Representing and reasoning with qualitative geospatial relationships among regions is an important task in many geospatial applications. In this paper, we present a Datalog + rulebase, GeospatialRules, which can be used for this task. The rulebase is built on top of the Region Connection Calculus (RCC). It includes rules, facts, and queries. The rules in GeospatialRules consist of a set of rules that are equivalent to the Datalog+ fragment of the RCC axioms and additional rules which express part of the RCC knowledge that are not captured by the Datalog + fragment. The XML version of the rulebase complies to the Deliberation RuleML 1.01 standard, so that it allows the use of RuleML-compatible implementations for geospatial reasoning.
Introduction
Datalog+
Datalog+ is an extension of Datalog with existential quantiers, equality, and
falsity in rule conclusions. This extension addresses the modeling needs of many
applications, e.g. of knowledge-based data access [
Region Connection Calculus
The Region Connection Calculus (RCC) is an axiomization of spatial relations
between regions in rst-order logic [
On top of C, several binary spatial relations can be dened. The meanings and rst-order denitions of these relations are shown in Table 1. All top-level free variables are implicitly universally quantied. Among these relations, P,PP,TPP, and NTPP are asymmetric and their inverses can be dened. The notation i is used to represent the inverses for 2 fP; PP; TPP; NTPPg, e.g. PPi is the inverse of PP. The rst-order denition for the inverses are i(x; y) def (y; x). 1 http://deliberation.ruleml.org/1.01
Among these relations, the set fDC; EC; PO; EQ; TPP; NTPP; PPi; NTPPig are
provably jointly exhaustive and pairwise disjoint [
The rst-order denitions of Table 1 with inverses imply a lattice of all dened
relations, as shown in Figure 2 [
To assist with consistency checking and query answering in RCC, composition
tables are used [
Development of GeospatialRules
The GeospatialRules rulebase contains rules, facts and queries. The rules consist
of two parts of knowledge: (1) the Datalog + fragment of the rst-order RCC
denitions; (2) rules which express part of the RCC knowledge that are not
captured by (1). We will explain all the parts in the following paragraphs. The
rulebase will be shown here in the presentation syntax of PSOA RuleML [
In GeospatialRules, we use RCC-prexed CamelCase for the relation names to improve readability. The mapping between RCC relations and GeospatialRules relations is shown in Table 2:
The rst part of rules in GeospatialRules is the Datalog + fragment of the axioms (1)(12) and the denitions of inverse RCC relations shown in Section 2.2. Axioms (1) and (2) are kept unchanged in GeospatialRules: Forall ?X ( RCCConnected(?X ?X) ) Forall ?X ?Y ( RCCConnected(?Y ?X) :- RCCConnected(?X ?Y) ) Axiom (4) is equivalent to
(P(x; y) ! 8z[C(z; x) ! C(z; y)]) ^ (8z[C(z; x) ! C(z; y)] ! P(x; y)) The second conjunct is non-Horn, hence cannot be expressed in Datalog +. The rst conjunct can be transformed to the equivalent Datalog rule
C(z; x) ^ P(x; y) ! C(z; y); which is written in GeospatialRules as
The correctness of the transformation can be shown by the following rst-order equivalences:
The axioms (3),(5)(12), and the denitions of the inverse relations are all of the form r Vim=1 pi ^ Vjn=1 :qj, where pi and qj are relational atoms or existential formulas. Such a formula is equivalent to the conjunction of two formulas (r ! m n m n ^ pi ^ ^ :qj) ^ (^ pi ^ ^ :qj ! r) i=1 j=1 i=1 j=1 The rst formula is further separated into m + n rules fr ! pigim=1 and fr ^ qj ! ?gjn=1, where ? denotes falsity (represented by Or() in the presentation and XML syntaxes in a queryable manner). The second formula is in Datalog+ only if there are no negations in the premise, i.e. for n = 0. In such a case the formula is of the form Vim=1 pi ! r. For any pi being an existential formula, we move all premise-side existentially quantied variables to top-level universally quantied variables. Following are the rules belonging to this part. The rules employing Or() conclusions are negative-constraint rules which can be used for checking the consistency of the rulebase by querying Or(). ) )
Or() :- And(RCCConnected(?X ?Y) RCCDisconnected(?X ?Y)) ) Forall ?X ?Y ?Z (
RCCOverlapped(?X ?Y) :- And(RCCPartOf(?Z ?X) RCCPartOf(?Z ?Y)) ) Forall ?X ?Y (
Exists ?Z (And(RCCPartOf(?Z ?X) RCCPartOf(?Z ?Y))) :
RCCOverlapped(?X ?Y) The above existential rule is transformed from axiom (5) in Table 1. The rule says if a region ?X overlaps with a region ?Y, then there exists a region ?Z that is part of both ?X and ?Y.
Forall ?X ?Y ( Or() :- And(RCCDiscrete(?X ?Y) RCCOverlapped(?X ?Y)) ) Forall ?X ?Y (
RCCExternallyConnected(?X ?Y) :
And(RCCConnected(?X ?Y) RCCDiscrete(?X ?Y)) ) Forall ?X ?Y ( RCCConnected(?X ?Y) :- RCCExternallyConnected(?X ?Y) ) Forall ?X ?Y ( RCCDiscrete(?X ?Y) :- RCCExternallyConnected(?X ?Y) ) Forall ?X ?Y ( RCCEqual(?X ?Y) :- And(RCCPartOf(?X ?Y) RCCPartOf(?Y ?X)) ) Forall ?X ?Y ( RCCPartOf(?X ?Y) :- RCCEqual(?X ?Y) ) Forall ?X ?Y ( RCCPartOf(?Y ?X) :- RCCEqual(?X ?Y) ) Forall ?X ?Y ( RCCOverlapped(?X ?Y) :- RCCPartiallyOverlapped(?X ?Y) ) Forall ?X ?Y ( Or() :- And(RCCPartiallyOverlapped(?X ?Y) RCCPartOf(?X ?Y)) ) Forall ?X ?Y ( Or() :- And(RCCPartiallyOverlapped(?X ?Y) RCCPartOf(?Y ?X)) ) Forall ?X ?Y ( RCCPartOf(?X ?Y) :- RCCProperPartOf(?X ?Y) ) Forall ?X ?Y ( Or() :- And(RCCProperPartOf(?X ?Y) RCCPartOf(?Y ?X)) ) Forall ?X ?Y ?Z (
RCCTangentialProperPartOf(?X ?Y) :
And(RCCProperPartOf(?X ?Y)
RCCExternallyConnected(?Z ?X)
RCCExternallyConnected(?Z ?Y)) ) Forall ?X ?Y (
Exists ?Z (
And (RCCExternallyConnected(?Z ?X) RCCExternallyConnected(?Z ?Y)) ) :- RCCTangentialProperPartOf(?X ?Y)
Or() :
And(RCCNonTangentialProperPartOf(?X ?Y)
RCCExternallyConnected(?Z ?X)
RCCExternallyConnected(?Z ?Y)) ) Forall ?X ?Y ( RCCProperPartOf(?X ?Y) :- RCCTangentialProperPartOf(?X ?Y) ) Forall ?X ?Y ( RCCProperPartOf(?X ?Y) :- RCCNonTangentialProperPartOf(?X ?Y) ) The following rules are transformed from the denitions of the inverses of nonsymmetric RCC relations. For each relation 2 fP; PP; TPP; NTPPg, there are two rules (x; y) ! i(y; x) and i(x; y) ! (y; x).
Forall ?X ?Y ( RCCInversePartOf(?Y ?X) :- RCCPartOf(?X ?Y) ) Forall ?X ?Y ( RCCPartOf(?Y ?X) :- RCCInversePartOf(?X ?Y) ) Forall ?X ?Y ( RCCInverseProperPartOf(?Y ?X) :- RCCProperPartOf(?X ?Y) ) Forall ?X ?Y ( RCCProperPartOf(?Y ?X) :- RCCInverseProperPartOf(?X ?Y) ) Forall ?X ?Y (
RCCInverseTangentialProperPartOf(?Y ?X) :- RCCTangentialProperPartOf(?X ?Y) ) Forall ?X ?Y (
RCCTangentialProperPartOf(?Y ?X) :- RCCInverseTangentialProperPartOf(?X ?Y) ) Forall ?X ?Y (
RCCInverseNonTangentialProperPartOf(?Y ?X) :- RCCNonTangentialProperPartOf(?X ?Y) ) Forall ?X ?Y (
RCCNonTangentialProperPartOf(?Y ?X) :- RCCInverseNonTangentialProperPartOf(?X ?Y)
The second part of rules in GeospatialRules contain rules that express part of the RCC knowledge that are not captured by the extracted Datalog + fragment. We rst add all subsumptions in Figure 2 that are not implied by those extracted rules: Forall ?X ?Y ( RCCConnected(?X ?Y) :- RCCOverlapped(?X ?Y)) Forall ?X ?Y ( RCCOverlapped(?X ?Y) :- RCCPartOf(?X ?Y)) Forall ?X ?Y ( RCCDiscrete(?X ?Y) :- RCCDisconnected(?X ?Y) ) Besides these subsumption rules, we also add rules for inferring new geospatial relations among entities from the relations explicitly expressed as facts in the rulebase. These rules are the Datalog + subset of the rst-order form of the RCC8 composition table.
Forall ?X ?Y ?Z (
RCCDiscrete(?X ?Z) :
And(RCCTangentialProperPartOf(?X ?Y) RCCExternallyConnected(?Y ?Z)) ) Forall ?X ?Y ?Z (
RCCProperPartOf(?X ?Z) :
And(RCCTangentialProperPartOf(?X ?Y) RCCTangentialProperPartOf(?Y ?Z)) ) Forall ?X ?Y ?Z (
RCCNonTangentialProperPartOf(?X ?Z) :
And(RCCPartOf(?X ?Y) RCCNonTangentialProperPartOf(?Y ?Z)) ) Forall ?X ?Y ?Z (
RCCDisconnected(?X ?Z) :
And(RCCNonTangentialProperPartOf(?X ?Y) RCCExternallyConnected(?Y ?Z)) ) Forall ?X ?Y ?Z (
RCCNonTangentialProperPartOf(?X ?Z) :
And(RCCNonTangentialProperPartOf(?X ?Y)
RCCTangentialProperPartOf(?Y ?Z)) ) Forall ?X ?Y ?Z (
RCCProperPartOf(?X ?Z) :
And(RCCProperPartOf(?X ?Y) RCCProperPartOf(?Y ?Z)) )
The rules in the second part compensate for a portion of information that is lost when the Datalog + fragment is extracted. These rules do not recover all the lost knowledge since some of it cannot be expressed in Datalog +. However, they ensure that all subsumptions between the RCC relations are preserved in our rulebase. Moreover, the rules from the RCC8 composition table allow the users to make inferences that are usually needed in geospatial applications, which we will show in the next paragraph.
The remaining of GeospatialRules consists of example facts and queries. The facts in the KB describe geospatial relationships among certain regions in North America, as well as the Pacic and the Atlantic oceans: RCCProperPartOf(USA NorthAmerica) RCCProperPartOf(California USA) RCCProperPartOf(Virginia USA) RCCExternallyConnected(Pacific California) RCCExternallyConnected(Atlantic Virginia) RCCExternallyConnected(Pacific USA) RCCExternallyConnected(Atlantic USA) State(California) State(Virginia) Ocean(Pacific) Ocean(Atlantic) Based on the above rules and facts, the following example queries can be asked: 1. RCCProperPartOf(California NorthAmerica)
The ground query asks whether California is a proper part of North America. This query atom can be inferred by applying the rule
RCCProperPartOf(?X ?Z) :
And(RCCProperPartOf(?X ?Y) RCCProperPartOf(?Y ?Z)) to the facts RCCProperPartOf(California USA) and RCCProperPartOf(USA NorthAmerica). 2. And(State(?St) RCCTangentialProperPartOf(?St USA))
The query asks which states in the rulebase are tangential proper parts of the USA. From the geospatial relations among Pacific, California, and USA, and the denition rule of RCCTangentialProperPartOf : the atom RCCTangentialProperPartOf(California USA) can be derived. Hence California is an answer of the query. Similarly, another answer, Virginia, can also be derived. 3. And( State(?St)
Ocean(?O) RCCProperPartOf(?St NorthAmerica)
RCCConnected(?St ?O)) This query asks for any state ?St which is a proper part of North America and connects with an ocean ?O. From the rst query, we know that California is a proper part of North America. And since RCCConnected is a super-relation of RCCExternallyConnected , RCCConnected(California Pacific) can also be inferred. Therefore, one answer of the query is ?St=California, ?O=Pacific . Another answer, ?St=Virginia,?O=Atlantic , can be obtained in a similar way.
The full RuleML/XML version of the rulebase, 2 which complies to the Deliberation RuleML 1.01 specication, is included in Appendix A. It uses an abbreviated encoding allowed by the specication, which omits some edge tags and default attribute values to make the document more compact and easier to read. The compact version can be expanded to a normalized version using the Deliberation RuleML Normalizer XSLT stylesheet for Disjunctive Hornlog+ with Naf and Neg.3 To increase readability and comprehensibility of the rulebase, we use proper indentations and add verbal denitions to each of the RCC relation as comments. 4
Conclusions and Future Work In this paper, we describe a Datalog + rulebase, GeospatialRules, which can be used in geospatial applications for reasoning with geospatial relations. The rulebase is built on top of RCC relations. It contains rules, facts, and queries, where the rules consist of: (1) a Datalog + fragment of the rst-order RCC relation denitions; (2) additional rules expressing part of the RCC knowledge that are not captured in (1), which ensure all subsumptions between RCC relations are preserved. The rulebase enables the use of RuleML-compatible implementations for qualitative geospatial reasoning.
In Semantic Web applications, graph representations are widely used for
modeling knowledge and answering queries. In these representations, an object
consists of a globally unique Object IDentier (OID) typed by a class and
described by an unordered collection of n attribute-value slots, where the value can
again identify an object. Future versions of the rulebase can represent regions
by OIDs with slots describing its connections with other regions, using graph
versions of RCC relations as slot names, e.g. dened in the GeoSPARQL
standard [
Further evaluation of the eciency of reasoning in GeospatialRules is also an important research topic for the wide use of the rulebase. More facts and test queries will need to be developed for the evaluation. The rulebase and queries 2 The le can be validated at and downloaded from http://deliberation.ruleml.
org/1.01/exa/RulebaseCompetition2014/GeospatialRulesRCC.ruleml
3 http://wiki.ruleml.org/index.php/Specification_of_Deliberation_RuleML_
1.01#Appendix_4:_Normalizing_or_Upgrading_RuleML_Files_with_XSLT
will be posed to existing implementations of RuleML, e.g. PSOATransRun [
GeospatialRules in RuleML/XML <Var>X</Var> </Atom> </Forall> </Assert> <Assert> <!-- Symmetry of RCCConnected --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCConnected</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCConnected</Rel> <Var>Y</Var> <Var>X</Var> </Atom> </then> </Implies> </Forall> </Assert> <!-- In RCC, a region X is part of a region Y if and only if all regions
that connect to X also connect to Y. <Var>X</Var> </Atom> </And> </if> <then> <Atom> <Rel>RCCEqual</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCEqual is a subrelation of RCCPartOf --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCEqual</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCEqual</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCPartOf</Rel> <Var>Y</Var> <Var>X</Var> </Atom> </then> </Implies> </Forall> </Assert> <!-- In RCC, two region X and Y are partially overlapped if and only if X and Y are overlapped, X is not part of Y, and Y is not part of Y. --> <Assert> <!-- RCCPartiallyOverlapped is a subrelation of RCCOverlapped --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies>
<if> <Assert> <!-- RCCProperPartOf is a subrelation of RCCPartOf --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <And> <Atom> <Rel>RCCProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> <Atom> <Rel>RCCPartOf</Rel> <Var>Y</Var> <Var>X</Var> </Atom> </And> </if> <then>
<Or/> </then> </Implies> </Forall> </Assert> <!-- In RCC, a region X is a tangential proper part of a region Y if and only if
X is a proper part of Y and there exists a region Z such that Z is externally connected to both X and Y. --> <Assert> <Forall> <Var>X</Var> <Var>Y</Var> <Var>Z</Var> <Implies> <if> <And> <Atom> <Rel>RCCProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Var>Z</Var> <Var>X</Var> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Var>Z</Var> <Var>Y</Var> </Atom> </And> </if> <then> <Atom> <Rel>RCCTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Exists> <Var>Z</Var> <And> <Atom> <Rel>RCCExternallyConnected</Rel> <Var>Z</Var> <Var>X</Var> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Var>Z</Var> <Var>Y</Var> </Atom> </And> </Exists> </then> </Implies> </Forall> </Assert> <!-- In RCC, a region X is a nontangential proper part of a region Y if and only if X is a proper part of Y and there does not exist a region Z such that Z is externally connected to both X and Y. <Rel>RCCExternallyConnected</Rel> <Var>Z</Var> <Var>X</Var> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Var>Z</Var> <Var>Y</Var> </Atom> </And> </if> <then>
<Or/> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCTangentialProperPartOf is a subrelation of RCCProperPartOf --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCNonTangentialProperPartOf is a subrelation of RCCProperPartOf
--> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCNonTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </then> </Implies> </Forall> </Assert> <!-- Definitions of the inverses of non-symmetric RCC relations: RCCPartOf,
RCCProperPartOf, RCCTangentialProperPartOf, and RCCNonTangentialProperPartOf. --> <Assert> <Assert> <!-- RCCInverseProperPartOf and RCCProperPartOf --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCInverseProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCProperPartOf</Rel> <Var>Y</Var> <Var>X</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCTangentialProperPartOf and RCCInverseTangentialProperPartOf --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCInverseTangentialProperPartOf</Rel> <Var>Y</Var> <Var>X</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCTangentialProperPartOf and RCCInverseTangentialProperPartOf --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCInverseTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCTangentialProperPartOf</Rel> <Var>Y</Var> <Var>X</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCNonTangentialProperPartOf and RCCInverseNonTangentialProperPartOf --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCNonTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCInverseNonTangentialProperPartOf</Rel> <Var>Y</Var> <Var>X</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCNonTangentialProperPartOf and RCCInverseNonTangentialProperPartOf --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCInverseNonTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCNonTangentialProperPartOf</Rel> <Var>Y</Var> <Var>X</Var> </Atom> </then> </Implies> </Forall> </Assert> <!-- Additional rules expressing subsumptions between RCC relations that are not implied by the Datalog+ fragment. </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCPartOf is a subrelation of RCCOverlapped --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCOverlapped</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- RCCDisconnected is a subrelation of RCCDiscrete --> <Forall> <Var>X</Var> <Var>Y</Var> <Implies> <if> <Atom> <Rel>RCCDisconnected</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </if> <then> <Atom> <Rel>RCCDiscrete</Rel> <Var>X</Var> <Var>Y</Var> </Atom> </then> </Implies> </Forall> </Assert> <!-- Additional rules extracted from the RCC8 composition table. --> <Assert> <Forall> <Var>X</Var> <Var>Y</Var> <Var>Z</Var> <Implies> <if> <And> <Atom> <Rel>RCCPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> <Atom> <Rel>RCCDisconnected</Rel> <Var>Y</Var> <Var>Z</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <Forall> <Var>X</Var> <Var>Y</Var> <Var>Z</Var> <Implies> <if> <And> <Atom> <Rel>RCCPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> <Atom> <Rel>RCCNonTangentialProperPartOf</Rel> <Var>Y</Var> <Var>Z</Var> </Atom> </And> </if> <then> <Atom> <Rel>RCCNonTangentialProperPartOf</Rel> <Var>X</Var> <Var>Z</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <Forall> <Var>X</Var> <Var>Y</Var> <Var>Z</Var> <Implies> <if> <And> <Atom> <Rel>RCCNonTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Var>Y</Var> <Var>Z</Var> </Atom> </And> </if> <then> <Atom> <Rel>RCCDisconnected</Rel> <Var>X</Var> <Var>Z</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert>
<Forall> <Var>X</Var> <Var>Y</Var> <Var>Z</Var> <Implies> <if> <And> <Atom> <Rel>RCCNonTangentialProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> <Atom> <Rel>RCCTangentialProperPartOf</Rel> <Var>Y</Var> <Var>Z</Var> </Atom> </And> </if> <then> <Atom> <Rel>RCCNonTangentialProperPartOf</Rel> <Var>X</Var> <Var>Z</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <Forall> <Var>X</Var> <Var>Y</Var> <Var>Z</Var> <Implies> <if> <And> <Atom> <Rel>RCCProperPartOf</Rel> <Var>X</Var> <Var>Y</Var> </Atom> <Atom> <Rel>RCCProperPartOf</Rel> <Var>Y</Var> <Var>Z</Var> </Atom> </And> </if> <then> <Atom> <Rel>RCCProperPartOf</Rel> <Var>X</Var> <Var>Z</Var> </Atom> </then> </Implies> </Forall> </Assert> <Assert> <!-- Facts of the rulebase --> <Atom> <Rel>RCCProperPartOf</Rel> <Ind>USA</Ind> <Ind>NorthAmerica</Ind> </Atom> <Atom> <Rel>RCCProperPartOf</Rel> <Ind>California</Ind> <Ind>USA</Ind> </Atom> <Atom> <Rel>RCCProperPartOf</Rel> <Ind>Virginia</Ind> <Ind>USA</Ind> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Ind>PacificOcean</Ind> <Ind>California</Ind> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Ind>AtlanticOcean</Ind> <Ind>Virginia</Ind> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Ind>PacificOcean</Ind> <Ind>USA</Ind> </Atom> <Atom> <Rel>RCCExternallyConnected</Rel> <Ind>AtlanticOcean</Ind> <Ind>USA</Ind> </Atom> <Atom> <Rel>State</Rel> <Ind>California</Ind> </Atom> <Atom> <Rel>State</Rel> <Ind>Virginia</Ind> </Atom> <Atom> <Rel>Ocean</Rel> <Ind>PacificOcean</Ind> </Atom> <Atom> <Rel>Ocean</Rel> <Ind>AtlanticOcean</Ind> </Atom> </Assert> <Query> <!-
Is California a proper part of North America? Answer: True The query atom is obtained by applying the rule Forall ?X ?Y ?Z (
RCCProperPartOf(?X ?Z) :- And(RCCProperPartOf(?X ?Y) RCCProperPartOf(?Y ?Z)) ) to the facts RCCProperPartOf(California USA) and RCCProperPartOf(California USA). --> <Atom> <Rel>RCCProperPartOf</Rel> <Ind>California</Ind> <Ind>NorthAmerica</Ind> </Atom> </Query> <Query> <!-
Which states are tangential proper part of the USA? Answers: St: California
St: Virginia ) to the facts of relations RCCProperPartOf and RCCExternallyConnected.