Knowledge Repn. & Reasoning Lec #11+13: Frame Systems and Description Logics UIUC CS 498: Section EA Professor: Eyal Amir Fall Semester 2004
Jan 18, 2018
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
Frame Systems
Person
Man Woman
Concept frames
Jane
Object frames
Frame Systems
Person
Man Woman
Jane
Object frames
child
ageRoles
child
age
Jill,John
26
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 Reasoning
TBox
ABox
Description Logics: Language
.
• Example definition: C = AпB
• Example assertion: C(John), CпD = AпB
DescriptionLanguage Reasoning
TBox
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-elements
hasChild.Female
AL Description Logic: Language
• AL: C,D A | (atomic concept)T, | (universal, bottom)A, CпDR.C | (value restrict.)R.T | (limited existential quantific.)
• The concept including all elements that have role R filled by some element
hasChild.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 operations
п
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 ManC’= 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 satisfiability
1. 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 backtrackp (~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]
Example
• (R.A) п (R.B) R.(A п B)п
?
Example 2
• (R.A) п (R.B) п (≤1R) R.(A п B)п
?
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
Possible Projects
• Resolution-style algorithm for ALCN
Description Logics: Language
• REMEMBER:1. Beth’s definability and TBox/Abox
distinction
• Example definition: пU
• Assertions are made at an assertional part of the KB (Abox)