diff --git a/spec/semantics.html b/spec/semantics.html index 142ef97..68dd226 100644 --- a/spec/semantics.html +++ b/spec/semantics.html @@ -306,8 +306,8 @@

Terms, Triples and Graphs

-->
- In this context, an RDF graph is a graph that does not contain graph terms nor lists, does not have literals as subjects or predicates, does not have variables as predicates, and for which $U=\emptyset$. - The blank nodes occurring in the graph would be labeled and stored in $E$. + In this context, an RDF graph is a graph that does not contain graph terms nor lists; does not have literals as subjects nor predicates; does not have variables as predicates; for which $U=\emptyset$; + and $E$ is the set of blank nodes occurring in the graph.
The translation of nested blank nodes as explained here and for example applied in Example 3 follows @@ -320,7 +320,7 @@

Terms, Triples and Graphs

as
$\qquad (\{\},\{x\},\{(\text{:socrates}, \text{:says}, <\{\},\{\},\{(x,\text{rdf:type}, \text{:Mortal} )\}>) \})$
and not as
- $(\{\},\{\},\{(\text{:socrates}, \text{:says}, <\{\},\{x\},\{(x,\text{rdf:type}, \text{:Mortal} )\}>) \})$.
+ $\qquad (\{\},\{\},\{(\text{:socrates}, \text{:says}, <\{\},\{x\},\{(x,\text{rdf:type}, \text{:Mortal} )\}>) \})$.
A solution for that problem is currently being discussed. The difference, however, does not influence the formal semantics of the abstract syntax, as quantification is always explicit there.
@@ -387,11 +387,11 @@

Variables

Note that the definition makes a difference between graphs and graph terms. A graph term is a direct constituent of the graph ("containing graph") in which it occurs as a triple component. However, the direct constituents of the graph term's graph, and their direct constituents (and so on), are not direct constituents of the containing graph. Rather, they are constituents of the containing graph. - This difference is crucial for the definitions below: a graph is not the same as a graph term.

+ This difference between graph and graph term is crucial for the definitions below.

With that definition, we now introduce free variables: these are variables which occur directly or nested in the graph but which are not covered by a quantification set.

- The set of free variables ($FV$) of a term or a set of triples is defined as follows: +

The set of free variables ($FV$) of a term or a set of triples is defined as follows:

- The set $FV$ for a graph $G=(U,E,F)$ and a variable $v\in V\cap C(G)$: +

Consider a graph $G=(U,E,F)$ and a variable $v\in V\cap C(G)$:

-

+

We denote the set of variables which are free in $G$ as $FV(G)$.

Examples:

    @@ -421,8 +420,8 @@

    Variables

This enables us to introduce the notion of closed graphs and ground terms.

-

Note that the images of $Q_I$ are syntactic elements (graph terms), but they can (and most likely will) also be elements of the domain $Δ_I$. - Note however that not all interpretations are required to have graph terms in their domain, because $Q_I$ is allowed to be the empty mapping. +

Note that the images of $\Gamma_I$ are syntactic elements (graph terms), but they can (and most likely will) also be elements of the domain $Δ_I$. + Note however that not all interpretations are required to have graph terms in their domain, because $\Gamma_I$ is allowed to be the empty mapping. Such interpretations could still satisfy a graph that has no graph term constituent.

@@ -653,8 +653,8 @@

Base Semantics

The basic interpretation maps graph terms to an isomorphic copy of themselves. As a consequence, two non-isomorphic graph terms cannot have the same meaning. For the abstract versions of the graph terms {_:x :p :o}, {_:y :p :o} and {_:x :p :o. _:y :p :o}, we thus get:

@@ -683,7 +683,7 @@

Base Semantics

--> -will be true if $(D_I(\text{:bob}), D_I(\text{:says}), Q_I(<\{\}, \{\}, \{(\text{:bob}, \text{:is}, \text{:wise})\}>)\in EXT_I$. +will be true if $(D_I(\text{:bob}), D_I(\text{:says}), \Gamma_I(<\{\}, \{\}, \{(\text{:bob}, \text{:is}, \text{:wise})\}>)\in EXT_I$.

If we revert back to our original example with variables, a regular valuation function (such as the mapping A used for blank nodes in [[RDF11-MT]]) will simply replace both occurrences of variable $x$ with a domain element $\delta_i$ from $\Delta_I$. - In that case, $Q_I$ will have to map the graph term $<\{\}, \{\}, \{(\delta_i, \text{:is}, \text{:wise})\}>$ + In that case, $\Gamma_I$ will have to map the graph term $<\{\}, \{\}, \{(\delta_i, \text{:is}, \text{:wise})\}>$ to an isomorphic copy. However, the isomorphism is only defined for concrete N3 terms, and not domain elements. - If not applying the valuation function to the graph term, $Q_I$ will have to map the graph term $<\{\}, \{\}, \{(x, \text{:is}, \text{:wise})\}>$, + If not applying the valuation function to the graph term, $\Gamma_I$ will have to map the graph term $<\{\}, \{\}, \{(x, \text{:is}, \text{:wise})\}>$, which is however no longer ground as its graph is no longer closed - ($x$ is quantified outside of the graph). In our basic interpretation, the mapping $Q_I$ is only defined for ground graph terms. + ($x$ is quantified outside of the graph). In our basic interpretation, the mapping $\Gamma_I$ is only defined for ground graph terms. We thus need to first ground these kinds of leaked variables, i.e., variables that become free when considering the graph term in isolation (as is required when determining isomorphisms).

@@ -767,7 +767,7 @@

Graphs with variables

The mapping $A_2$ assigns variables to terms, and will be used to ground free variables within graph terms. Note that given an interpretation $I$ and two assignments $A$ for a set $V_1$, and $B$ for a set $V_2$ of variables, the combination $A\bullet B$ is again an assignment for $V_1\cup V_2$.

-