This document mixes HTML and
FL
(a KRL, i.e., a Knowledge Representation Language; it is here used in the
preformatted text parts).
It is an input+backup+presentation file for the
Multi-Source Ontology (MSO).
The MSO i) aligns and extends several ontologies, and
ii) can be extended by Web users via the shared KB server
WebKB-2 and its
shared KB (knowledge base) editing protocols.
In this document, for supertype links,
"/^" is used instead of the more common "<" symbol.
Similarly, for subtype links, "\." is used instead of the more common ">" symbol.
The ontology in this document is original. Some content is added to it from time to time.
It is part of a set of complementary "ontologies on formal languages":
The goals, underlying ideas, conventions and content of these ontologies are explained and illustrated in this article.
Table of contents
1. Between a property AE and an RDF reified statement
2. Between frames and conjunctions of binary relations from a same source
3. Between binary relations and non-binary ones
4. Between a property AE and an RDF reified statement
In each rule below, ":-" means "<=" and
the AE in bold font is the AEs (partially) defined by the rule.
Each r_descr relation relates a Descr_content (thing) to an AE.
//Note: in FL, by default, variables prefixed by "%" are universally quantified
// in their minimal context (e.g., a rule if they are in a rule)
[%prop /^^ Property r_op: %op,
r_args: .[(%propSource r_descr of: %describedPropSource)
(%propDest r_descr of: %describedPropDest) ]
<=> [%prop rdf#predicate: %op, rdf#subject: %describedPropSource,
rdf#object: %describedPropDest ];
/* In RIF-BLD/RIF-PS:
Forall ?prop ?op ?propSource ?describedPropSource ?propDest ?describedPropDest (
?prop[rdf:type Property r_op->?op r_args->List(?propSource ?propDest)]
:- And (?prop [rdf:predicate->?op rdf:subject->?describedPropSource
rdf:object->?describedPropDest ]
?describedPropSource[r_descr->?propSource]
?describedlPropDest[r_descr->?propDest] ) )
Forall ?prop ?op ?propSource ?describedPropSource ?propDest ?describedPropDest (
?prop [rdf:predicate->?op rdf:subject->?describedPropSource
rdf:object->?describedPropDest ]
:- And( ?prop[rdf:type Property r_op->?op r_args->List(?propSource ?propDest)]
?describedPropSource[r_descr->?propSource]
?describedlPropDest[r_descr->?propDest] ) )
*/
The next rule relates the structures of two AEs (%c and %f) that are
description instruments of something that is both a
Thing_that_can_be_represented_via_binary-relations_from_a_same_source
and a Thing_that_can_be_represented_via_a_frame.
By default, in KRLO, these two types are related by an equivalence relation: see
this Figure 4.2 of this article.
However, as explained in Section 3 of te same article, if the represented knowledge
has more expressiveness than OWL-RL and RIF-BLD, this equivalence is not (always) true.
Hence, to get the translation results she wants, i.e., for the next rule to apply only
to certain types of frames and certain types of conjunctions of binary relations from
a same source, the end-user can modify this equivalence relation by specializing its
source type or destination type, hence restricting them. She can also remove this
equivalence or keep it and have more complete results even if they may not always
be correct.
Note: alone, the next rule is not sufficient to correctly handle quantifiers around
the conjunction or the use of cardinality restrictions within the frame. This is
handled by other rules.
And{[%relSource = %f_head]
[%halfProperties r_half-property-item_matching_this_binary-relation: %binRel]
[%binaryRelations r_binary-relation_item_matching_this_half-property: %halfProperty] }
<= [%t /^^ Thing_that_can_be_represented_via_binary-relations_from_a_same_source
Thing_that_can_be_represented_via_a_frame,
r_descr_instrument: (%c /^^ Conjunction_of_binary-relations_from_a_same_source,
r_args: (%binaryRelations pred#list-contains:
(%binRel r_1st_arg: %relSource) ) )
r_descr_instrument: (%f /^^ Frame_as_non-token-phrase, r_op: %f_head,
r_args: (%halfProperties pred#list-contains: %halfProperty) ) ];
//With the following definitions:
Thing_that_can_be_represented_via_a_frame
<=> Thing_that_can_be_represented_via_binary-relations_from_a_same_source;
r_half-property-item_matching_this_binary-relation ~[?halfProperties, ?binRel]
:= [?halfProperties pred:list-contains:
(a thing ?halfProperty r_op: (an Operator ?relType r_op of: ?binRel),
r_1st_arg: (a Relation_node ?relDest r_2nd_arg of: ?binRel) ) ];
r_binary-relation_item_matching_this_half-property ~[?binaryRelations, ?halfProperty]
:= [?binaryRelations pred:list-contains:
(a thing ?binRel r_op: (an Operator ?relType r_op of: ?halfProperty),
r_2ng_arg: (a Relation_node ?relDest r_1st_arg of: ?halfProperty) ) ];
[%at r_1st_arg: %arg1] <=> [%at r_args: List(%arg1 | %l)];
[%at r_2nd_arg: %arg2] <=> [%at r_args: List(%arg1 %arg2 | %l)];
/* In RIF-BLD/RIF-PS:
Forall ?t ?c ?binaryRelations ?binRel ?f ?f_head ?halfProperties ?halfProperty (
And( ?relSource = ?f_head
r_half-property-item_matching_this_binary-relation(?halfProperties ?binRel)
r_binary-relation_item_matching_this_half-property(?binaryRelations ?halfProperty) )
:- And( ?t[rdf:type->Thing_that_can_be_represented_via_binary-relations_from_a_same_source
rdf:type->Thing_that_can_be_represented_via_a_frame
r_descr_instrument->?c r_descr_instrument->?f ]
?c[rdf:type->Conjunction_of_binary-relations_from_a_same_source
r_args->?binaryRelations ]
?f[rdf:type->Frame_as_non-token-phrase r_op->?f_head
r_args->?halfProperties ]
External(pred:list-contains(?binaryRelations ?binRel))
?binRel[r_1st_arg->?relSource]
External(pred:list-contains(?halfProperties ?halfProperty)) ) )
//With the following definitions:
Forall ?t (
?t[rdf:type->Thing_that_can_be_represented_via_a_frame]
:- ?t[rdf:type->Thing_that_can_be_represented_via_binary-relations_from_a_same_source] )
Forall ?t (
?t[rdf:type->Thing_that_can_be_represented_via_binary-relations_from_a_same_source]
:- ?t[rdf:type->Thing_that_can_be_represented_via_a_frame] )
Forall ?halfProperties ?halfProperty ?binRel ?relType ?relDest (
r_half-property-item_matching_this_binary-relation(?halfProperties ?binRel)
:- And( External(pred:list-contains(?halfProperties ?halfProperty))
?halfProperty[r_op->?relType r_1st_arg->?relDest]
?binRel[r_op->?relType r_2nd_arg->?relDest] ) )
Forall ?binaryRelations ?halfProperty ?binRel ?relType ?relDest (
r_binary-relation_item_matching_this_half-property(?binaryRelations ?halfProperty)
:- And( External(pred:list-contains(?binaryRelations ?binRel))
?binRel[r_op->?relType r_2nd_arg->?relDest]
?halfProperty[r_op->?relType r_1st_arg->?relDest] ) )
Forall ?at ?arg1 ?l ( ?at[r_1st_arg->?arg1]
:- ?at[r_args->List(?arg1 | ?l) ] )
Forall ?at ?arg1 ?arg2 ?l ( ?at[r_2nd_arg->?arg2]
:- ?at[r_args->List(?arg1 ?arg2 | ?l) ] )
*/
The next implication relates the structures of two AEs that are description instruments of
something that is both
a Thing_that_can_be_represented_via_a_non-binary_relation and
a Thing_that_can_be_represented_via_a_binary_relation.
By default, in KRLO, these two types are related by an equivalence relation
(the next rule can be used for translations in both directions).
And{ [%binRelArg1 = %nbrArg1] [%binRelArg2 = %nbrArgsExcept1st] }
<= [%t /^^ Thing_that_can_be_represented_via_a_non-binary_relation
Thing_that_can_be_represented_via_a_binary_relation,
r_descr_instrument: (%nonBinRel /^^ Non-binary_relation,
r_args: List(%nbrArg1 | %nbrArgsExcept1st) ),
r_descr_instrument: (%binRel /^^ Binary_relation,
r_1st_arg: %binRelArg1, r_2nd_arg: %binRelArg2 ) ];
//With the following definitions:
Thing_that_can_be_represented_via_a_non-binary_relation
has_descr_instrument<=: a Non-binary_relation, //actually "=:", not just "<=:" but "=: a ..."
<=> (Thing_that_can_be_represented_via_a_binary_relation, // cannot be translated to RIF-BLD
has_descr_instrument<=: a Binary_relation ); // without skolem functions
/* In RIF-BLD/RIF-PS:
Forall ?t ?binRel ?binRelArg1 ?binRelArg2 ?nonBinRel ?nbrArg1 ?nbrArgsExcept1st (
And ( ?binRelArg1 = ?nbrArg1 ?binRelArg2 = ?nbrArgsExcept1st )
:- And( ?t [rdf:type->Thing_that_can_be_represented_via_a_non-binary_relation
rdf:type->Thing_that_can_be_represented_via_a_binary_relation
r_descr_instrument->?nonBinRel r_descr_instrument->?binRel ]
?binRel[rdf:type->Binary_relation r_1st_arg->?binRelArg1
r_2nd_arg->?binRelArg2]
?nonBinRel[rdf:type->Non-binary_relation
r_args->List(?nbrArg1 | ?nbrArgsExcept1st) ] ) )
//With the following definitions:
Forall ?t ?nbr (
?t[rdf:type->Thing_that_can_be_represented_via_a_non-binary_relation]
:- And ( ?t[has_descr_instrument->?nbr] ?nbr[rdf:type->Non-binary_relation] ) )
Forall ?t ?br (
?t[rdf:type->Thing_that_can_be_represented_via_a_binary_relation]
:- And ( ?t[has_descr_instrument->?nbr] ?br[rdf:type->Binary_relation] ) )
Forall ?t (
?t[rdf:type->Thing_that_can_be_represented_via_a_binary_relation]
:- ?t[rdf:type->Thing_that_can_be_represented_via_a_non-binary_relation] )
*/
And{ [%fctArgs = %fRelArg1] [%fctResult = %fRelArgsExcept1st] }
<= [%t /^^ Thing_that_can_be_represented_via_a_functional_relation
Thing_that_can_be_represented_via_a_function,
r_descr_instrument: (%fRel /^^ Functional_relation, r_op: %fRelType,
r_args: List(%fRelArg1 | %fRelArgsExcept1st) ),
r_descr_instrument: (%fct /^ Function, r_op: %fctType,
r_args: %fctArgs, r_result: %fctResult ) ];
//With the following definitions:
Thing_that_can_be_represented_via_a_functional_relation
has_descr_instrument<=: a Functional_relation, //actually "=:", not just "<=:" but "=: a ..."
<=> (Thing_that_can_be_represented_via_a_function, // cannot be translated to RIF-BLD
has_descr_instrument<=: a Function ); // without skolem functions
/* In RIF-BLD/RIF-PS:
Forall ?t ?fct ?fctType ?fctArgs ?fctResult ?fRel ?fRelType ?fRelArg1 ?fRelArgsExcept1st (
And ( ?fctArgs = ?fRelArg1 ?fctResult = ?fRelArgsExcept1st )
:- And( ?t [rdf:type->Thing_that_can_be_represented_via_a_functional_relation
rdf:type->Thing_that_can_be_represented_via_a_function
r_descr_instrument->?fct r_descr_instrument->?fRel ]
?fRel[r_op->?fRelType r_args->List(?fRelArg1 | ?fRelArgsExcept1st)]
?fct[r_op->?fctType r_args->?fctArgs r_result->?fctResult] ) )
//With the following definitions:
Forall ?t ?fRel (
?t[rdf:type->Thing_that_can_be_represented_via_a_functional_relation]
:- And ( ?t[has_descr_instrument->?fRel] ?fRel[rdf:type->Functional_relation] ) )
Forall ?t ?fct (
?t[rdf:type->Thing_that_can_be_represented_via_a_function]
:- And ( ?t[has_descr_instrument->?fct] ?fct[rdf:type->Function] ) )
Forall ?t (
?t[rdf:type->Thing_that_can_be_represented_via_a_functional_relation]
:- ?t[rdf:type->Thing_that_can_be_represented_via_a_function] )
Forall ?t (
?t[rdf:type->Thing_that_can_be_represented_via_a_function]
:- ?t[rdf:type->Thing_that_can_be_represented_via_a_functional_relation] )