Formalizing SPARQL rules

Noodling on using MathML for the definitions in the SPARQL spec, and deriving N3 from it.

RDF Term

The set of RDF Terms, RDF-T, is RDF-U union RDF-L union RDF-B

$$

Query Variable

Let V be the set of all query variables. V and RDF-T are disjoint.

hmm... mathml can say "the intersection of V and RDF_T is empty" but not "V and RDF_T are disjoint." Maybe it's extensible... ah yes... csymbol. Hmm... encodings aren't grounded in URI space.

$\mathrm{disjointFrom}(V, \mathrm{RDF\_T})$

Triple Pattern

The set of triple patterns is (RDF-U union RDF-B union V) x (RDF-U union V) x (RDF-T union V)

sent comment: let's use the same terms everywhere@@. Term Pattern is RDF-T union V.

@@hmm... Triple Pattern isn't needed is, it? We can skip right to Graph Pattern, no?

$$ $$

Binding

skipping. obsoleted by substitution

Substitution

A substitution S is a partial functional relation from variables to RDF terms or variables. We write S[v] for the RDF term that S pairs with the variable v and define S[v] to be v where there is no such pairing.

@@more on: extending substitution of terms to triples and graphs. Oops! so codomain isn't right; we want onProperty/allValuesFrom

@@move this up: For substution S and Triple Pattern T, S(T) is the triple pattern forms by replacing any variable v in T with S[v].

Triple Pattern Matching

Triple Pattern T matches RDF graph G with substitution S, if S(T) is a triple of G.

$\forall T, G, S, \mathrm{TriplePattern}(T)\land \mathrm{RDF\_Graph}(G)\land \mathrm{Substitution}(S)\colon S(T)\in G\implies \mathrm{matches}(S)(T, G)$

Graph Pattern (Partial Definition) -- Conjunction

A set of triple patterns is a graph pattern GP.

$\forall \mathrm{GP}\colon (\mathrm{GP}\subseteq \mathrm{TriplePattern})\implies \mathrm{GraphPattern}(\mathrm{GP})$

Graph Pattern Matching

Graph Pattern GP matches RDF graph G with substitution S if S(GP) is a subgraph of G.

$\forall \mathrm{GP}, G, S, \mathrm{GraphPattern}(\mathrm{GP})\land \mathrm{RDF\_Graph}(G)\land \mathrm{Substitution}(S)\colon \mathrm{entailsSimply}(G, S(\mathrm{GP}))\implies \mathrm{matches}(S)(\mathrm{GP}, G)$

Pattern Solution

A Pattern Solution of Graph Pattern GP on graph G is any substitution S such that GP matches G with S.

For a graph pattern GP formed as a set of triple patterns, S(G), has no variables and is a subgraph of G.

$\forall \mathrm{GP}, G, S, \mathrm{GraphPattern}(\mathrm{GP})\land \mathrm{RDF\_Graph}(G)\land \mathrm{Substitution}(S)\colon \mathrm{matches}(S)(\mathrm{GP}, G)\implies \mathrm{patternSolution}(G)(\mathrm{GP}, S)$

Query Results

The Query Results, for a given graph pattern GP on G, is written R(GP,G), and is the set of all query solutions such that GP matches G.

R(GP, G) may be the empty set.

@@hmm... I don't think we can write "the set of all S such that..." in N3 yet.

$\mathrm{FunctionalProperty}(R)$ $\forall \mathrm{GP}\colon \mathrm{GraphPattern}(\mathrm{GP})\implies \mathrm{FunctionalProperty}(R(\mathrm{GP}))$ $\forall \mathrm{GP}, G, S, \mathrm{GraphPattern}(\mathrm{GP})\land \mathrm{RDF\_Graph}(G)\land \mathrm{Substitution}(S)\colon S\in R(\mathrm{GP})(G)\implies \mathrm{matches}(S)(\mathrm{GP}, G)$

Optional Matching

Given graph pattern GP1, and graph pattern GP2, let GP= (GP1 union GP2).

The optional match of GP2 of graph G, given GP1, defines a pattern solution PS such that:

If GP matches G, then the solutions of GP is the patterns solutions of GP else the solutions are the pattern solutions of GP1 matching G.

$\forall \mathrm{GP1}, \mathrm{GP2}, \mathrm{GP}, G, \mathrm{PS}, \mathrm{GraphPattern}(\mathrm{GP1})\land \mathrm{GraphPattern}(\mathrm{GP2})\land \mathrm{RDF\_Graph}(G)\land \mathrm{Substitution}(S)$

Appendix: MathML as a semantic web rules language

Let's take the classic syllogism, represent it in MathML, and then convert it to N3 via XSLT (mathmlRules.xsl), and then back using the n3.n3@@link parser

• • All men are mortal
  • Socrates is a man
• Socrates is mortal

All men are mortal:

Socrates is a man:

another take at socrates, generated from N3: