Transcript Document

Ontology as a logic of intensions
Marie Duží
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Content
 Ontology and Knowledge Representation
 Languages for Ontology Specification
 Ontology Content
 Logic of Intensions (TIL in Brief)
Requisite Relation
 Part-whole Relation
 Integrity Constraints – inference rules

(presupposition vs. mere entailment)
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Ontology and Knowledge Representation
Why do we need an ontology?
 To make hidden knowledge explicit and
logically tractable.
 How do we build an ontology?
 By applying an expressive semantic
framework in order to make all the
semantically salient features of
knowledge specification explicit and
logically tractable.

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Languages for Ontology Specification
 F-calculi
 DL – description logic
 RDF – Resource Description Framework
•
OIL, DAML-OIL, DAML+OIL
 OWL – Ontology Web Language based on DL
 SKIF (Possibility to mention properties)
 SWRL – Semantic Web Rule Language
•
OWL and RuleML
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Well-defined ontology should serve as:
 universal library, the backdrop work of computational agents
 integrating a knowledge base and proces development
However, current ontology languages do not make it possible to
 express modalities (what is necessary and what is contingent),
 to distinguish three kinds of context, viz.
 extensional level of objects like individuals, numbers, functions
(-in-extension),
 intensional level of properties, propositions, offices and roles,
and finally
 hyperintensional level of concepts (i.e. algorithmically structured
procedures).
 Concepts of n-ary relations are unreasonably modelled by
properties.
 Ontology language should be universal, highly expressive, with
transparent semantics and meaning driven axiomatisation.
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Procedural Semantics of
Hyperintensional Logic (TIL)
 Procedural semantics contrasts with set-theoretical
denotational semantics.
 denotational approach  the meaning of ‘E’ = the extralinguistic entity denoted (or referred to) by ‘E’.
 hyperintensional procedural semantics 
expressions encode algorithmically structured
procedures producing either extensional or intensional
entities or lower-order procedures as their products.
 Algorithmic or computational turn: the early 1970s, Tichý
introduced his notion of construction as abstract
procedure (see also Moschovakis, 1994).
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TIL constructions
(are structured procedures)
a)
Atomic constructions (consisting of just one constituent: itself): supply
objects on which molecular constructions operate
Variables x, w, t, … v(aluation)-construct entities
Trivialization 0X constructs X
b)
Molecular constructions (consisting of other constituents than
themselves)
Composition [X X1…Xn] v-constructs the value of f at a
f
a
otherwise (v-)improper
Closure [x1…xn X] v-constructs a function f
Double Execution 2X: X  Y, Y  Z, then 2X  Z;
otherwise (v-)improper
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TIL types: ramified hierarchy
1.
Types of order 1 (non-constructions)


2.
3.
Base of atomic types: {, , , }
Functional types: ( 1… n), i.e. the set of partial
functions (1  …  n)  
Constructions of order n: v-construct objects of
types of order n
Types of order n+1 (constructions and functions
involving constructions in their domain or range)


The collection of constructions of order n, n, is the type
of order n+1
( 1… n) involving n is the type of order n+1
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Example
 ‘Dividing any number by 0 is improper’
Improper/(1) – the class of constructions of order
1 that are v-improper for any valuation v
Divide/() – the function of dividing; x  ; 0/:
[0Improper 0[0Divide x 00]]/2, v-constructs True
 ‘Tom knows that dividing any number by 0 is
improper’;
 Know/(((3))), (3)
wt [0Knowwt 0Tom 0[0Improper 0[0Divide x 00]]]
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Ontology Content
Conceptual (terminological) dictionary

primitive concepts

compound concepts (ontological definitions of entities)

the most important descriptive attributes, in particular
identification of entities
2. Conceptual Relations

contingent empirical relations between entities, in particular
the part-whole relation

analytical relations between intensions, i.e., requisites and
essence, which give rise to ISA hierarchy
3. Integrity constraints (inference rules)

Analytically necessary rules

Nomologically necessary rules

Common rules of ‘necessity by convention’
1.
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1. Conceptual dictionary

primitive concepts


…
compound concepts (ontological definitions
of entities)



0Car, 0Vehicle, 0Road, 0Junction, 0Driver,
‘driver is a person with a driving license’
0Driver = wt x [[0Person
wt x] 
[0Havewt x 0Driving_License]]
the most important descriptive attributes, in
particular identification of entities
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2. Conceptual Relations
 analytical (necessary) relations between
intensions, i.e., requisites and essence,
which give rise to ISA hierarchy


[0Requisite 0Vehicle 0Car]: necessarily, if
something is a car then it is a vehicle:
wt x [[0Carwt x]  [0Vehiclewt x]]
Requisite/(() ()); Vehicle, Car/()
 contingent typical empirical relations
between entities, in particular the partwhole relation
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3. Integrity constraints
 Analytically necessary rules


Necessarily, no car is a ship
wt [0No 0Carwt 0Shipwt]
 Nomologically necessary rules


No distinct physical objects can occur in the
same place (at the same time)
wt xy [x  y  [0Locwt x] = [0Locwt x]]
 Common rules of ‘necessity by convention’


wt x [C …x …]
Use the right-hand side lane (if possible)
 The degree of necessity decreasing top-down 
agents’ reasoning
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Logic of Intensions: requisite relation
obtains between intensions of any types; the most
important types:
Req1/(()()): an individual property is a requisite of another
property.
Req2/(): an individual office is a requisite of another such office.
Req3/(()): an individual property is a requisite of an individual
office.
Req4/(()): an individual office is a requisite of an individual
property.

Definition: “Y is a requisite of X” iff
“necessarily whatever occupies/instantiates X at w, t it also
occupies/instantiates Y at this w, t.”
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Requisite relations between properties
Req1 /(()()): basic relation that
gives rise to ISA taxonomies


explicitly record in ontology
hierarchies of intensions based on
requisite relations establish inheritance
of attributes and possibly also of
operations
Claim 1 Req1 is a quasi-order on the set of properties.
Proof obvious
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Requisite relations between properties
 Due to partiality – not anti-symmetric
(the property of having stopped smoking):


X = wt x [0StopSmokewt x]
Y = wt x [0Truewt wt [0StopSmokewt x]]
 In order to obtain week partial order, we need
antisymmetry; apply the usual “trick”: factor set of
equivalent classes defined as follows:
 0Eq =
pq [x [[0Truewt wt [pwt x]] = [0Truewt wt [qwt x]]]].
 [p]eq = q [0Eq p q] and [Req1’ [p]eq [q]eq] = [Req1 p q].
Claim 2 Req1’ is a weak partial order on the factor set of
the set of -properties with respect to Eq.

Proof obvious
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Part-whole relation (modest
individual anti-essentialism)
 If an individual i has a property P necessarily (in all worlds
and times), then P is a constant or partly constant function.
In other words, the property has a non-empty essential core
Ess, where Ess is a set of individuals that have the property
necessarily, and i is an element of Ess.
 frequently voiced objection:
If, for instance, Tom’s only car is disassembled into its elementary
physical parts, then Tom’s car no longer exists; hence, the
property of being a car is essential of the individual referred to by
‘Tom’s only car’.
First, what is denoted (as opposed to referred to) by ‘Tom’s only car’
is not an individual, but an individual office/role / .
Second, the individual referred to as ‘Tom’s only car’ does not cease
to exist even after having been taken apart into its most
elementary parts. It has simply lost some properties, among them
the property of being a car, the property of being composed of its
current parts, etc, while acquiring some other properties.
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Part-whole relation
Question
 Which parts are essential
for an individual in order
to have a property P?
 For instance, the property
of having an engine is
essential for the property of
being a car
 We have an instance of a
requisite relation between
intensions
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Part-whole relation
 Part-whole relation obtains contingently between individuals
which consist of other individuals and thereby create a
mereological sum.
 Being a part of is a relation between individuals, not between
intensions.
 From a logical point of view a car is not a structured whole
that organizes its parts in a particular manner.
 There is no inheritance or implicative relation between the
respective properties ascribed to a whole and its individual
parts.
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Some other properties of intensions
 Some higher-order properties of intensions are
necessarily valid due to the way they are
constructed.
 Since we explicate concepts as closed
constructions modulo - and -transformation,
i.e., procedurally isomorphic constructions, we
can also speak about mutual relations between
and properties of concepts which define
particular intensions, in particular:
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Relations between concepts
 Incompatibility of concepts; the populations of the defined properties are
necessarily disjoint;

Example: bachelor vs. married man
 Equivalence of concepts; the defined properties are one and the same
property (in particular ontological definitions);

Example: bachelor is an unmarried man
 Week-equivalence of concepts, the defined properties are ‘almost the same’;

Example: we echo the relation Eq between individual properties defined
above
 Functionality of a relation-in-intension; necessarily, in each w, t-pair, a
given relation R  Awt  Bwt is a mapping fR: Awt  Bwt assigning to each
element of A at most one element of B

Example: Each person has at most one driving license
 Inverse functionality of a relation-in-intension; necessarily, in each w, t-pair,
a given relation-in-extension R  Awt  Bwt is a mapping fR–1: Bwt  Awt
assigning to each element of Bwt at most one element of Awt.
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Reasoning of agents based on ontology
 It is useful to include into ontology important inference
rules, in particular the relations between hyperpropositions of (mere) entailment and presupposition
P is a presupposition of S
S |= P and non-S |= P
Corollary:
If non-P then neither S nor non-S is true  truth-value gap
S merely entails P
S |= P and neither (non-S |= P) nor (non-S |= non-P)
(entailement: necessarily, P is true whenever S is true)
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Topic-focus articulation
Sentences communicate something (focus F) about something (topic T).

schematic structure: F(T).
 The topic T of a sentence S is often associated with a presupposition
P of S  P is entailed both by S and non-S.
(1) “The critical situation on the highway D1
was caused by the agent a”.

presupposes that there be a critical situation on D1
 wt [if 0Crisiswt then [0Causewt 0a 0Crisis] else Fail]
(2) “The agent a caused
the critical situation on the highway D1”.

merely entails that there be a critical situation on D1
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If-then-else (the strict definition)
 Non-adequate analysis:
 [(Crisis  Caused-by-a) & (Crisis  Fail)]
 The whole Composition fails even if it is the case of crisis
 Mechanism of lazy evaluation:
 The procedural semantics of TIL operates smoothly
even at the hyper-intensional level of constructions:
 The analysis of “If P then C, else D” is a procedure that
decomposes into two phases:
1. on the basis of the condition P v , select one of C, D
as the procedure to be executed.
2. execute the selected procedure.
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If-then-else (the strict definition)
1. The selection is realized by the Composition
 [0the_only c [[P  [c=0C]]  [P  [c=0D]]]]
2. the chosen construction c is executed (Double
Execution)
The schematic analysis of “If P then C else D”:
2[0the_only
c [[P  [c=0C]]  [P  [c=0D]]]].
“If P then C else Fail”: 2[0the_only c [P  [c=0C]]]
If Crisis then Caused by a else Fail
wt 2[0the_only c
[0Crisiswt  [c = 0[0Causewt 0a 0Crisis]]]]
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Analytic schema of a sentence with a
presupposition P





“If P then S else Fail.”
The corresponding schematic TIL construction
wt 2[0c [Pwt  [c=0Swt]]].
In general, logic cannot disambiguate a sentence. Yet
our logical analysis can substantially contribute to the
disamiguation by making all the possible readings
explicit and logically tractable.
Thus the agent can ask: “What do you mean? This or
that?”
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Conclusion
‘Logic and AI for Multi-Agent Systems’
(http://labis.vsb.cz/)
Development of FIPA compliant computational variant of TIL,
the TIL-Script language
continue development into its full-fledged version equivalent
to TIL calculus.
Implementation of a method that decides a subset of the TILScript language computable by Prolog
now the subset equivalent to standard FOL.
We developed an extension of the editor Protégé-OWL so that to
create an interface between OWL and TIL-Script.
Sample test: 5 mobile agents (cars), 3 car parks and a GIS agent.
The GIS agent provided the mobile agents with ‘visibility’.
Communicated in TIL-Script and started with minimal (but not
overlapping) ontologies.
During the test they learned new concepts and enriched their
ontology. The agents’ goal was to find a vacant parking lot (out of 3
available) and park the car – succeeded.
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Reference
 Duží, M., Jespersen, B., Materna, P. (2010):
Procedural Semantics for Hyperintensional
Logic, Berlin, Springer.
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Thank you for your attention
If questions then answers else Fail

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