, y[In(x, y) ∧ ¬Investigate(x, y) ∧ Color(x, Black)→ ¬SuspectIn(y)](19)

We now obtain syntactic definitions for SuspectIn and Investigate according to
the description provided in Section 4.4.3 and obtain the results shown in (20). The
positive part of the minimized SuspectIn relation is simply those tuples explicitly
stored as positive in the extensional database, while the negative part contains the restof the tuples, while the definition of the varied relation Investigate is more complex.
SuspectIn : SuspectIn+(y)¬SuspectIn : SuspectIn--(y)Investigate : Investigate+(x,y) |In+(x,y) & SuspectIn+(y) & Color+(x,Black)¬Investigate : Investigate-(x,y) | In-(x,y) |exists z [Color+(x,z) & z!=Black](20)

Observe that the definitions contain unbound variables, and presenting them to theknowledge base as queries will produce exactly the tuples satisfying the new relation
definitions. To evaluate a complex query containing the minimized or varied relations
it suffices to replace those occurrences with their syntactic definitions and pass themodified query to the intensional database layer. Evaluating the definitions in our
examples produces the tuples in (21), including new tuples produced by the I DB rule.
In(x, y) :¬In(x, y) :SuspectIn(y) :¬SuspectIn(y) :Investigate(x, y) :¬Investigate(x, y) :C1, R1C1, R2 , C1, R3
R1 , R2
R3
C1, R1C1, R2 , C1, R3 , C2, R1 ,C2, R2 , C2, R3(21)

Although the I DB rule excluded the possibility of C 1 being anywhere else than in
R1 , it remains unknown which regions the other cars are in. Minimizing SuspectIn

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removes R3 from the set of suspected regions since there is no reason to believe otherwise, while varying Investigate prompts the UAV to search for C 1 in region R1
since we know it is a black car located in a region which we suspect the stolen car to
be in. In addition, the UAV concludes that it is not necessary to look for C 1 anywhere
else, using the I DB rule and the part of the theory stating that it should not investigate
a region, looking for a car it knows is not there. Car C 2 can be in any of the regions but
there is no point looking for it as it has the color Red, different from Black. Finally,
it remains unknown, even after applying the closure policy, if searching for the car C 3
in any of the regions is necessary.
Now, assume the UAV takes action, flying over region R 1 looking for C 1 , and that
it finds the car but it is not the stolen car we are looking for. It updates the knowledge
base by removing R 1 from the list of suspected regions and adding the fact that, while
searching R1 for C1 , the car C3 was not encountered, expressed by In-(C3,R1). Using
the same syntactic definitions of relations, we reevaluate the queries in light of these
new facts.
In(x, y) :¬In(x, y) :SuspectIn(y) :¬SuspectIn(y) :Investigate(x, y) :¬Investigate(x, y) :C1, R1C1, R2 , C1, R3 , C3, R1
R2
R1 , R3(22)

C1, R2 , C1, R3 , C2, R1 ,C2, R2 , C2, R3 , C3, R1

The In tuples in (22) changed to incorporate the fact that C 3 has not yet beenfound, and the R 1 tuple in the SuspectIn relation has moved to reflect the fact thatno stolen car was found there, but the varied Investigate relation has changed too.
The UAV has already searched region R 1 for C1 , and it concludes that it is no longer
necessary to investigate whether C 3 is in R1 , but it is still unknown if the UAV shouldlook for C3 in one of the other regions.
Notice that without changing the definitions, the query results have changed toreflect the new knowledge situation. This will stay true until we modify the closure
policy or the logical theory describing the mission, in which case the definitions must
be recalculated. As long as the policy and theory stay the same the A KDB caches thecalculated definitions, improving efficiency.
In its current state of uncertainty, the UAV might either explain the two remaining
possibilities to a mission operator, asking for new information or advice on which
action to take, or continue on itself, e.g. by systematically searching for C 3 , first in
region R2 and then in R 3 . Assuming the latter alternative, and that the stolen car is infact located in one of the regions, the UAV will find it and successfully complete themission.

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Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

7. Conclusions
We have developed an experimental environment in which the ideas and techniques summarized in [DOH 06] can be investigated and explored. The system, called
the Approximate Knowledge Database, is implemented in JAVA and consists of a layered architecture based on a plug-in S QL database. The A KDB may be used either
as a service through an interface, or stand-alone through file input or a graphical userinterface, the Graphical Database Design Tool. This environment is intended to offer robust support to users wishing to explore the use of approximate databases forknowledge representation. A number of functionalities, not normally associated with
databases are also included such as quantifier elimination algorithms and generators
for W SC s and S NC s. The techniques used are tractable, but still allow for advanced
types of non-monotonic reasoning, including variants of default logics, limited circumscriptive reasoning as well as reasoning with the local closed world assumption.
The system is highly portable and may be embedded in robotic systems such as
the UAV platforms we use in our research. We are currently investigating the use of
A KDB s with planning algorithms which deal with incomplete information.

Acknowledgements
This work is partially funded by the Wallenberg Foundation under the WITASUAV Project, the NFFP03-539 COMPAS Project and the grant 3 T11C 023 29 of thePolish Ministry of Science and Information Society Technologies.

Appendix
A. Additional Functionality
This appendix describes some additional functionality available in the current version of our A KDB package.

A.1. Graphical Database Design
Even if not a necessary functional part of a system, a graphical user interface canoften substantially simplify experimentation. Since we built an experimental platform
for a collection of new deductive database techniques not yet extensively explored,such functionality is beneficial. This is the idea behind the Graphical Database Design
Tool, shown in Figure 3, for the A KDB that provides an environment where knowledge
bases, complete with relation definitions, facts, rules, and theories, can be created,changed, or removed. The interface is built upon a window system, where each relation, theory, policy, or query, has its own window. The windows can then be connected
to link a query with a policy and a theory.

Approximate databases

27

Figure 3. A screenshot of the Graphical Database Design user interface.
A.2. DLS∗D LS ∗ is substantial for many solutions offered by A KDBs. It is also possible tocall the D LS ∗ algorithm using a www interface (see [MAG 05b]).

A.3. Weakest Sufficient and Strongest Necessary Conditions
Weakest sufficient (W SC) and strongest necessary (S NC) conditions have been introduced in [LIN 00] in the framework of the classical propositional logic and thengeneralized to the first-order case in [DOH 00b]. W SC s and S NCs have many important applications, including 10 building communication interfaces between agents,
modularization and information hiding, knowledge compilation and theory approximation, abduction and hypotheses generation, reasoning with reduced data sets.
10. For a deeper discussion see, e.g., [LIN 00, DOH 00b, DOH 06, DOH 04b, DOH 04e].

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Journal of Applied Non-Classical Logics — January 5, 2006, 2nd submission.

W SC s and S NC s can be calculated by right-clicking the title bar of a theory windowand selecting [Calculate WSC] or [Calculate SNC], respectively. The background theory of the WSC or SNC will consist of the conjunction of the formulas in the theory
window, while the formula to be explained ( formula ) is entered into a dialog box,
as is a list of relations ( rel-sym-list ) that should be projected out of the resulting
explanation. The computation method used here has been proposed in [DOH 00b],where W SC s and S NC s are first characterized by means of the second-order formulas
and then reduced by applying the D LS ∗ . For a large class of formulas (semi-Horn)
reduction of W SC s and S NC s to the first-order or fixpoint logic is guaranteed.

A.4. Default Rules
An important functionality depends on providing default rules in the spirit ofdefault logic of Reiter [REI 80] and followers. Unfortunately, the original logic isvery complex. To make it tractable, we use the idea of approximations, as initiatedin [DOH 02]. 11 The idea is to formulate Reiter’s default rules A : B as intensionalC
rules of the form A+ ∧B ⊕ → C + .
The A KDB engine allows one to deal with such rules. However, in order to facilitate the use of default rules, a part of the graphical interface specialized to deal withdefaults is under development.

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