More on Description Logic(s) Frederick Maier

Note Added 10/27/03 So, there are a few errors that will be obvious to some: 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). 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). Instance checking is not reducible to subsumption in every case (e.g., see this).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. (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). 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.

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

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

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

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

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

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

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

Defining Descriptions (ALC, a typical language) A description C or D can be: A description C or D can be: A an atomic concept. * T (top) the universal concept.*  (bottom) the null concept *  Ca negated concept * C 1 ∏ D 1 the intersection of concepts. * C 1  D 1 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]

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

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

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

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

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

The Language AL A description C or D can be: A description C or D can be: A an atomic concept. T (top) the universal concept.  (bottom) the null concept  Ca negated Atomic concept C 1 ∏ D 1 the intersection of concepts.  R.C (restriction)  R.T (Limited existential quantification).

A family of languages OperationNotation Union (U) C  B Complementation (C)  C (Any Concept) Full Existential Quantification (E)  R.C Cardinality (N)≥ nR, ≤nR Qualified Cardinality (Q)≥ nR.C, ≤nR.C Enumeration (O){a,b,…} Selection (F)f:C

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

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

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

 C} Satisfiability of a Concept or KB {C,  C} Instance CheckingFather(john)? ≡ EquivalenceCreatureWithHeart ≡ CreatureWithKidney ∏ D Disjointness C ∏ D RetrievalFather(X)? X = {john, robert} RealizationX(john)? X = {Father} Other Problems:

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

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

Complexity LanguageSubsumptionInstance Checking FL - PP ALPP ALENPPSPACE ALCPSPACE ALCOPSPACE SHIQEXPTIME KL-ONEundecidable OWL-Lite??

Inference Mechanisms ALC is equivalent to L 2 and so, theoretically, we could translate all the expressions of the DL into L 2 and then use resolution or some algorithm as a decision procedure. ALC is equivalent to L 2 and so, theoretically, we could translate all the expressions of the DL into L 2 and then use resolution or some algorithm as a decision procedure. However, it is generally the case that Tableau algorithms are computationally less expensive. However, it is generally the case that Tableau algorithms are computationally less expensive.

Tableau algorithms They work by systematically building up a tree of possible models to for a KB. 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. 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. Tableau proofs are sound and complete for many languages, including ALC.

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 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(2 p(n) ) time, where p(n) is a polynomial function of n. In complexity theory, EXPTIME is the set of all decision problems solvable by a deterministic Turing machine in O(2 p(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 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

References The Description Logic Website: The Description Logic Website: Presentations from Enrico Franconi’s DL course*: Presentations from Enrico Franconi’s DL course*: Chapter 2 of the Description Logic Handbook: Chapter 2 of the Description Logic Handbook: *Upon which this presentation is mostly based.