2. A tableau algorithm for ALC with TBoxes, number restrictions, and inverse roles. Extend ALC-tableau algorithm from first session with

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1 2. A tableau algorithm for ALC with TBoxes, number restrictions, and inverse roles Extend ALC-tableau algorithm from first session with 1 general TBoxes 2 inverse roles 3 number restrictions Goal: Design sound and complete desicion procedures for satisfiability (and subsumption) of DLs which are well-suited for implementation purposes 1

2 A tableau algorithm for ALC with general TBoxes Remember: A concept equation is of the form C. = D for C, D (complex) concepts A (general) TBox is a finite set of concept equations I satisfies C =. D iff C I = D I. I is a model of TBox T = {C i = Di 1 i n} iff I satisfies each concept equation in T Extend ALC tableau algorithm to TBox transform a TBox. T = {C i = Di 1 i n}, into a single, equivalent concept equation. = 1 i n ( C i D i ) (C i D i ) 1 i n (C i D i ) =: C T, 2

3 A tableau algorithm for ALC with general TBoxes Extend ALC-tableau algorithm with the following rule: CE : If C T L(x) Then L(x) L(x) {C T } Example: Consider TBox {C. = R.C}. Is C satisfiable w.r.t. this TBox? Tableau algorithm no longer terminates! Reason: The role depth of concepts no longer decreases along paths in a completion tree 3

4 A tableau algorithm for ALC with general TBoxes Regain termination with cycle-detection Blocking: if x and one of its ancestors y are very similar, only extend y: x is directly blocked if it has an ancestor y with L(x) L(y) In this case and if y is the closest such node to x, we say that x is blocked by y A node is blocked if it is directly blocked or one of its ancestors is blocked restrict the application of all rules to nodes which are not blocked Tableau algorithm for ALC w.r.t. TBoxes 4

5 A tableau algorithm for ALC with general TBoxes Lemma: Let T be a general ALC-Tbox and C 0 an ALC-concept. Then 1. the algorithm terminates when applied to T and C 0 and 2. the rules can be applied such that they generate a clash-free and complete completion tree iff C 0 is satisfiable w.r.t. T. Corollary: Satisfiability of ALC-concept w.r.t. general TBoxes is decidable and ALC has the finite model property. 5

6 A tableau algorithm for ALC with general TBoxes Proof: 1.(Termination): blackboard 2. Soundness: for T a complete, clash-free completion tree for C 0 and T, construct a finite (non-tree) model as follows: := {x x is a node in T, x is not blocked} A I := {x I A L(x)} for concept names A I R I := { x, y I2 x, y E(R) or x, y E(R) and y blocks y } for role names R Prove (by induction on the structure of concepts) that, for all x I, C sub(c 0, T ): C L(x) implies x C I. Then, since C 0 is in the label of the root node and C T I is indeed a model of C 0 and T. in the label of all nodes, 6

7 A tableau algorithm for ALC with general TBoxes 3. (Completeness): use model I of C 0 and T to steer the application of the -rule. Inductively define a mapping π : nodes of completion tree I and show that L(x) {C π(x) C I } if x, y E(R), then π(x), π(y) R I ( ) 7

8 A tableau algorithm for ALCI with general TBoxes ALCI is the extension of ALC with inverse roles R in the place of role names: (R ) I := { y, x x, y R I }. Motivation: with inverse roles, one can use both has-child and is-child-of has-part and is-part-of and their interaction Example: w.r.t. T = {. = parent. }, does parent. child.blond Blond? Example: T = { C =.. R.C = T. S. R.C} and C 0 = R. S. T.A 8

9 A tableau algorithm for ALCI with general TBoxes Modifications necessary to handle inverse roles: 1 extend edges E(.) in c-trees to inverse roles, 2 call y an R-neighbour of x if either y is an R-successor of x or x is an R successor of y, 3 substitute R-successor in the - and -rule with R-neighbour (still create an R-successor in case no R-neighbour exists for an R.C R -successor in case no R -neighbour exists for an R.C). Example: Is A satisfiable w.r.t. {A. = R.A ( R.( A S.B))}? Does this suffice? No! Example: A R. A R.A and T = {A. = X} 9

10 A tableau algorithm for ALCI with general TBoxes subset-blocking (with L(x) L(x )) no longer suffices: 4 A node x is called directly blocked if it has an ancestor y with L(x) = L(y). Lemma: Let T be a general ALCI-Tbox and C 0 an ALCI-concept. Then 1. the algorithm terminates when applied to T and C 0, 2. the rules can be applied such that they generate a clash-free and complete completion tree iff C 0 is satisfiable w.r.t. T. 10

11 A tableau algorithm for ALCI with general TBoxes Proof: 1.(Termination): identical to the ALC case. 2. (Soundness): again, construct a finite (non-tree) model from a complete, clash-free c-tree T for C 0 and T (take into account edges labelled with inverse roles): := {x x is a node in T, x is not blocked} A I := {x I A L(x)} for concept names A I R I := { x, y I2 x, y E(R), y, x E(R ) or x, y E(R) and y blocks y or y, x E(R ) and x blocks x } for role names R Again, prove that, for all x I : C L(x) implies x C I. 11

12 A tableau algorithm for ALCI with general TBoxes 3. (Completeness): again, use model I of C 0 and T and mapping π to steer the application of the -rule. Corollary: Satisfiability of ALCI-concept w.r.t. general TBoxes is decidable and ALCI has the finite model property. It can be shown that pure ALCI-concept satisfiability is PSpace-complete, just like ALC. 12

13 A tableau algorithm for ALCQI with general TBoxes Further add qualifying number restrictions ( nr.c) and ( nr.c): ( nr.c) I := {x I #{y x, y r I and y C I } n} ( nr.c) I := {x I #{y x, y r I and y C I } n} ALCQI is ALCI extended with qualifying number restrictions. Observation: ALCQI with general TBoxes looses finite model property: C 0 := A, T := {. = R.A ( 1R. )} C 0 is satisfiable w.r.t. T, but only in infinite models 13

14 A tableau algorithm for ALCQI with general TBoxes Obvious: 2 new rules for tableau algorithm : : If ( nr.c) L(x), x is not blocked, and x has less than n R-neighbours y i with C L(y i ) Then create n new R-successor y 1,..., y n of x with L(y i ) := {C} for each 1 i n If ( nr.c) L(x), x is not indirectly blocked, x has n + 1 R-neighbours y 0,..., y n with C L(y i ), and there are i, j with y j is not an ancestor of y i Then L(y i ) L(y i ) L(y j ), make y j s successors to successors of y i, remove y j from the tree 14

15 A tableau algorithm for ALCQI with general TBoxes. Use explicit inequality = to prevent yoyo effect : : If ( nr.c) L(x), x is not blocked, and x has less than n R-neighbours y i with C L(y i ) Then create n new R-successor y 1,..., y n of x with. L(y i ) := {C} for each 1 i n and y i = y j for all i j If ( nr.c) L(x), x is not indirectly blocked, x has n + 1 R-neighbours y 0,..., y n with C L(y i ), and. there are i, j with not y i = y j and y j is not an ancestor of y i Then L(y i ) L(y i ) L(y j ), make y j s successors to successors of y i,.. add y i = z for each z with y j = z, remove y j from the tree 15

16 A tableau algorithm for ALCQI with general TBoxes Extend definition of a clash to NRs: Additionally, x contains a clash if ( nr.c) L(x) and. x has more than n R-neighbours y 0,..., y n with y i = y j for all i j. Does this suffice? No: ( 1R.A) ( 1R. A) ( 3R.B) is unsatisfiable, but the algorithm would answer satisfiable Reason: if ( nr.c) L(x) and x has an R-neighbour y, we need to know whether x is a C or x is a C. 16

17 A tableau algorithm for ALCQI with general TBoxes Solution: 1 use C as an abbreviation for the NNF of C, 2 extend the node labels to 2 add a third new rule: choice : cl(c 0, T ) := sub(c 0, T ) { C C sub(c 0, T )} If ( nr.c) L(x), x is not indirectly blocked, x has an R-neighbour y with {C, C} L(y) = Then L(y) L(y) {D} for some D {C, C} Does this suffice? No... 17

18 A tableau algorithm for ALCQI with general TBoxes Example: C 0 := A ( S.D) D := A ( 1S. ) ( S. A) T := {. = S.D} x L(x) = {C 0, A, ( S.D)} s y L(y) = {D, A, ( 1 S ), ( S. A), ( S.D)} s s z L(z) = {D, A, ( 1 S ), ( S. A), ( S.D)} z would block y but we cannot construct a model from this! 18

19 A tableau algorithm for ALCQI with general TBoxes Solution: y is directly blocked if there are ancestors x, x, and y of y with x is predecessor of y, x is predecessor of y, E( x, y ) = E( x, y ), L(x) = L(x ), and L(y) = L(y ). Lemma: Let T be a general ALCQI-Tbox and C 0 an ALCQI-concept. Then the 1. algorithm terminates when applied to T and C 0, 2. the rules can be applied such that they generate a clash-free and complete completion tree iff C 0 is satisfiable w.r.t. T. Proof: 1. (Termination): tree is no longer built monotonically,. but = prevents yoyo effect 19

20 A tableau algorithm for ALCQI with general TBoxes 2. (Soundness): a complete, clash-free T can be unravelled into (infinite tree) model: elements of the model are paths starting from the root instead of going to a blocked node, go to its blocking node p A I if A L(Tail(p)) roughly speaking, set (p, p y) R I if y is an R-successor of Tail(p) (and similar for inverse roles), taking care of blocked nodes danger: suppose two successors y, y of x are blocked by the same node z: Standard unravelling yields one path [... xz] for both nodes [... x] might not have enough R-successors for some ( nr.c) L(x) Solution: annotate points in the path with blocked nodes: [... x x y ] [... x x z z ] y 20

21 A tableau algorithm for ALCQI with general TBoxes 2. (Completeness): Identical to the proof for ALCI, but for stricter invariance condition on π: L(x) {C π(x) C I } if x, y E(R), then π(x), π(y) R I. if x = y, then π(x) π(y) ( ) 21

22 Tableau algorithms for ABox consistency Tableau algorithms for ABox consistency either use pre-completion (for DLs without inverse roles): reduce ABox-consistency to (several) satisfiability tests by completing the ABox using all but generating rules (i.e.,,, ) Example: { a, b : R, a : (A R. S.( A B)) b, a : S, b : (A S. B)} or work on completion forests (for DLs with inverse roles), where root nodes can be related Example: { a, b : R, a : (A R. S.( A B)) b, a : S, b : (A S.( S. R. A))} 22

23 A tableau algorithm for SHIQ The DL SHIQ extends ALCQI with transitive roles and role hierarchies: transitive roles: certain role names must be interpreted as transitive relations e.g., ancestor, has-part, part-of, etc. role hierarchy: set of implications R S, which require R I S I e.g., daughter child, has-component has-part roles in NRs are simple (don t have transitive subroles) Note: If Trans(S) and R S, then S I is a transitive relation containing R I (R + ) I is the smallest transitive relation containing R I 23

24 A tableau algorithm for SHIQ Known: SAT(SHIQ) w.r.t. TBoxes is ExpTime-complete FaCT and Racer are highly optimised SHIQ-implementations The tableau algorithm for SHIQ is similar to the one of ALCQI but for transitivity and role hierarchies: relational structure of the completion tree is only skeleton (Hasse-Diagram) of the relational structure of the model to be built (transitive edges are left out) edges are labelled with sets of role names e.g., S 1, S 2 R and ( 1R. ) ( 1S 1.A) ( 1S 2.B) if S.C L(x), R L(E( x, y ), R S, and Trans(S), then R.C L(x) 24