Tim BernersLee, August 2005
$Revision: 1.152 $ of $Date: 2011/09/27 22:31:21 $
Status: An early draft of a semiformal semantics of the N3
logical properties.
Up to
Design Issues
An RDF language for the Semantic Web
Notation 3 Logic
This article gives an operational
semantics for Notation3 (N3) and some RDF properties for
expressing logic. These properties, together with N3's
extensions of RDF to include variables and nested graphs, allow
N3 to be used to express rules in a web environment.
This is an informal semantics in that should be understandable by
a human being but is not a machine readable formal
semantics. This document is aimed at a logician wanting to a
reference by which to compare N3 Logic with other languages, and
at the engineer coding an implementation of N3 Logic and who
wants to check the detailed semantics.
These properties are not part of the N3 language, but are
properties which allow N3 to be used to express rules, and rules
which talk about the provenance of information, contents of
documents on the web, and so on. Just as OWL is expressed
in RDF by defining properties, so rules, queries, differences,
and so on can be expressed in RDF with the N3 extension to
formulae.
The log: namespace has functions, which have builtin meaning
for CWM and other software.
See also:
The prefix log: is used below as shorthand for the
namespace <http://www.w3.org/2000/10/swap/log#>.
See the schema
for a summary.
Motivation
The motivation of the logic was to be useful as a tool in in open
web environment. The Web contains many sources of
information, with different characteristics and relationships to
any given reader. Whereas a closed system may be built
based on a single knowledge base of believed facts, an open
webbased system exists in an unbounded sea of interconnected
information resources. This requires that an agent be aware of
the provenance of information, and responsible for its
disposition. The language for use in this environment
typically requires the ability to express what document or
message said what, so the ability to quote subgraphs and match
them against variable graphs is essential. This
quotation and reference, with its inevitable possibility of
direct or indirect selfreference, if added directly to first
order logic presents problems such as paradox traps. To avoid
this, N3 logic has deliberately been kept to limited expressive
power: it currently contains no general first order
negation. Negated forms of many of the builtin
functions are available, however.
A goal is that information, such as but not limited to rules,
which requires greater expressive power than the RDF graph,
should be sharable in the same way as RDF can be
shared. This means that one person should be able to
express knowledge in N3 for a certain purpose, and later
independently someone else reuse that knowledge for a different
unforeseen purpose. As the context of the later use is
unknown, this prevents us from making implicit closed assumptions
about the total set of knowledge in the system as a whole.
Further, we require that other users of N3 in the web can express
new knowledge without affecting systems we have already
built. This means that N3 must be fundamentally monotonic:
the addition of new information from elsewhere, while it might
cause an inconsistency by contradicting the old information
(which would have to be resolved before the combined system is
used), the new information cannot silently change the meaning of
the original knowledge.
The nonmonotonicity of many existing systems follows from a form
of negation as failure in which a sentence is deemed false if it
not held within (or, derivable from) thecurrent knowledge
base. It is this concept of current knowledge
base, which is a variable quantity, and the ability to
indirectly make reference to it which causes the
nonmonotonicity. In N3Logic, while a current
knowledge base is a fine concept, there is no ability to make
reference to it implicitly in the negative. The
negation provided is the ability only for a specific given
document (or, essentially, some abstract formula) to objectively
determine whether or not it holds, or allows one to derive, a
given fact. This has been called Scoped Negation As Failure
(SNAF).
Formal syntax
The syntax of N3 is defined by the contextfree
grammar This is available in machinereadable form in
Notation3
and RDF/XML.
The toplevel production for an N3 document is
<http://www.w3.org/2000/10/swap/grammar/n3#document>.
In the semantics below we will consider these productions using
notation as follows.
Production 
N3 syntax examples 
notation below for instances 
symbol 
<foo#bar>
<http://example.com/> 
c d e f 
variable 
Any symbol quantified by @forAll or @forSome in the
same or an outer formula. 
x y z 
formula 
{ ...
} or an entire document 
F G H K 
set of universal variables of F 
@forAll :x,
:y. 
uvF 
set of existential variables of F 
@forSome :z,
:w. 
evF 
set of statements of F 

stF 
statement 
<#myCar>
<#color> "green". 
Fi or
{s p o} 
string 
"hello
world" 
s 
integer 
34 
i 
list 
( 1 2 ?x <a> ) 
L M 
Element i of list L 

Li

length of list 

L 
expression 
see grammar 
n m 
Set* 
{$ 1, 2, <a> $} 
S T

*The set syntax and semantics are not part of the current
Notation3 language but are under consideraton.
Semantics
Note. The Semantics
of a generic RDF statement are not defined here. The
extensibility of RDF is deliberately such that a document may
draw on predicates from many sources. The statement {n
c m} expresses that the relationship denoted by c holds between
the things denoted by n and m. The meaning of
the statement {n c m} in general is defined by any
specification for c. The Architecture of the WWW specifies
informally how the curious can discover information about
the relation. It discusses how the architecture and management of
the WWW is such that a given social entity has jurisdiction over
certain symbols (though for example domain name ownership). This
philosophy and architecture is not discussed further
here. Here though we do define the semantics of
certain specific predicates which allow the expression of the
language. In analyzing the language the reader is
invited to consider statements of unknown meaning ground
facts. N3Logic defines the semantics of certain
properties. Clearly a system which recognizes further logical
predicates, beyond those defined here, whose meaning introduces
greater logical expressiveness would change the properties of the
logic.
Simplifications
N3 has a number of types of shortcut
syntax and syntactic sugar. For simplicity, in this article
we consider a language simpler the full N3 syntax referenced
above though just as expressive, in that we ignore most syntactic
sugar. The following simplifications are made.
We ignore syntactic sugar of comma and semicolon as shorthand
notations. That is, we consider a simpler language in
which any such syntax has been expanded out. Loosely:
A sentence of the form 
becomes two sentences 
subject stuff ; morestuff . 
subject stuff
. subject morestuff . 
subject predicate stuff , object . 
subject predicate stuff subject
predicate object . 
For those familiar with N3, the other simplifications in the
language considered here are as follows.
 prefixes have been expanded and all qualified names
replaced with symbols using full URIs between angle
brackets.
 The path syntax which uses "!" and "^" is
assumed expanded into its equivalent blank node form;
 The "is ... of " backwards construction has been replaced
by the equivalent forwards direction syntax.
 The "=" syntax is not used as shorthand for owl:sameAs. In
fact, we use = here in the text for value equality.
 @keywords is not used
 The @a shorthand for rdf:type is replaced with
a direct use of the full URI symbol for rdf:type
 all ?x forms are replaced with explicit universal
quantification in the enclosing parent of the current
formula.
Notation3 has explicitly quantified existential variables as well
as blank nodes. The description below does not mention
blank nodes, although they are very close in semantics to
existentially quantified variables. We consider for
now a simpler language in which blank nodes have been replaced by
explicitly named variables existentially quantified in
the same formula.
We have only included strings and integers, rather than the whole
set of RDF types an userdefined types.
These simplifications will not deter us from using N3 shorthand
in examples where it makes them more readable, so the reader is
assumed familiar with them.
Defining N3 Entailment
The RDF specification defines a
very weak form of entailment, known as RDF entailment or simple
entailment. He we define the equivalent very simple
N3entailment. This does not provide us with useful powers of
inference: it is almost textual inclusion, but
just has conjunction elimination (statement removal) ,
universal elimination, existential introduction and variable
renaming. Most of this is quite traditional. The
only thing to distinguish N3 Logic from typical logics is
the Formula, which allows N3 sentences to make statements about
N3 sentences. The following details are included for
completeness and may be skipped.
Substitution
Substitution is defined to recursively
apply inside compound terms, as is usual. Note only
that substitution does descend into compund terms, while
substitution of owl:sameAs, discussed later, does
not.
We define a substitution operator
σ_{x/m}
which replaces occurrences of the variable x. with the expression m. For
compound terms, substitution of a compound term (list,
formula or set) is performed by performing substitution of
each component, recursively.
Abbreviating the substitution
σ_{x/m}
as σ , we define substitution operator as
usual:
σx = m
(x is
replaced by m)
σy = y (y not
equal to x)
σa = a (symbols and literals are
unchanged)
σi = i
σs = s
σ( a b ... c ) = ( σa σb ...
σc )
(substitution goes into compound
terms)
σ{$ a, b, ... c $} = {$ σa,
σb, ... σc $}
uv σF = σ uvF
ev σF = σ evF
st σF = σ stF
In general a substitution operator is the sequential application
of single substitutions:
σ = σ_{x1/m1}σ_{x2/m2}σ_{x2/m2} ...
σ_{xn/mn}
Value equality
Value equality between terms is
defined in an ordinary way, compatible with RDF.
For concepts which exist in RDF, we use RDF equality. This
is RDF node equality. These atomic concepts have a simple
form of equality.
For lists, equality is defined as a pairwise matching.
For sets, equality is defined as a mapping between equal terms
existing in each direction.
For formulae, equality F = G is defined as a
substitution σ existing mapping variables to
variables. (Note that as here RDF Blank Nodes are
considered as existential variables, the substitution will map
bnodes to bnodes.)
The table below is a summary for completeness.
Production 
Equality 
symbol 
uri is equal unicode string 
variable 
variable name is equal unicode string 
formula 
F = G iff stF = stG and there is some
substitution σ such
that (∀i . ∃j . σFi = σGj. ) 
statement 
Subjects are equal, predicates are equal, and
objects are equal 
string 
equal unicode string 
integer 
equal integer 
list L = M 
L = M
& (∀i . Li = Mi ) 
set S = T 
(∀i
. ∃j
. Si =
Tj. )
& (∀i . ∃j . Si = Tj. ) 
formula F = G 
∃σ. σ F
= σ G 
unicode string 
Unicode strings should be in canonical form. They are
equal if the corresponding characters have numerically
equal code points. 
Conjunction
N3, like
RDF, has an implied conjunction, with its normal properties,
between the statements of a formula.
The semantics of a formula which has no quantifiers (@forAll or
@forSome) are the conjunction of the semantics of the statements
of which it is composed.
We define the conjunction elimination operator ce(i) of removing
the statement Fi from
formula F. By the conventional semantics of conjunction,
the ce(i) operator is truthpreserving. If you take a
formula and remove a statement from it it is still true.
CE: From F follows ce(i)
F
Existential quantification
Existential quantifiers and Universal
quantifiers have the usual qualities
Any formula, including the root
formula which matches the "document" production of the
grammar, may have a set of existential variables indicated
by an @forSome
declaration. This indicates that, where the formula
is considered true, it is true for at least one substitution
mapping the existential variables onto nonvariables.
As usual, we define a truthpreserving Existential
Introduction operator on formulae, that of introducing an
existentially quantified variable in place of any term. The
operation ei(x, n) is defined as
 Creation of a new variable x which occurs nowhere
else
 The application of σ_{x/n} to F
 The addition ofx
to evF.
EI: From F follows ei(x,n)
F for any x not occurring anywhere else
Universal quantification
Any formula, (including the root formula), may have a set
of universal variables. These are indicated by
@forAll
declarations. The scope of the @forAll is outside the
scope of any @forSome.
If both universal and existential quantification are specified
for the same context, then the scope of the universal
quantification is outside the scope of the existentials:
{ @forAll <#h>. @forSome <#g>. <#g> <#loves> <#h> }.
means
∀<#h> ( ∃<#g> ((
<#g>
<#loves> <#h> ))
The semantics of @forAll is that for any substitution
σ = subst(x, n)
where x member of uvF, if F is true then
σF is also true. Any @forAll declaration may also be
removed, preserving truth. Combining these, we define a
truthpreserving operation ue(x, n) such that
ue(x, n) F is formed by
 Removal of x from evF
 Application of subst(x, n)
We have the axiom of universal elimination
UE: From F follows
ue(x, n) F for all x in evF
As the actual variable used in a formula is quite irrelevant to
its semantics, the operation of replacing that variable with
another one not used elsewhere within the formula is
truthpreserving.
Variable renaming
We define the operation of variable renaming
vr(x,y) on F when x is a
member of uvF or is a member of evF.
VR: From F follows
vr(x, y) F
where x is
in uvF or evF and y does
not occur in F
Occurrence in F is defined recursively in the same way as
substitution: x
occurs in F iff σ_{x/n}F is not equal to F for
arbitrary n.
Union of formulae
The union H = F∪G of two formulae F
and G is formed, as usual, as follows.
A variable renaming operator is applied to G such that the
resulting formula G' has no variables which occur unquantified
or differently quantified or existentially quantified in F, and
viceversa. (F and G' may share universal
variables).ied or existentially quantified in F, and
vicever
F∪G is then defined by:
st(F∪G) = stF ∪ st G' ; ev(F∪G)
= evF ∪ evG' ; uv(F∪G) =
uvF ∪ uv G'
N3 entailment
The operators conjunction elimination, existential
elimination, universal introduction and variable
renaming are truth preserving. We define an
N3 entailment operator (τ) as any operator which is the
successive application of any sequence (possibly empty) of
such operators. We say a formula F n3entails a
formula τ F. By a combination
of SE, EI, UE and VR, τ F logically
follows from F.
Note.
RDF Graph is a subclass of N3 formula. If F and G are
RDF graphs, only CI and EI apply and n3entailment
reduces to simple entailment from RDF Semantics. (@@check
for any RDF weirdnesses)
We have now defined this simple form of
N3entailment, which amounts to little more than textual
inclusion in one expression of a subset of another. We
have not defined the normal collection of implication,
disjunction and negation which first order logic, as N3logic does
provide for first order negation. We have, in the
process, defined a substitution operation which we can now
use to define implication, which allows us to express
rules.
Logic properties and builtin functions
We now define the
semantics of N3 statements whose predicate is one of a small set
of logic properties. These are statements whose truth can
be established by performing calculations, or by accessing the
web.
One of our objectives was to make it possible to make statements
about, and to query, other statements such as the contents of
data in information resources on the web. We have, in
formulae, the ability to represent such sets of statements.
Now, to allow statements about them, we take some of the
relationships we have defined and give them URIs so that these
statements and queries can be written in N3.
While the properties we introduced can be used simply as ground
facts in a database, is very useful to take advantage of
the fact that in fact they can be calculated. In some
cases, the truth or falsehood of a binary relation can be
calculated; in others, the relationship is a function so one
argument (subject or object of the statement) can be calculated
from the other.
We now show how such properties are defined, and give examples of
how an inference system can use them. A motivation
here is to do for logical information what RDF did for data: to
provide a common data model and a common syntax, so that
extensions of the language are made simply by defining new
terms in an ontology. Declarative programing languages
like scheme[@@] of course do this. However, they differ in
their choice of pairs rather than the RDF binary relational model
for data, and lack the use of universal identifiers as
symbols. The goal with N3 was to make a
minimal extension to the RDF data model, so that the
same language could be used for logic and data, which in practice
are mixed as a colloidal solution in many real
applications.
Calculated entailment
We introduce also a set of properties whose truth may be
evaluated directly by machine. We call these
"builtin" functions. The implementation as builtin
functions is not in general required for any
implementation of the N3 language, as they can always soundly be
treated as ground facts. However, their usefulness
derives from their implementation. We say that for example
{ 1 math:negation 1 } is entailed by
calculation. Like other RDF properties,
the set is designed to be extensible, as others can use URIs for
new functions. A much larger set of such properties is described for
example in the CWM bulttins list, and the semantics of those
are not described here.
When the truth of a statement can be deduced because its
predicate is a builtin function, then we call the derivation
of the statement from no other evidence calculated entailment.
We now define a small set of such properties which provide the
power of N3 logic for inference on the web.
log:includes
If a formula G n3entails another
formula F, this is expressed in N3 logic as
F log:includes
G.
Note. In deference to the
fact that RDF treats lists not as terms but as things constructed
from first and rest pairs, we can view formulae which include
lists as including rdf:first and rdf:rest statements. The
effect on inclusion is that two other entailment operations are
added: the addition of any statement of the form
L rdf:first
nwhere n is the first
element of L, or L rdf:rest K where K is list forming the
remaining nonfirst elements of L. This is not essential
to a further understanding of the logic, nor to the operation of
a system which does not contain any explicit mention of the terms
rdf:first or rdf:rest.
For the discussion of n3entailment, clearly:
From F and F log:includes G
logically follows G
This can be calculated, because it is a mathematical operation on
two compound terms. It is typically used in a query to test
the contents of a formula. Below we will show how it can be
used in the antecedent of a rule.
log:notIncludes
We write of formulae F and G: F log:notIncludes G if it is
not the case that G
n3entails F.
As a form of negation, log:notincludes is completely monotonic.
It can be evaluated by a mathematical calculation on the
value of the two terms: no other knowledge gained can influence
the result. This is the scoped negation as failure mentioned
in the introduction. This is not a nonmonotonic negation
as failure.
Note on computation: To
ascertain whether G n3entails F in the worst case involves
checking for all possible n3entailment transformations
which are combinations of the variables which occur in G. This
operation may be tedious: it is strictly graph isomorphism
complete. However the use of symbols rather than variables
for a good proportion of nodes makes it much more tractable for
practical graphs. The ethos that it is a good idea to
give name things with URIs (symbols in N3) is a basic meme of web
architecture [AWWW]. It has direct practical
application in the calculation of n3entailment, as comparison of
graphs whose nodes are labelled is much faster (of order n log
(n)))
The
log:implies property relates two formulae, expressing
implication. The shorthand notation for log:implies is
=>
. A statement using log:implies, unlike log:includes,
cannot be calculated. It is not a builtin function,
but the predicate which allows the expression of a rule.
The semantics of implication
are standard, but we elaborate them now for
completeness.
F log:implies G is true if and only if when the formula F is true
then also G is true.
MP: From F and
F => G follows G
A statement in formula H is of the form F=>G can be
considered as rule, in which case, the subject F is the premise
(antecedent) of the rule, and the object G is the
consequent.
Implication is normally used within a formula with universally
quantified variables.
For example, universal quantifiers
are used with a rule in H as follows. Here H is
the formula containing the rules, and K the formula upon which
the rules are applied, which we can call the knowledge
base.
If F => G is in H, and then for every σ which
is a transformation composed of universal eliminations of
variables universally quantified in H, then it also
follows that σF => σG. Therefore, for
every σ such that K includes σF,
σG follows from K.
In the particular case that H and K are both the knowledge base,
or formula believed true at the top level, then
GMP: From F => G
and σF follows σG
if σ is a transformation composed of
universal eliminations of variables universally quantified at the
top level.
Filtering
When a knowledge base (formula) contains a lot
of information, one way to filter off a subset is to run a set of
rules on the knowledge base, and take only the new data which is
generated by the rules. This is the filter
operation.
When you apply rules to a knowledge base, the filter result of rules in H applied
to K is the union of all σG for every statement F
=> G which is in H, for every σ which s
a transformation composed of universal eliminations of variables
universally quantified in H such that K includes σF.
Repeated application of rules
When rules are added back
repeatedly into the same knowledge base, in order to
prevent the unnecessary extra growth of the knowledge base,
before adding σG to it, there is a check to see
whether the H already includes σG, and if it does, the
adding of σG is skipped.
Let the result of rules in H applied to K,
ρ_{H}K, be the union of K with
all σG for every statement F => G which is in
H, for every σ which is a transformation
composed of universal eliminations of variables universally
quantified in H, such that K includes σF, and K does not
n3entail σG.
Note. This form of rule allows
existentials in the consequent: it is not datalog. It
is is clearly possible in a forwardchaining reasoner to generate
an unbounded set of conclusions with rules of the form
(using shorthand)
{ ?x
a :Person } => { ?x :mother [ a :Person]
}.
While this is a trap for the
unwary user of a forwardchaining reasoner, it was found to be
essential in general to be able to generate arbitrary RDF
containing blank nodes, for example when translating information
from one ontology into another.
Consider the repeated application of rules in H to K,
ρ^{i}_{H}K. If there
are no existentially quantified variables in the consequents of
any of the rules in H, then this is like datalog, and there will
be some threshold n
above which no more data is added, and there is a closure:
ρ^{i}_{H}K =
ρ^{n}_{H}K for all
i>n. In fact in many practical
applications even with the datalog constraint removed, there is
also a closure. This ρ^{∞}_{H}K
is the result of running a forwardchaining reasoner on H and
K.
Rule Inference on the knowledge base
In the case in which
rules are in the same formula as the data, the single rule
operation can be written ρ_{K}K, and the
closure under rule application
ρ^{∞}_{K}K
Cwm note: the rules command line
option calculates ρ_{K}K,
and the think calculates ρ^{∞}_{K}K.
The filter=H calculates the filter result of H on the
knowledge base.
Examples
Here a
simple rule uses log:implies.
@prefix log: <http://www.w3.org/2000/10/swap/log#>.
@keywords.
@forAll x, y, z. {x parent y. y sister z} log:implies {x aunt z}
This N3 formula has three universally quantified variables and
one statement. The subject of the statement,
{x parent y. y sister z}
is the antecedent of the rule and the object,
{x aunt z}
is the conclusion. Given data
Joe parent Alan.
Alan sister Susie.
a rule engine would conclude
Joe aunt Susie.
As a second example, we use a rule which looks inside a
formula:
@forAll x, y, z.
{ x wrote y.
y log:includes {z weather w}.
x livesIn z
} log:implies {
Boston weather y
}.
Here the rule fires when x is bound to a symbol denoting some
person who is the author of a formula y, when the formula makes a
statement about the weather in (presumably some place) z, and x's
home is z. That is, we believe statements about the
weather at a place only from people who live there. Given
the data
Bob livesIn Boston.
Bob wrote { Boston weather sunny }.
Alice livesIn Adelaide.
Alice wrote { Boston weather cold }.
a valid inference would be
Boston weather sunny.
log:supports
We say that F log:supports G if there is some sequence of
rule inference and/or calculated entailment and/or n3
entailment operators which when applied to F produce G.
log:conclusion
The log:conclusion property expresses the relationship between a
formula and its deductive closure under operations of
n3entailment, rule entailment and calculated entailment.
As noticed above, there are circumstances when this will not be
finite.
log:conclusion is the transitive closure of log:supports.
log:supports can be written in terms of log:conclusion and
log:includes.
{ ?x log:supports ?y } if and only dan {
?x log:conclusion [ log:includes ?y ]}
However, log:supports may be evaluated in many cases without
evaluating log:conclusion: one can determine whether y can be
derived from x in many ways, such as backward chaining, without
necessarily having to evaluate the (possibly infinite) deductive
closure.
Now we have a system which has the capacity to do inference using
rules, and to operate on formulae. However, it
operates in a vacuum. In fact, our goal is that the
system should operate in the context of the web.
Involving the Web
We therefore expose the web as a
mapping between URIs and the information returned when such a URI
is dereferenced, using appropriate protocols. In
N3, the information resource is identified by a
symbol, which is in fact is its URI. In N3, information is
represented in formulae, so we represent the information
retrieved as a formula.
Not all information on the web is, of course in N3. However the
architecture we design is that N3 should here be the interlingua.
Therefore, from the point of view of this system, the semantics
of a document is exactly what can be expressed in N3, no more and
no less.
log:semantics**
c log:semantics F is true iff c is a document whose
logical semantics expressed in N3 is the formula F.
The relation between a document and the logical expression
which represents its meaning expressed as N3. The
Architecture of the World Wide Web [AWWW] defines algorithms by
which a machine can determine representations of
document given its symbol (URI). For
a representation in N3, this is the formula which corresponds to
the document production
of the grammar. For a representation in
RDF/XML it is the formula which is the entire graph
parsed. For any other languages, it may be calculated
in as much a specification exists which defines the
equivalent N3 semantics for files in that language.
On the meaning of N3 formula
This is not of course the semantics of the document in
any absolute sense. It is the semantics expressed in
N3. In turn, the full semantics of an N3 formula are
grounded, in the definitions of the properties and classes
used by the formula. In the HTTP space in which
URIs are minted by an authority, definitive information about
those definitions may be found by dereferencing the URIs. This
information may be in natural language, in some
machineprocessable logic, or a mixture. Two patterns
are important for the semantic web.
One is the grounding of properties and classes by defining
them in natural language. Natural language, of course,
is not capable of giving an absolute meaning to anything in
theory, but in practice a well written document, carefully
written by a group of people achieves a precision of definition
which is quite sufficient for the community to be able to
exchange data using the terms concerned. The other
pattern is the raftlike definition of terms in terms of related
neighboring ontologies.
@@@@ A full discussion of the grounding of meaning in a
web of such definitions is beyond the scope of this
article. Here we define only the operation semantics
of a system using N3.
@@@@ Edited up to here
The log:semantics of an N3
document is the formula achieved by parsing representation of the
document.
(Cwm note: Cwm knows how to go get a document and parse N3 and
RDF/XML , in order to evaluate this. )
Other languages for web documents may be defined whose N3
semantics are therefore also calculable, and so they could be
added in due course.
See for example [GRDDL], [RDF/A], etc
However, for the purpose of the analysis of the language, it
is a convenient to consider the semantic web simply as a
binary 1:1 relation between a subset of symbols and
formulae.
For a document in Notation3, log:semantics is the
log:parsedAsN3 of the log:contents of the document.
log:says
log:says is defined by:
F log:says G iff ∃ H .
F log:semantics
H and H log:includes G
In other words, loosely a document says something if a
representation of it in the sense of the Architecture of the
World Wide Web [AWWW] N3entails it.
The semantics of log:says are similar to that of says in
[PCA].
Miscellaneous
log:Truth
This is a class of true formulae.
From { F rdf:type log:Truth } follows
F
The cwm engine will process rules in the (indirectly
commandline specified) formula or any formula which that
declares to be a Truth.
The dereifier will output any described formulae which are
described as being in the class Truth.
This class is not
at all central to the logic.
Working with OWL
@@ Summary
 owl:sameAs considered the same as N3 value equality for
data values. Axioms of
equality. log:equalTo and
log:notEqualTo compared with owl:SameAs. Compare
math and string equality, and SPARQL equality.
 Operating in equalityaware mode.
 No attempt at connecting OWL DL language with the N3
logic.
 Use of functional properties of a datatype conflicting with
OWL DL.
Conclusion
The semantics of N3 have been defined, as have some builtin
operator properties which add logical inference using rules to
the language, and allow rules to define inference which can be
drawn from specific web documents on the web, as a function of
other information about those documents.
The language has been found to have some useful practical
properties. The separation between the Notation3
extensions to RDF and the logic properties has allowed N3 by
itself to be used in many other applications directly, and to be
used with other properties to provide other functionality such as
the expression of patches (updates) [Diff].
The use of log:notIncludes to allow default reasoning without
nonmonotonic behavior achieves a design goal for distributed
rule systems.
**[Footnote: Philosophers may be distracted here into worrying
about the meaning of meaning. At least we didn't call this
function "meaning"! In as much as N3 is used as an interlingua
for interoperability for different systems, this for an N3 based
system is the meaning expressed by a document. One
reviewer was aghast at the definition of semantics as being that
of retrieval of a representation, its parsing and assimilation in
terms of the local common logical framework. I suspect however
that the meaning of the paper to the reviewer could be considered
quite equivalently the result of the process of
retrieval of a representation of the paper, its parsing by the
review, and its assimilation in terms of the reviewer's local
logical framework: a similar though perhaps imperfect
process.
Of course, the semantics of many documents are not expressible in
logics at all, and many in logic but not in N3. However, we are
building a system for which a prime goal is the reading and
investigation of machinereadable documents on the web. We use
the URI log:semantics for this function and apologize for any
heartache it may cause.]
F = G iff stF
= stG and there is some substitution σ such
that (∀i
. ∃j .
σFi =
σGj. )
formatting XHTML 1 with
nvu
yes, discuss notational
abbreviation, but not abstract syntax
hmm... are log:includes,
log:implies and such predicates? relations? operators?
properties?
To do: describe the syntactic
sugar transformations formally to close the loop.