dl by qingyunliuliu


									More on Description Logic(s)

         Frederick Maier
               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
      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.

   The language basics
   Interpretations
   A Family of Languages
   Subsumption
   And other problems
   Complexity
 We must understand the Syntax,
 Semantics, and Inference Mechanisms
 of these languages if we are to use
 them effectively.

   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.
           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.
        Basic Concepts of a DL

   Individuals (such as john and mary)
   Concepts (such as Man and Woman).
   Roles (such as hasChild).
         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.

   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”
    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]
      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.
                        and 
    and  deserve special attention.
   Note that they only can come before a Role:

      HasChild.Girl         isEmployedBy.Farmer

   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.

   Add or Eliminate certain operators (e.g., , ),
    and the statements that can be expressed are
    increased/reduced in number.

   Higher expressiveness implies higher complexity.
              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).
                  A family of languages

Operation                             Notation

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

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

Determining whether one concept logically
 contains another is called the subsumption
               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}

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

The Subsumption Problem:
 It’s undecidable for reasonably expressive
 It’s non-polynomial for fairly restricted
Language   Subsumption   Instance Checking
FL-        P             P
AL         P             P
ALE        NP            PSPACE
ALC        PSPACE        PSPACE
KL-ONE     undecidable   undecidable
OWL-Lite   ?             ?
         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
   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.
   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.
              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]
   The Description Logic Website:

   Presentations from Enrico Franconi’s DL

   Chapter 2 of the Description Logic Handbook:

    *Upon which this presentation is mostly based.

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