ITA P12 Task2
The Representation of Special Logics in Semantic Web languages v0.2
1 Purpose
To define how logical statements from specialist logics may be explicitly represented
in ontologies expressed in semantic web technologies. This document builds upon the
representation of basic logical statements [BASLOG]
This work is a deliverable of the ITA programme, project 12 task 2 “Collective
Intelligence and the Web”, for Q2 of the BPP09.
2 Introduction
As noted in the basic document [BASLOG], we take an ontology to include
information on structure (such as classes and properties) together with logical
relationships between structural elements (such as temporal relations between tasks to
be undertaken in parallel). Such logical relationships provide a reasoner with a
“licence to entail” defining what new information may be inferred from any given
state of knowledge and allows the developer of applications or computer agents to
know what may or may not be inferred from other data. We also noted that the inbuilt
logical relationships available within the representation capabilities of the semantic
web technology family of languages, OWL, are not sufficient to represent all that is
needed in a model of planning.
The basic document defined how logical statements needed in a planning model may
be represented and embedded within an OWL ontology, but concentrated on the basic
logic of First Order Predicate Logic (FOPL).
There is a need to address how more complex “specialist logics” may be expressed,
such as temporal logics, assumption-based logics, situation calculus. These logics are
useful in order to express logical statements in a particular specialist subdomain such
as time, decision making and action-planning, respectively. Whereas the logical
statement could be expressed in basic FOPL, nevertheless the resulting statements can
be very verbose, and thereby difficult to understand. A specialist logic provides a new
higher-level logic vocabulary that can express the concepts more succinctly.
The work undertaken in Q2 of the BPP09 has focussed on the main and most
significant issue left open in the basic document, that of the representation of
negation. It is recognised that further work is still required to extend the consideration
of specialist logics into the areas noted above.
It should be noted that representing the logic of negation is a very complex and deep
philosophical and logical area, and this document is only an informal starting point for
the purpose of discussion. In addition, we have recently learnt that there is an RIF
initiative to define a dialect based on default negation, and it is appropriate to review
whether this dialect could be adopted instead of the approach proposed here (or
whether we might indeed contribute to the development of this dialect).
3 However, although the definition of negation
(especially in the situation of uncertain knowledge)
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The Representation of Special Logics in Semantic Web languages v0.2
is complex with competing alternative semantics,
nevertheless it is necessary to provide the users of
CPM with a facility for expressing negative
knowledge, and for using negative predicates in
inference rules. They must be able to reason about
situations where knowledge is absent or where
something is specifically known not to be true
(such as “if there is no enemy in area A then it is
safe to move forwards”). It is also important to
ensure that whatever mechanism is chosen, that
the semantics is clearly defined, so that a system
(or human) can know what is and is not permissible
to be inferred. It is in this spirit that the problem of
defining negation has been tackled. It is expected
that further use of such mechanisms will provide
feedback as to their suitability in the domain of
planning and other types of problem solving. The
representation of Negation
This document focuses on the main issue left open in the basic document, that of the
representation of negation, and its use with rules of inference. Negation, of course,
specifies that a proposition, such as “there is a bridge at the river” is not true.
However there are differing ways of “knowing” that something is not true, including:




I have an explicit fact in my knowledgebase stating that the proposition is
false, ie “there is no bridge at the river”
I have no such explicit fact, but I also know that all information is present in
the knowledgbase, so if something is missing then its negation is true (the
Closed World Assumption or CWA)
I have no such explicit fact, and I have tried to prove this proposition from my
existing knowledge, but have failed to do so even after exploring all possible
inferences, therefore I conclude the proposition is false (Default Negation,
with a varient called Negation as Failure, or NAF)
I have no such explicit fact, and I have tried to prove this proposition from my
existing knowledge, but have failed to do so because I have run out of
computing resources (e.g. time) therefore I conclude the proposition is false
(we suggest this may be called Negation as Timeout, or NAT)
Each of these possible interpretations of negation has different semantics, and would
in some circumstances arrive at different conclusions. The first three are well known
in theories of logic.
There is a fifth possibility, that we do not permit inferences from negated propositions
when there is no direct evidence of the negation:
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The Representation of Special Logics in Semantic Web languages v0.2

I have no such explicit fact, therefore I do not know whether the proposition is
false or not, and I therefore cannot make any inferences from this.
This may be the simplest situation, not requiring any complex logic, but in some
circumstances is very weak, limiting the amount of information that can be inferred.
It is therefore necessary to decide upon the interpretation (or interpretations) of
negation that is to be used within the CPM. Since the basic proposal is based upon
RIF, it is also necessary to define a new RIF dialog to represent the interpretation of
negation (but as noted above, we have recently heard that such an initiative is being
undertaken though no details are available as at the time of writing). More
specifically, the following must be defined:



The syntax for use in the RIF abstract syntax to define negation
The formal model-theoretic semantics for each of these syntactic terms
A XML representation of these terms
Since the proposal also integrates a CNL and a representation of rationale, these must
also be defined, although this is not a requirement of a RIF dialect.
3.1 Choosing an Interpretation of Negation
In fact RIF-FLD describes two different types of negation that may be defined:


symmetric (captured by the syntax Neg)
default (captured by the syntax Naf)
3.1.1 Symmetric Negation
Symmetric negation includes classical negation, strong negation and explicit negation.
Classical negation defines the term ¬ A to mean that A is false. In addition, classical
negation uses the law of excluded middle, i.e (A V ¬A) is true for all A.
However, it has been argued in [Alferes] that the use of classical negation leads to
counter-intuitive results in situations of incomplete knowledge (such as may be found
in the semantic web), but that default negation (discussed below) also leads to
problems, and hence they have proposed strong and explicit negation. This is based
upon the notion that some terms A may have explicit negative versions –A (eg for
achievable(Plan) there is defined the negated term unachievable(Plan)). Then they add
the logical constraint that A AND –A is inconsistent, for all such terms. This is not the
same as classical negation, since the law of excluded middle is not applied and it is
permissible that neither A nor –A be true.
This document, in its current state makes a proposal only for classical negation, and
further discussion is required before the acceptance of the need to use strong or
explicit negation.
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The Representation of Special Logics in Semantic Web languages v0.2
It is proposed that classical negation be represented by the Neg operator. This utilises
the law of excluded middle. However if there is a logical inference that has Neg A as
a precondition, then this may only succeed if Neg A is specifically in the
knowledgbase.
3.1.2 Default Negation
A key aspect of the CPM is that it is intended for use within a semantic web context.
There is a body of opinion that the semantic web in general terms is an “open world”,
and that the CWA cannot be made in the general case. There may be local subsets of
the semantic web whereby the CWA may be made, and indeed in the more controlled
environment of military planning, there may be situations where the CWA can be
made.
There are two approaches to logic that are related to these considerations:


Assumption-based reasoning [ABR]
Default logic [DEFLOG]
Assumption-based reasoning considers that it is permissible to create certain
propositions that are “assumed”, that is they do not follow logically from inferences
over the knowledgebase, but are in some sense created from “extra-logical”
considerations (e.g. intuition, or a desire to ask “what if”). Thus a problem solver may
create an assumption, and determine, via rules of inference, what follows from the
assumed proposition. Should an inconsistency be derived from a set of one or more
assumptions, then it may be concluded that the conjunction of the assumed
propositions is false. Such logic may be used to perform “what-if” reasoning or to
support search algorithms. Mechanisms have been proposed to hold databases of
alternative “possible worlds” indexed by the assumptions [ATMS]. These
considerations actually underpin the existing CPM3 representations of possible
worlds, choice points and assumptions [CPM3]
Default logic is intended to capture human-like reasoning, where propositions are
concluded “by default” in the absence of other countermanding information. Rules
may be specified that state that “in a given context (the prerequisite) whilst a
particular set of propositions (the justifications) are consistent with the current state
of knowledge, then one is entitled to infer another set of propositions (the
consequent).
It should be noted that the use of such rules (unlike classical logic) leads to nonmonotonicity of the knowledgebase. It is possible that at time t, the justification of a
default rule is consistent, so the consequent is added; then at a later time t1, something
is added to the knowledge base that makes the justification inconsistent, leading to the
retraction of the consequent. Thus something that was true at time t is no longer true
at time t1.
The application of a default rule may be seen as the making of an assumption (or one
assumption for each justification), followed by the use of the rule as an inference from
preconditions (the prerequisite and the justifications) to conclusions (the consequent).
Indeed the use of assumption based reasoning may also be non-monotonic if only a
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The Representation of Special Logics in Semantic Web languages v0.2
single knowledge base is maintained, to which the propositions are added (and
subsequently retracted). However if a global database of possible worlds indexed by
the assumptions is used, as in an ATMS, then this global database is monotonic, even
if individual possible worlds are not.
Default negation may be seen in terms of default reasoning, where the justification
and the consequent are both the negated proposition. This means that unless the
negated proposition is inconsistent with the knowledgebase it may be asserted (and
the consequences of this inferred). This is in effect stating that if the proposition
cannot be proved then its negation is not inconsistent.
However, it should be noted that all of these logics, default logic, assumption based
reasoning and default negation, all rely upon the notion that a set of propositions is
inconsistent with the knowledgebase. However the computation of consistency is in
the general case computationally complex, and so different mechanisms have been
defined to compute the set of negations (or defaults) that may be made.
The simplest mechanism is that of Prolog, where the idea of Negation-as-Failure
originated, which merely tries to prove a goal, and if this fails, to act as if the negation
be true. Although this is computationally tractable, it is deeply flawed in the general
case, since the negation will never be retracted as a result of subsequent additions to
the fact base.
Other mechanisms build upon the notion of an “extension” to the knowledgebase, that
is how can the default rules (or assumptions) be continued to be applied until there are
no further rules whose justifications do not directly contradict the knowledgebase
(this being an easy computation). Unfortunately, different orderings of applying the
rules may lead to different extensions, and hence there may be no single “correct”
extension. One such mechanism, defined for default logic is that of Etherington
[Etherington], and this is the one that is proposed here.
It is proposed that default negation be represented by the Naf operator, with the
semantics based upon defining negation as a type of default assumption, with
semantics as defined by [Etherington].
3.1.3 Summary
In summary, in respect of the representation of negation, we propose:




There are different requirements for the interpretation of negation, some where
the CWA may be made, some where default negations are permissible, and
some where only classical negation may be used
Classical negation may be represented, using the Neg operation. This utilises
the law of excluded middle. If Neg A is not present in the knowledgebase,
then no inference may be made about A.
A negation in absence of information to the contrary may be represented as the
making of an assumption (that the negation is true); however the making of
such assumptions should not occur automatically, but must be under the
control of a problem solver, and must hence be explicitly encoded in inference
rules
The making of a negative assumption is equivalent to the use of default logic,
where there is a single justification and consequent, both being the negated
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The Representation of Special Logics in Semantic Web languages v0.2

proposition. This may be described as “default negation”. A default negation
(assumption) may only be made for as long as it is not inconsistent with the
current knowledgebase.
The semantics of Etherington will be used to define the extension of a set of
default negation rules, expressed as default rules. This provides a formal
definition of “inconsistent” as used in the previous bullet point; that there is no
extension of the current knowledgebase that contains the negation.
3.2 RIF Dialect
A new RIF dialect is required in order to represent negation, since RIF-BLD [RIFBLD] excludes negation, and RIF-FLD (on which the proposal is based) does not
provide a specific semantics.
Within this document, we will call this the RIF-NAA (Negation as assumptions)
dialect.
3.2.1 Truth Values
RIF-FLD requires a dialect to specify the set of possible truth values, with a partial
order on these truth values, defining a complete lattice. We propose the use of the
existing set of RIF-FLD truth values:



t (true)
f (false)
u (unknown)
with the existing RIF-BLD total order:

f<u<t
It is necessary to include an unknown value, to handle the situation where default
negation is not acceptable.
Further consideration should be given to whether a “i” value (for inconsistency)
should be added, as this would be in the spirit of an ATMS, which explicitly encodes
inconsistencies.
3.2.2 Additional Syntax
Although RIF-FLD does not provide specific semantics for negation, it does provide
two syntactic terms in the abstract syntax:


Neg, to mean symmetric negation
Naf, to mean default negation
These syntactic terms, as defined in RIF-BLD will be used in RIF-NAA, for classical
negation and default negation as defined in the section on formal semantics below.
3.2.3 Formal Semantics
A RIF dialect must, for each syntactic term, provide an interpretation of that term and
a definition of the intended semantic multi-structures.
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The Representation of Special Logics in Semantic Web languages v0.2
The definition of Neg will be that of classical negation in RIF-FLD. All models that
use Neg (and do not use Naf) are intended semantic multi-structures.
The definition of Naf will be that of RIF-FLD. In respect of intended semantic multistructures, only those maximal extensions, as defined by Ethridgeton’s formal
semantics, are intended.
3.2.4 XML representation
The existing RIF-BLD XML representation of Neg and Naf will be used.
3.3 Controlled English
3.3.1 Neg
The specification of classical negation (Neg) is to be represented in Controlled
English by the expression:
“it is false that <Proposition>”
For example:
it is false that there is a bridge named B that has the river R as location
This may be used in the premises and conclusions of rules.
3.3.2 Naf
The specification of default negation (Naf) is to be represented in Controlled English
by the expression:
“it is assumed false that <Proposition>”
This reflects the derivation of default negation from the making of assumptions.
3.4 Rationale
The truth status of a Proposition, in the CPM rationale system, is defined by the
PropositionIndex which is pointing to it. It is therefore necessary to add two further
PropositionIndex types to correspond to Neg and Naf.
The PropositionIndex subtype “NegPI” will represent the situation where the negation
of the associated Proposition is explicitly known to be false.
4 The PropositionIndex subtype “DefNegPI” will
represent the situation where the negation of the
associated Proposition is assumed. More
specifically, DefNegPI is also a subtype of
Assumption, reflecting the derivation of default
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The Representation of Special Logics in Semantic Web languages v0.2
negation from the making of
assumptions.Conclusion
A proposed RIF dialect has been outlined, tentatively called RIF-NAA. This permits
classical negation and default negation, handling situations where explicit negative
knowledge and implicit negative knowledge is available.
However, it is recognised that this is a complex area, and that the current document
has only scratched the surface. Therefore more discussion about this document is
required, and it is expected that modifications will be made in the light of these
discussions. Furthermore, should there be a dialect of RIF handling default negation
from the RIF community, then this will be reviewed to see if the proposals in this
document need to be merged or superceded with this dialect.
Currently, the following questions are known to be unresolved:


whether a more powerful form of symmetric negation (such as strong or
explicit) is required
whether there is a need for an inconsistent value in the truth set.
5 Further work will be undertaken in the BPP09,
based on feedback from attempts to represent
negative information in CPM plans. References
[ABR] de Kleer, J. Problem solving with the ATMS, Artificial Intelligence 28 (2) pp.
197-224. 1986
[Alferes] Alferes, J.J, Pereira, L.M., Przymusinski, T.C. Strong and Explicit Negation
in Non-Monotonic reasoning and Logic programming, Proceedings of the European
Workshop on Logics in Artificial Intelligence, 1996
[ATMS] de Kleer, J. An assumption-based TMS, Artificial Intelligence 28 (2) pp.
127-162 (1986)
[BASLOG] Mott, D., The representation of logic within semantic web languages,
August 2009, ITACS https://www.usukitacs.com/?q=node/4986
[DEFLOG] Reiter, Ray, A logic for default reasoning, Artificial Intelligence, 13:81137, 1980.
[Etherington] Etherington, D.W., A Semantics for Default Logic: Proc IJCAI-10,
1987
[RIF-BLD] RIF Basic Logic Dialect Harold Boley, Michael Kifer, eds. W3C Working
Draft, 3 July 2009, http://www.w3.org/TR/2009/WD-rif-bld-20090703/.
[RIF-FLD] RIF Framework for Logic Dialects Harold Boley, Michael Kifer, eds.
W3C Working Draft, 3 July 2009, http://www.w3.org/TR/2009/WD-rif-fld20090703/.
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