More on Description Logic(s) Frederick Maier

1. More on Description Logic(s)Frederick Maier

2. Note Added 10/27/03 • So, there are a few errors that will be obvious to some: • Some of the symbols used in expressions are not in the right font (or even of the right type in some cases). • Instance checking is not reducible to subsumption in every case (e.g., see this). • (The) typical means of proof is based upon satisfiability (as the slides on semantic tableaux indicate); I should have pointed this out more explicitly. • Again, most of the material is taken form Enrico Franconi’s course website (I believe he’s even the originator of the DL logo) . I’d like to take the presentation down, as it really offers nothing that couldn’t be found elsewhere just as readily, but I’ll wait until the end of the term.

3. Overview • The language basics • Interpretations • A Family of Languages • Subsumption • And other problems • Complexity

4. Assumption: • We must understand the Syntax, Semantics, and Inference Mechanisms of these languages if we are to use them effectively.

5. OWL • The language in which our ontologies are going to be written in is likely going to be OWL, or something like it. • And OWL is based in part on DL.

6. What are DL’s? Key features: • They are a family of Knowledge Representation languages with a formal semantics based largely on FOL. • They attempt to discover “implicitly represented knowledge” using efficient inference mechanisms. • The complexity of the inferences is an area of determined research.

7. Basic Concepts of a DL • Individuals (such as john and mary) • Concepts (such as Man and Woman). • Roles (such as hasChild).

8. Basic Concepts of a DL • Individuals are treated exactly the same as constants in FOL. • Concepts are exactly the same as Unary Predicates in FOL. • Roles are exactly the same as Binary Predicates in FOL.

9. Descriptions • Just Like in FOL, what we are dealing with (ultimately) are sets of individuals and relations between individuals. • The basic unit of semantic significance is the Description. “We are describing sets of individuals”

10. Defining Descriptions (ALC, a typical language) • A description C or D can be: A an atomic concept. * T (top) the universal concept. *  (bottom) the null concept * C a negated concept * C1 ∏ D1 the intersection of concepts. * C1 D1 the union of two concepts. R.C (restriction) * R.C (existential quantification). * [* present in AL. Only atomic concepts can be negated.  restricted to R.T]

11. Interpretations and Models • Mostly, the formal semantics of a DL follows FOL: • An individual is interpreted as an element from the universe of discourse. • A concept is interpreted as the set of elements from the universe to which the concept applies.

12.  and  •  and  deserve special attention. • Note that they only can come before a Role: HasChild.Girl isEmployedBy.Farmer • Remember, they describe sets of individuals.

13. and  HasChild.Girl would be interpreted as: The set{ x | (y)( HasChild(x,y)  Girl(y) ) } [Note the conditional: Am I in that set?].

14.  and  isEmployedBy.Farmer would be: The set{ x | (y)( isEmployedBy(x,y) & Farmer(y) ) }

15. A family of languages • The expressiveness of a description logic is determined by the operators that it uses. • Add or Eliminate certain operators (e.g., , ), and the statements that can be expressed are increased/reduced in number. • Higher expressiveness implies higher complexity.

16. The Language AL • A description C or D can be: A an atomic concept. T (top) the universal concept.  (bottom) the null concept C a negated Atomic concept C1 ∏ D1 the intersection of concepts. R.C (restriction) R.T (Limited existential quantification).

17. A family of languages

18. Axioms • We may assign names to complex descriptions: Bachelor ≡ Unmarried ∏ Male • Or assert that one concept is subsumed by another: C  D • These are Axioms of the system.

19. Subsumption A concept C subsumes a concept D iff I(D)  I(C) on every interpretation I. This means the same as the assertion: (x)(D(x)  C(x)) where D and C are complex statements

20. The Subsumption Problem C  D ? Determining whether one concept logically contains another is called the subsumption problem.

21. Other Problems: Satisfiability of a Concept or KB {C, C} Instance Checking Father(john)? Equivalence CreatureWithHeart ≡ CreatureWithKidney Disjointness C ∏ D Retrieval Father(X)? X = {john, robert} Realization X(john)? X = {Father}

22. Reduction • These problems can be reduced to subsumption (for languages with negation). • They can be reduced to the satisfiability problem, as well.

23. Complexity The Subsumption Problem: • It’s undecidable for reasonably expressive languages, • It’s non-polynomial for fairly restricted languages.

24. Complexity

25. Inference Mechanisms • ALC is equivalent to L2 and so, theoretically, we could translate all the expressions of the DL into L2 and then use resolution or some algorithm as a decision procedure. • However, it is generally the case that Tableau algorithms are computationally less expensive.

26. Tableau algorithms • They work by systematically building up a tree of possible models to for a KB. • If every branch of the tree possesses a contradiction, then the KB is unsatisfiable. • Tableau proofs are sound and complete for many languages, including ALC.

30. Complexity: Notes • In complexity theory the class PSPACE is the set of decision problems that can be solved by a Turing machine using a polynomial amount of memory, and unlimited time. • In complexity theory, EXPTIME is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. • EXPTIME is known to be a subset of EXPSPACE and a superset of PSPACE, NP-complete, NP, and P. That is significant because it is currently unknown which (if any) of those four sets are equal to each other. It is known however that P is a strict subset of EXPTIME [From www.wikipedia.org]

31. References • The Description Logic Website: http://dl.kr.org/ • Presentations from Enrico Franconi’s DL course*: http://www.inf.unibz.it/~franconi/dl/course/ • Chapter 2 of the Description Logic Handbook: http://www.inf.unibz.it/~franconi/dl/course/dlhb/dlhb-02.pdf *Upon which this presentation is mostly based.