Download - Knowledge Repn. Reasoning Lec #11+13: Frame Systems and Description Logics UIUC CS 498: Section EA Professor: Eyal Amir Fall Semester 2004

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)


.

• Definitions are stated in a terminological part of the KB (TBox)


DescriptionLanguage Reasoning

TBox

ABox


.

• Example definition: C = AпB

• Example assertion: C(John), CпD = AпB

DescriptionLanguage Reasoning

TBox

ABox


• 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: 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: 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: 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: 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


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


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


• 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)}


-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)}


• ≥-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


• REMEMBER:1. Beth’s definability and TBox/Abox

distinction

• Example definition: пU


Download - Knowledge Repn. Reasoning Lec #11+13: Frame Systems and Description Logics UIUC CS 498: Section EA Professor: Eyal Amir Fall Semester 2004

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