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Knowledge Repn. & Reasoning Lec #11+13: Frame Systems and Description Logics. UIUC CS 498: Section EA Professor: Eyal Amir Fall Semester 2004. Today. Restricting expressivity of FOL: DLs Description Logics (DLs) Language Semantics Inference. Description Logics (DLs). - PowerPoint PPT Presentation
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Knowledge Repn. & ReasoningLec #11+13: Frame Systems and
Description LogicsUIUC CS 498: Section EA
Professor: Eyal AmirFall Semester 2004
Today
• Restricting expressivity of FOL: DLs
• Description Logics (DLs)– Language– Semantics– Inference
Description Logics (DLs)• Originate in semantic networks (NLP), and
Frame Systems (KR)
• Hold information about concepts, objects, and simple relationships between them– Hierarchical information
• Many DLs, differing in their expressive power
Differences from DBs
• Hierarchical structure (?)
• Many times no closed-world assumption
• Values may be missing
• More expressive (?)
• Semantic structure between concepts and roles
• Typical reasoning tasks (satisfiability, generality/classification)
Description Logics: Language
• Formal language that can be analyzed
• Describe frame systems with attention to the expressive power needed
• Definitions are stated in a terminological part of the KB (TBox)
• Assertions are made at an assertional part of the KB (Abox)
Description Logics: Language
.
• Definitions are stated in a terminological part of the KB (TBox)
• Assertions are made at an assertional part of the KB (Abox)
DescriptionLanguage
ReasoningTBox
ABox
Description Logics: Language
.
• Example definition: C = AпB
• Example assertion: C(John), CпD = AпB
DescriptionLanguage
ReasoningTBox
ABox
AL Description Logic: Language
• AL: C,D A | (atomic concept)
T | (universal concept)
| (bottom concept)
A | (atomic negation)
CпD | (intersection)
R.C | (value restrict.)
R.T | (limited existential quantific.)
AL Description Logic: Language
• AL: C,D A | (atomic concept)
A Person | Female
• An atomic concept corresponds to a unary predicate symbol in FOL
• Extensionally, a set of world elements
AL Description Logic: Language
• AL: C,D A | (atomic concept)
T | (universal concept)
• Intuition: The universal concept corresponds in FOL to λx.TRUE(x), the unary predicate that holds for every object
AL Description Logic: Language
• AL: C,D A | (atomic concept)
T | (universal concept)
| (bottom concept)
• Intuition: The bottom concept corresponds in FOL to λx.FALSE(x), the unary predicate that holds for no object
AL Description Logic: Language
• AL: C,D A | (atomic concept)
T | (universal concept)
| (bottom concept)
A | (atomic negation)• The negation of A is the concept that is the
complement of A, i.e., contains all elements that A does not
Female, Person
AL Description Logic: Language
• AL: C,D A | (atomic concept)
T | (universal concept)
| (bottom concept)
A | (atomic negation)
CпD | (intersection)
• Intersection of concepts corresponds to set intersection of their elements
• Person п Female
AL Description Logic: Language
• AL: C,D A | (atomic concept)
T, | (universal, bottom)
A | (atomic negation)
CпD | (intersection)
R.C | (value restrict.)• All elements whose R is filled only by C-
elementshasChild.Female
AL Description Logic: Language
• AL: C,D A | (atomic concept)
T, | (universal, bottom)
A, CпD
R.C | (value restrict.)
R.T | (limited existential quantific.)• The concept including all elements that
have role R filled by some elementhasChild.T
AL DL: FOL Semantics
• Interpretation I maps Δ to nonempty set ΔI
and,– Every atomic concept A is mapped to AI ΔI
– TI = ΔI
I = Ø– (A)I = ΔI \ AI
– (CпD)I = CI п DI
– (R.C)I = {a ΔI | b. (a,b)RI b CI }
– (R.T)I = {a ΔI | b. (a,b)RI}
DLs that Extend ALR.C – full existential quantification
• (≥n R) - number restrictionsC – negation of arbitrary concepts
• CUD – union of concepts
• Trigger rules – CLASSIC (configuration of systems), LOOM
TBox: Terminological Axioms
• C D – The left-hand side is a symbol• R S – same• C D – same • R S – same
• Mother Woman п hasChild.Person• Parent Mother U Father• Grandmother Mother п hasChild.Mother
пп
Definitional / Nondefinitional
• Base interpretation for atomic concepts
• The TBox is definitional if every base interpretation has only one extension
• Observation: If the TBox has no cycles then it is definitional
ABox: Assertions About Elements
• Father(Peter) C(a)
• Grandmother(Mary) C(a)
• hasChild(Mary,Peter) R(b,c)
• hasChild(Mary,Paul) R(b,c)
• hasChild(Peter,Harry) R(b,c)
• C(a) – concept assertions
• R(b,c) – role assertions
ABox: Assertions About Elements
• UNA – Unique Names Assumption
• Interpretation I maps object names to elements in ΔI
• Some languages allow other statements, within a fragment of FOL.
• TBox,Abox equivalent to a set of axioms in FOL (with two variables, without functions)
Take a Breath
• So far: Language + Semantics
• From here:– Reasoning Tasks– Algorithms
• Later: NLP using Description Logics
TBox Reasoning Tasks
• Satisfiability of C:– A model I of T such that CI is nonempty
• Subsumption of C by D– For every model I of T, CI DI
• Equivalence of C and D
• Disjointness of C and D
п
Reductions to Subsumption
• C is unsatisfiable iff C • C,D equivalent iff C D, D C
• C,D disjoint iff CпD
• With an empty or nonempty TBox
• Assuming we have the needed operationsп
ппп
Reductions to Unsatisfiability
• C D iff CпD unsatisfiable
• C,D equivalent iff CпD , CпD unsatisfiable
• C,D disjoint iff CпD unsatisfiable
• With an empty or nonempty TBox
• Assuming we have the needed operations
п
Systems vs Reasoning
• CLASSIC, LOOM : Subsumption
• KRIS, CRACK, FACT, DLP, RACE: Satisfiability
• Subsumption is most general and therefore most expensive computationally
Eliminating the TBox
• Converting definitional TBox problems to concept problems
T={ Woman Person п Female
Man Person п Woman }
C = Woman п Man
C’= Person п Female п Person п
(Person п Female)
ABox Queries
• Consistency
• Instance check – A C(a)– “a” is an instance name– Reduces to concept satisfiability if “set” and
“fill” constructors are allowed
• Retrieval of all individuals satisfying C
• Find most specific concept for individual a╨
Structural Subsumption
• Language: FL0
– Concept conjunction C п D– Value restriction R.C
• Normal form of concepts in FL0
C A1 п … п Am п R1.C1 п … п Rn.Cn
D B1 п … п Bk п S1.D1 п … п Sl.Dl
• C D iffi≤k j≤m s.t. Bi = Aj
i≤l j≤n s.t. Si = Rj , Ci Dj
• Proof?
п
п• Proof?
Structural Subsumption Algorithm for FL0
1. Convert concepts to normal form
C A1 п … п Am п R1.C1 п … п Rn.Cn
D B1 п … п Bk п S1.D1 п … п Sl.Dl
2. Check recursively:
i≤k j≤m s.t. Bi = Aj
i≤l j≤n s.t. Si = Rj , Ci Dj
п
Extending FL0
• Language: FL0
– Concept conjunction C п D– Value restriction R.C
• Language: ALN– AL (C п D, R.C , T, , A, R.T)– Number restrictions (≥nR, ≤nR)
Structural Subsumption for ALN
• Language: ALN– AL (C п D, R.C , T, , A, R.T)– Number restrictions (≥nR, ≤nR)
• Normal form for ALNC L1 п … п Lm п R1.C1 п … п Rn.Cn
or C , – Li atomic concepts, their negation, or ≥nR,≤nR
– Ci in normal form, Ri, Ai distinct
Computing Normal Form for ALN
• C п D, R.C , T, , A, R.T, ≥nR, ≤nR
C L1п…пLm п R1.C1п…пRn.Cn or C1. Look at outermost connective
1. , T, , ≥nR, ≤nR, R.T : return concept2. R.C : C’ = recurse on C; return R.C’ 3. C п D – recurse on C,D, generating C’,D’; 4. If top level of C’ п D’ includes conflict (A,A;
; ≥nR,≤mR (n<m); ≥nR,R.), return 5. Return C’ п D’
Structural Subsumption Algorithm for ALN
1. Convert concepts to normal form
C L1 п … п Lm п R1.C1 п … п Rn.Cn
D N1 п … п Nk п S1.D1 п … п Sl.Dl
2. Check recursively:
i≤k j≤m s.t. Bi = Aj
i≤l j≤n s.t. Si = Rj , Ci Dj
with ≥nR ≥mR iff n≥m
п
п
Example
• C=Person п Female п hasChild.T п hasChild.Person п hasChild.Female п hasChild.hasChild.Female п hasChild.hasChild.Female
• D=Person п ≥1.hasChild
ON BOARD
Extending ALN
• Language: ALCN– ALN:
CпD, R.C , T, , A, R.T, ≥nR, ≤nR
– Arbitrary negation (complement) C
• Overall algorithm for satisfiability1. Convert to negation normal form (negation
in front of atoms only)
2. Use tableau theorem proving to find model
Principles of Tableau Reasoning
• Apply rules and build tree (defines model):
• When a branch of the tree is contradictory to itself (e.g., has A,A), we backtrack
p (~q ~p)
p
(~q ~p)
~q ~p
Tableau forPropositional logic:Rules for ,
Tableau-based Satisfiability Algorithm for ALCN
1. Want to show that C0 (in NNF) is satisfiable
2. We look for a model of Abox A = {C0(x0)}, with x0 a new constant symbol
1. Apply (consistency preserving) transformation rules
2. If at some point a “complete” ABox is generated, then C0 is satisfiable
3. If no complete ABox found, C0 unSAT
Tableau-based Satisfiability Algorithm for ALCN
• п-rule:– Condition: A contains (C1 п C2)(x), but neither
C1(x),C2(x)– Action: A’=A{C1(x),C2(x)}
• U-rule:– Condition: A contains (C1 U C2)(x), but
neither C1(x),C2(x)– Action (nondeterministically choose):
A’=A{C1(x)}, A’’=A{C2(x)}
Tableau-based Satisfiability Algorithm for ALCN
-rule:– Condition: A contains (R.C)(x), but there is
no individual name z s.t. C(z) and R(x,z) in A– Action: A’=A{C(y),R(x,y)} for y an individual
name not occuring in A
• -rule:– Condition: A contains (R.C)(x) and R(x,y),
but C(y) is not in A– Action: A’=A{C(y)}
Tableau-based Satisfiability Algorithm for ALCN
• ≥-rule:– Condition: A contains (≥nR)(x), but no individual
names z1,…, zn s.t. R(x,zj) (i≤n) and zj≠zj (i<j≤n)– Action: A’=A{R(x,yj)| i≤n}{yi≠yj| i<j≤n}, and y1,…,yn
distinct individual names not in A
• ≤-rule:– Condition: A contains distinct individual names y1,
…,yn+1 s.t. (≤nR)(x) and R(x,yi) (i≤n) in A, but yi≠yj not in A for some i≠j
– Action (nondeterministically choose j<i≤n with yi≠yj): A’=A[yi/yj]
Computational Properties
• Satisfiability (and subsumption) in ALCN is PSpace-complete
• This tableau algorithm takes time O(22^n)
• Small improvement gives a nondeterministic PSpace tableau algorithm which takes time O(22n)– n = length of concept/s
Related to DL
• Natural language processing
• Semantic web
• Complexity of reasoning and decidable first-order languages
• Conceptual modeling
• CYC
Summary So Far
• Description Logics provide expressivity / tractability tradeoff– ALN reasoning in polynomial time– ALCN reasoning in PSpace
• Next: Medical informatics
Application: Medical Informatics
• GALEN: A terminological knowledge base (TBox) of human anatomy
• Hierarchical display
• Multiple axes
• Simple combinations of concepts
• Automatic-dynamic classification of new concepts
• Aid in creating new concepts
Application: Medical Informatics
• Example: classification– Leg which
• hasLeftRightSelector leftSelection
– Leg п leftRightSelector.leftSelection, or– Leg п leftRightSelector.{leftSelection}
• The language does not include negation
• If have time – show demo