Modal Logic with Variable Modalities & its Applications to Querying Knowledge Bases

1. Modal Logicwith Variable Modalities&its Applications toQuerying Knowledge Bases Evgeny Zolin The University of Manchester zolin@cs.man.ac.uk

2. Talk Outline • Part 1. Logic with variable modalities • Standard modal logic • Variable modalities: • Syntax & Semantics • Expressivity & Complexity • Part 2. Querying KBs using ML • Answering unary queries • Answering boolean queries • 50% + 25% + 10%

3. Standard Modal Logic • (Multi-)modal language: • propositional variables: p0 , p1 , … • boolean connectives: ?, ! • modal operators (“modalities”): ¤1, … , ¤m • Modal formulas: • Other connectives are definable:

4. Kripke Semantics • Frame:F = hW ,R1, … ,Rmi, where Ri µW£W • Model:M = hF , i, where a valuation(pi)µW • A formula is true at a point e of a model M:M,e ² • Validity of a formula  at a point e of a frame F : F,e² iff M,e² for any model M based on F F² iff F,e ² for all points e in the frameF

5. Expressive power • Typical questions: • What property of frames does a modal formula express? • Which properties of frames are modally expressible? etc. • Typical answers: • p  ◊p! xRx(reflexivity) • ◊p  ◊◊p! xRy  yRz  xRz(transitivity) • ¤(¤p p)  ¤p!transitivity  no infinite ascending chains • Only relational first- or second-order properties…

6. Introducing Variable Modalities • The language is extended in two ways: • Modal formulas: • The dual variable modalities are defined as:

7. Semantics for Variable Modalities • Frame:F =hW ; V1 ,…,Vn; R1,…,Rm i, Vi µW, Ri µW£W • Model:M =hF ,; S0,S1 ,…i, (pi)µW ; SiµW£W • A formula is true at a point e of a model M:M,e ² • Validity of a formula  at a point e of a frame F: F,e ² iff M,e² for any model M based on F In other words:  is true at e for any interpretation of propositional variables pi and variable modalities ¡i

8. What can we express now? Ex.1: Formula ¤p!¡p. Frame for it: F = hW,Ri. Thus, “R is a universal relation” is expressible! Ex.2: Formula p!¡p. Frame for it: F = hWi. Ex.3: Question: complexity of reasoning for the new language?

9. Complexity and further examples Theorem.Satisfiability is PSPACE-complete. Just because the minimal logic K’ coincides with K. Ex.4: “Any element from A is reflexive” Ex.5: Ex.6: “All elements in A are visible from the point e”

10. Part 2. Querying KBs using ML Task 1: Find all individuals a such that KB ²a:C , i.e. answer the query q(x) Ãx:C over a given KB. Solution: KB ²a:C , KB [ { a::C } is unsatisfiable Task 2: Find all individuals a such that KB ²aRa, i.e. answer the query q(x) ÃxRx over a given KB. Solution a: KB ²aRa, KB ²a:9R.{a} Recall that q(x) (reflexivity) is expressed by p!◊p Solution b: KB ²aRa, KB ²a:(:Pt 9R.P) (P fresh)  

11. A S y x R Answering unary queries Task 3: Answer the query q(x) over a KB: q(x) Ã 9y ( xRy xSy  y:A ) This q(x) is expressed by a modal formula: ¤Rp!§S (pÆA) (where p is a variable, A a constant) Solution:KB ² q(a) , KB ² a: :8R.Pt 9S.(P u A) Idea: Given q(x), find a corresponding modal formula , and replace each pi with Pi (fresh concept names), ¤i with 8Ri and ¡i with 8Si (fresh role names). The resulting concept Cwill answer your query!

12. 50%+25%+10%, for unary queries Definition. q(x)locally corresponds to  : if for any frame F and its point e, Definition. A query q(x) is answered by a concept C: q(x) ¼ C, if for any KB and a, KB ² q(a) , KB ²a:C Theorem (50%) Theorem (25%) If then for any F and e, Theorem (10%) If (and no ¡ in ), then for finitely branching frames:

13. Answering boolean queries Task 1. How to check whether KB ² Reflexive(R) ? Solution 1: check KB[{:aRa} for unsatisfiability (a fresh), where :aRa is a shortcut for a: :9R.{a} Solution 2: KB ²a: :Pt 9R.P (a,P are fresh) Task 2. How to check whether KB ² Transitive(R) ? Solution: KB ²a: :9R.Pt 9R.9R.P (a,P are fresh) Task 3. How to check whether KB ²RvS ? Solution: KB ²a: :9R.Pt 9S.P (a,P are fresh) And so on: R1±R2vR3±R4±R5; Commute(R,S); … Recall that “global” reflexivity is expressed by p!◊p Recall that transitivity is expressed by ◊p!◊◊p Recall that role inclusion is expressed by ◊Rp!◊S p

14. 50%+25%+10%, for boolean queries Definition. qglobally corresponds to  : if for any frame F , we have: Definition. A concept Canswers a boolean query q : q¼ C, if for any KB, KB ² q, KB ²a:C (a – fresh) Theorem (50%) Theorem (25%) If then for any F, Theorem (10%) If then for any finite frame F,

15. Mary Likes All Cats Task: KB ² “Mary likes all cats” Mary (individual), Likes (role), Cat (concept) Solution 1: KB ² Cat v9 Likes—.{Mary} Need to introduce inverse roles and nominals… Solution 2: KB ² Mary:8:Likes.:Cat Need to introduce role complement (ExpTime) Recall: Solution 3: KB ² Mary: :8Likes.Pt 8S.(:Catt P)

16. d c a b Modal validity vs. entailment from a KB • Validity of a modal formula ≈ closed world assumption Example:F = hW,Ri, where W = {a,b,c,d }, R = {ha,b i, ha,c i, hc,d i }. • F,b²:◊> (b has no R-successors) • F,c²◊p! □p (R is functional at the point c) • Entailment from a KB ≈ open world assumption KB= hT, A i, TBox T is empty, Abox A = { aRb, aRc, cRd }

17. Conclusions and outlook • New modal language, more expressive, but the same complexity • Its expressive power can be used for querying KBs Questions left open: • Whether the remaining 15% holds? • In particular, any negative results? “Genuinely” cyclic queries? • Automatic correspondence: given q(x), how to build  ? • Extension to Sahlqvist & Kracht theorem, etc. Thank you!