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) 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 ManWoman Concept frames Jane Object frames

Frame Systems Person ManWoman Jane Object frames child age Roles 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) Description Language Reasoning TBox ABox

Description Logics: Language. Example definition: C = Aп B Example assertion: C(John), CпD = Aп B Description Language 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, CпD  R.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 Δ I and, –Every atomic concept A is mapped to A I  Δ I –T I = Δ I –  I = Ø –(  A) I = Δ I \ A I –(CпD) I = C I п D I –(  R.C) I = {a  Δ I |  b. (a,b)  R I  b  C I } –(  R.T) I = {a  Δ I |  b. (a,b)  R I }

DLs that Extend AL  R.C – full existential quantification (≥n R) - number restrictions  C – negation of arbitrary concepts C U D – 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 п п

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 Δ I 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 C I is nonempty Subsumption of C by D –For every model I of T, C I D I 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 пп п

Reductions to Unsatisfiability C D iff Cп  D unsatisfiable C,D equivalent iff Cп  D,  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: FL 0 –Concept conjunctionC п D –Value restriction  R.C Normal form of concepts in FL 0 C  A 1 п … п A m п  R 1.C 1 п … п  R n.C n D  B 1 п … п B k п  S 1.D 1 п … п  S l.D l C D iff  i≤k  j≤m s.t. B i = A j  i≤l  j≤n s.t. S i = R j, C i D j Proof? п п

Structural Subsumption Algorithm for FL 0 1. Convert concepts to normal form C  A 1 п … п A m п  R 1.C 1 п … п  R n.C n D  B 1 п … п B k п  S 1.D 1 п … п  S l.D l 2. Check recursively:  i≤k  j≤m s.t. B i = A j  i≤l  j≤n s.t. S i = R j, C i D j п

Extending FL 0 Language: FL 0 –Concept conjunctionC 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 ALN C  L 1 п … п L m п  R 1.C 1 п … п  R n.C n or C  , –L i atomic concepts, their negation, or ≥nR,≤nR –C i in normal form, R i, A i distinct

Computing Normal Form for ALN C п D,  R.C, T, ,  A,  R.T, ≥nR, ≤nR C  L 1 п…пL m п  R 1.C 1 п…п  R n.C n or C  1.Look at outermost connective 1. , T, , ≥nR, ≤nR,  R.T : return concept 2.  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  L 1 п … п L m п  R 1.C 1 п … п  R n.C n D  N 1 п … п N k п  S 1.D 1 п … п  S l.D l 2. Check recursively:  i≤k  j≤m s.t. B i = A j  i≤l  j≤n s.t. S i = R j, C i D j 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 backtrack p  (~q  ~p) p (~q  ~p) ~q ~p Tableau for Propositional logic: Rules for , 

Tableau-based Satisfiability Algorithm for ALCN 1.Want to show that C 0 (in NNF) is satisfiable 2.We look for a model of Abox A = {C 0 (x 0 )}, with x 0 a new constant symbol 1.Apply (consistency preserving) transformation rules 2.If at some point a “complete” ABox is generated, then C 0 is satisfiable 3.If no complete ABox found, C 0 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 z 1,…, z n s.t. R(x,z j ) (i≤n) and z j ≠z j (i<j≤n) –Action: A’=A  {R(x,y j )| i≤n}  {y i ≠y j | i<j≤n}, and y 1,…,y n distinct individual names not in A ≤ -rule: –Condition: A contains distinct individual names y 1,…,y n+1 s.t. ( ≤ nR)(x) and R(x,y i ) (  i≤n) in A, but y i ≠y j not in A for some i≠j –Action (nondeterministically choose j<i≤n with y i ≠y j ): 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(2 2^n ) Small improvement gives a nondeterministic PSpace tableau algorithm which takes time O(2 2n ) –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: п U Assertions are made at an assertional part of the KB (Abox)