### Fuzzy Logic theory -- Jos De Roo

@prefix log: <http://www.w3.org/2000/10/swap/log#>.
@prefix math: <http://www.w3.org/2000/10/swap/math#>.
@prefix list: <http://www.w3.org/2000/10/swap/list#>.
@prefix owl: <http://www.w3.org/2002/07/owl#>.
@prefix xsd: <http://www.w3.org/2001/XMLSchema#>.
@prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>.
@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>.
@prefix e: <http://eulersharp.sourceforge.net/2003/03swap/log-rules#>.
@prefix fl: <http://eulersharp.sourceforge.net/2003/03swap/fl-rules#>.


### classes
fl:Lower a rdfs:Class;
	rdfs:comment "class of things that are the first element of e:valuation object".

fl:Upper a rdfs:Class;
	rdfs:comment "class of things that are the last element of e:valuation object".


### properties
fl:mu a rdf:Property;
	rdfs:domain rdf:List;
	rdfs:range xsd:decimal;
	rdfs:comment "to express fuzzy set membership e.g. (:x :C) fl:mu 0.8 says that :x a :C to a degree of 0.8".

fl:sigma a rdf:Property;
	rdfs:domain rdf:List;
	rdfs:range xsd:decimal;
	rdfs:comment "to express fuzzy subsethood e.g. (:C :D) fl:sigma 0.9 says that :C rdfs:subClassOf :D to a degree of 0.9".

fl:pi a rdf:Property, e:Builtin;
	rdfs:domain rdf:List;
	rdfs:range xsd:decimal;
	rdfs:comment "builtin to calculate fuzzy set membership according to Elpiniki Papageorgiou method".


### conjunction according to Łukasiewicz
{?C owl:intersectionOf ?E. ?E e:distinct [ list:first ?A; list:rest ?B].
 (?X ?A) fl:mu ?I. ?D owl:intersectionOf ?B. (?X ?D) fl:mu ?J. (?I ?J -1.0) math:sum ?K. (0.0 ?K) e:max ?M} => {(?X ?C) fl:mu ?M}.

### disjunction according to Łukasiewicz
{?C owl:unionOf ?E. ?E e:distinct [ list:first ?A; list:rest ?B].
 (?X ?A) fl:mu ?I. ?D owl:unionOf ?B. (?X ?D) fl:mu ?J. (?I ?J) math:sum ?K. (1.0 ?K) e:min ?M} => {(?X ?C) fl:mu ?M}.

### negation according to Łukasiewicz
{?C owl:complementOf ?A. (?X ?A) fl:mu ?I. (1.0 ?I) math:difference ?M} => {(?X ?C) fl:mu ?M}.

### implication
{?A rdfs:subClassOf ?B} => {(?A ?B) fl:sigma 1.0}.
{(?B ?A) fl:sigma ?I} => {(?A ?A) fl:sigma 1.0}.
{?C owl:complementOf ?A. (?B ?A) fl:sigma ?I. (1.0 ?I) math:difference ?M} => {(?B ?C) fl:sigma ?M}.


### Fuzzy Description Logic to Fuzzy Cognitive Map translation theory
{{?P a ?Q} e:valuation (?L ?U)} => {(?P (?Q fl:Lower)) fl:mu ?L}.
{{?P a ?Q} e:valuation (?L ?U)} => {(?P (?Q fl:Upper)) fl:mu ?U}.

{{?X rdfs:subClassOf ?Y} e:valuation (?L ?U)} => {((?X fl:Lower) (?Y fl:Lower)) fl:sigma ?L}.
{{?X rdfs:subClassOf ?Y} e:valuation (?L ?U)} => {((?X fl:Upper) (?Y fl:Upper)) fl:sigma ?U}.

{{?X!owl:complementOf rdfs:subClassOf ?Y} e:valuation (?L ?U). (1.0 ?L) math:difference ?M} => {((?Y fl:Lower) (?X fl:Lower)) fl:sigma ?M}.
{{?X!owl:complementOf rdfs:subClassOf ?Y} e:valuation (?L ?U). (1.0 ?U) math:difference ?M} => {((?Y fl:Upper) (?X fl:Upper)) fl:sigma ?M}.

{(?P (?C fl:Lower)) fl:pi ?X. (?P (?C fl:Upper)) fl:pi ?Y.
 ?Z e:optional {?X math:notGreaterThan ?Y. ?S log:equalTo (?X ?Y)}, {?X math:greaterThan ?Y. ?S log:equalTo (?Y ?X)}}
 =>
 {{?P a ?C} e:valuation ?S}.


### support
owl:Thing owl:intersectionOf ().
owl:Nothing owl:unionOf ().

{?C owl:intersectionOf [ list:rest ?B]} => {_:D owl:intersectionOf ?B}.
{?C owl:unionOf [ list:rest ?B]} => {_:D owl:unionOf ?B}.

{(?X ?A) fl:mu ?I} => {(?X owl:Thing) fl:mu 1.0}.
{(?X ?A) fl:mu ?I} => {(?X owl:Nothing) fl:mu 0.0}.