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

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]

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)