Ontology as a logic of intensions

1. Ontology as a logic of intensions Marie Duží, Martina Číhalová, Marek Menšík VSB–Technical University Ostrava, Department of Computer ScienceFEI, Silesian University in Opava, FPF, Institute of Computer Science Czech Republic

2. 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)

3. 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.

4. 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

5. Well-defined ontology should serve as: • universal library, thebackdrop work of computational agents • integrating a knowledge base and proces development However, current ontology languagesdo 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.

6. Procedural Semantics of Hyperintensional Logic (TIL) Procedural semantics contrasts with set-theoretical denotational semantics. denotational approach the meaning of ‘E’ = the extra-linguistic entity denoted (or referred to) by ‘E’. hyperintensional procedural semantics expressions encode algorithmically structured proceduresproducing either extensional or intensional entities or lower-order procedures as their products. Algorithmic or computational turn: the early 1970s, Tichý introduced his notion of constructionas abstract procedure (see also Moschovakis, 1994).

7. TIL constructions (are structured procedures) Atomic constructions (consisting of just one constituent: itself): supply objects on which molecular constructions operate Variables x, w, t, … v(aluation)-construct entities Trivialization 0Xconstructs X 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

8. TIL types: ramified hierarchy Types of order 1 (non-constructions) 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

9. 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/: [0Improper0[0Divide x 00]]/2, v-constructs True • ‘Tom knows that dividing any number by 0 is improper’; • Know/(((3))), (3) wt [0Knowwt 0Tom0[0Improper0[0Divide x 00]]]

10. Ontology Content Conceptual (terminological) dictionary primitive concepts compound concepts (ontological definitions of entities) the most important descriptive attributes, in particular identification of entities 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 Integrity constraints (inference rules) Analytically necessary rules Nomologically necessary rules Common rules of ‘necessity by convention’

11. 1.Conceptual dictionary primitive concepts 0Car, 0Vehicle, 0Road, 0Junction, 0Driver, … compound concepts (ontological definitions of entities) ‘driver is a person with a driving license’ 0Driver =wtx [[0Personwt x]  [0Havewt x 0Driving_License]] the most important descriptive attributes, in particular identification of entities

12. 2. Conceptual Relations analytical (necessary) relations between intensions, i.e., requisites and essence, which give rise to ISA hierarchy [0Requisite 0Vehicle0Car]: 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 part-whole relation

13. 3. Integrity constraints Analytically necessary rules Necessarily, no car is a ship wt[0No 0Carwt0Shipwt] 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

14. Logic of Intensions: requisite relation obtains between intensions of any types; the most important types: Req1/(()()): an individual property is a requisite of anotherproperty. Req2/(): an individual office is a requisite of another such office. Req3/(()): an individual property is a requisite of an individualoffice. Req4/(()): an individual office is a requisite of an individualproperty. 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.”

15. 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 inheritanceof attributes and possibly also of operations Claim 1 Req1 is a quasi-order on the set of -properties. Proof obvious

16. 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] = [Req1pq]. Claim 2Req1’ is a weak partial order on the factor set of the set of -properties with respect to Eq. • Proof obvious

17. Part-whole relation (modest individual anti-essentialism) • If an individualihas 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.

18. Part-whole relation Question Which parts are essentialfor an individual in order to have a propertyP? 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

19. 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.

20. 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:

21. 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Bwtassigning 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Awtassigning to each element of Bwtat most one element of Awt.

22. Reasoning of agents based on ontology • It is useful to include into ontology important inference rules, in particular the relations between hyper-propositions of (mere) entailment and presupposition P is a presupposition of S S |=P and non-S|=P Corollary: If non-P then neitherS nornon-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)

23. Topic-focus articulation Sentences communicate something (focus F) about something (topic T). • schematic structure: F(T). • The topic Tof a sentence S is often associated with a presuppositionPof SP is entailed both by S and non-S. • “The critical situationon the highway D1 was caused by the agent a”. • presupposes that there be a critical situation on D1 • wt [if0Crisiswtthen[0Causewt0a 0Crisis] else Fail] • “The agent acaused the critical situation on the highway D1”. • merely entails that there be a critical situation on D1

24. 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 “IfP thenC, elseD”is a procedure that decomposes into two phases: • on the basis of the condition P v , select one of C, D as the procedure to be executed. • execute the selected procedure.

25. If-then-else (the strict definition) • The selection is realized by the Composition • [0the_only c [[P  [c=0C]]  [P  [c=0D]]]] • 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]]] IfCrisis thenCaused by a elseFail wt 2[0the_only c [0Crisiswt[c = 0[0Causewt0a 0Crisis]]]]

26. Analytic schema of a sentence with a presupposition P • “If P then S else Fail.” • The corresponding schematic TIL construction • wt2[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?”

27. 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 TIL-Script 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.

28. Reference • Duží, M., Jespersen, B., Materna, P. (2010):Procedural Semantics for Hyperintensional Logic,Berlin, Springer.

29. Thank you for your attention If questions then answers else Fail 