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Definitions
The definitions below formalize the semantic model elements and provide interpretation graph patterns for the SDG. Examples of extracted graphs and the
abstract graph patterns can be found in Figure 1.
Named and Non-named Entities: will be represent respectively by ne and
∼ ne. We use e to denote indistinguishably named/non-named entities. They
are interpreted as finite subsets of an infinite set U of IRIs. Thus, [[ne]] ∈ U ,
[[∼ ne]] ∈ U and [[e]] ∈ U .
Basic Triple: a triple tr = (es , p, eo ) where es , eo represent entities associated
respectively with the subject (s) and object (o), and p represents a relation
between es and eo . A basic triple is called core triple (denoted by trc ) when
es = nes and eo = neo are both named entitites. Otherwise, it is called semicore triple and denoted by trsc . The interpretation of a basic triple is such that
[[tr]] = ([[es ]], [[p]], [[eo ]]) ∈ U × U × U .
Reification Triple: a triple trrei = (tr, reilink , reiobj ) where tr represents a
basic triple, reilink represents relation and reiobj represents a reification object
(i.e., an entity, a value or a triple). A temporal reification is a special kind
of reification triple where reilink has a special stamp (time) and reiobj represents explicit or implicit data references. The interpretation of a reification triple
[[trrei ]] = ([[tr]], [[reilink ]], [[reiobj ]]) ∈ U 3 × U × U means that the basic triple
tr is reified in reiobj through relation reilink .
Quantifier Operators & Generic Operators: a triple opt = (eo , oplink , op)
where eo is the object element in a basic triple tr, op represents a specific operator of eo wrt oplink . The interpretation of a quantifier or generic operator
is [[opt]] = ([[proj3 (tr)]], [[oplink ]], [[op]]) = ([[eo ]], [[oplink ]], [[op]]) ∈ U × U × U ,
where proj3 (tr) is a projection map that takes an element (es , p, eo ) to the value
eo meaning that the quantifier or generic operator op is applied to object eo
through link oplink .
Co-Reference Operators
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– Conjunctive Co-Reference: set of triples ccr = i=0 {(e, conjlinki , nei )}
which means that entity e is composed by the conjunction of named entities
nei . The interpretation of a conjunctive co-reference is [[ccr]]
Vn = {([[e]],
[[conjlinki ]], [[nei ]]) ∈ U ×U ×U : [[e]] = [[proj3 (tr)]] and i=0 [[nei ]] same as
[[e]]}, meaning that the entity e is related through p to ei+1 which is formed
by the conjunction of (conjlink ) named entities ne0 , ne1 , · · · , nen .
– Possessive/Reflexive/Demonstrative Co-Reference: set of triples pcr =
{(∼ nei , coreflink , pr) , (pr, coreflink , ej )} where coreflink associates nonnamed entities ∼ nei with ej through pronoun pr if there is a basic triple
tr = (∼ nei , p, ej ). The interpretation of this kind of co-reference is [[pcr]] =
{([[proj1 (tr)]], [[coreflink ]], [[pr]]), ([[pr]], [[coreflink ]], [[proj3 (tr)]]) ∈ U ×U ×
U : tr = (∼ nei , p, ej )}
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With these elements we can define an extracted graph G from a given
corpus as a set of (basic and reified ) triples and (generic, quantifier and coreference) operators. In an extracted graph:
basic path Pb : is a sequence of basic triples Pb =< tr1 , tr2 , · · · , trn > such
that tri = (ei , rellinki , ei+1 ) for all i ∈ [1, n]. The interpretation of a basic path
[[Pb ]] =< [[e0 ]], [[rellink0 ]], · · · , [[rellinkn−1 ]], [[en ]] > is such that for tri and tri+1
∀i ∈ [1, n − 1], we have [[proj3 (tri )]] = [[proj1 (tri+1 )]].
reified path Pr : is a basic path such that there are reified triples associated
with some of tri ’s in the sequence Pb =< tr1 , tr2 , · · · , trn >. The interpretation
of a reified path is such that for a basic triple tri and reified triples trreij and
trreij+1 and their respectively interpretations, we have one of the following cases:
– [[proj3 (tri )]] = [[proj1 (trreij )]] if j = i + 1,
– [[proj3 (trreij )]] = [[proj1 (tri )]] if j = i − 1 (in this case, the interpretation
of reified triple in the path should be completed before the interpretation of
the next basic triple in Pb ),
– [[proj3 (trreij )]] = [[proj1 (trreij+1 )]]. (in this case, the interpretation of reified
triple trreij in the path should be completed before the interpretation of
trreij+1 in Pb ).
operational path Po : is a basic path such that there are operators associated
with some of entities ei ’s in triples tri ’s in the sequence Pb =< tr1 , tr2 , · · · , trn >.
The interpretation of a operational path is such that for a basic or reified triples
tri and tri+1 ∈ Po with their respectively interpretations, we have one of the
following cases:
– [[proj3 (tri )]] = [[proj1 (tri+1 )]] and ([[opi ]] = ([[proj3 (tri )]], [[quantlink ]],
[[quant]]) or [[opi+1 ]] = ([[proj3 (tri+1 )]], [[quantlink ]], [[quant]]), or
– [[proj1 (ccr)]] = [[proj3 (tri )]] = [[proj1 (tri+1 )]] or [[proj1 (ccr)]] = [[proj3 (tri+1 )]],
or
– ∀t ∈ ccr [[proj1 (t)]] = [[proj3 (tri )]] = [[proj1 (tri+1 )]] or [[proj1 (t)]] =
[[proj3 (tri+1 )]], or
– (([[proj1 (tri )]], [[coreflinki ]], [[pr]]) and ([[pr]], [[coreflinki ]], [[proj3 (tri )]]) =
([[pr]], [[coreflinki ]], [[proj1 (tri+1 )]])) or
(([[proj1 (tri+1 )]], [[coreflinki ]], [[pr]]) = ([[proj3 (tri )]], [[coreflinki ]], [[pr]])
and ([[pr]], [[coreflinki ]], [[proj3 (tri+1 )]])).
complex path Pc : contains both reified and operational paths.
The interpretation of a basic triple with operators should be done before the
reification when it is also a reified triple.
We can agreggate some context to extracted graphs introducing:
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– Context Triple: a triple context = (tr, contextlink , ct) which indicates that
a basic triple tr can be associated with a specific context ct.
The interpretation of [[context]] = ([[tr]], [[contextlink ]], [[ct]]) ∈ U 3 × U × U .
Note that the notion of context spreads to all elements involved in this representation. If we consider only one context, all definition above can be considered
on a specific (unique and implicit) context. In the case we have more than one
context, definitions above can be generalized as follows:
– multi-context graph: is an extracted graph with more than one context
associated to its triples. For example, given triples tri , tri+1 , trj and trj+1 in
a extracted graph we can have:
• ct1 6= ct2
• (tri , contextlink , ct1 ), (tri+1 , contextlink , ct1 ), where we have a path associated with context ct1
• (trj , contextlink , ct2 ), (trj+1 , contextlink , ct2 ), where we have a path associated with context ct2
• proj3 (tri ) = proj1 (tri+1 ) = proj3 (trj ) = proj1 (trj+1 ), where we have an
entity in two different contexts
In this case, we can define two new co-reference operators based on the
context.
• Possessive/Reflexive/Demonstrative Co-Reference: triples (ei+1 ,
coreflink , pr) and (pr, coreflink , ej ) where relation coreflink links entity
ei+1 of some object of a basic triple tri associated with class context cti
((tri , contextlinki , cti )) with the entity ej of a basic triple trj associated
with class context ctj ((trj , contextlinkj , ctj )) where cti 6= ctj .
• Non-Pronominal Co-Reference: a triple (ei+1 , coreflink , ej ) where
ei+1 is an object of some basic triple tri associated with class context cti
((tri , contextlinki , cti )) and ej is an entity of a basic triple trj associated
with class context ctj ((trj , contextlinkj , ctj )) where cti 6= ctj .
In a multi-context graph, given a specific (basic, reified, operational or complex) path, we have:
• if all basic triples in a path belong to an unique (same) context, we
call this path an unique context (basic, reified, operational or
complex) path.
• otherwise, we call this path a multi-context (basic, reified, operational or complex) path.
Note that a multi-context graph G induces a context relation graph as
follows:
• let P = {Gcti : i > 1 and ∀tr ∈ Gcti , tr belongs to the context class cti }.
That is, each Gcti ⊆ G is a subgraph of G restrict to context cti .
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• for each Gcti , let out(Gcti ) be the set of contexts ctj such that there is
a link between one element of Gcti to at least one element of Gctj .
• the context relation graph GCR is a set of triples (cti , ctlink , ctj ) such
that ctj ∈ out(Gcti ), ∀cti ∈ P.
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Fig. 1: Examples of extracted sentence graphs from the Wikipedia article
Barack Obama (top) and depiction of semantic model elements and the graph
interpretation patterns (bottom).