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This talk explores the integration of variable modalities within modal logic and their applications in querying knowledge bases (KBs). In Part 1, we discuss the syntax, semantics, expressivity, and complexity of standard modal logic, followed by the introduction of variable modalities and their extended expressiveness. In Part 2, we demonstrate how to answer unary and boolean queries using modal logic. The presentation includes practical examples, complexity considerations, and findings about reflexivity and transitivity in frames, empowering attendees to leverage these concepts in knowledge representation and reasoning.
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Modal Logicwith Variable Modalities&its Applications toQuerying Knowledge Bases Evgeny Zolin The University of Manchester zolin@cs.man.ac.uk
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%
Standard Modal Logic • (Multi-)modal language: • propositional variables: p0 , p1 , … • boolean connectives: ?, ! • modal operators (“modalities”): ¤1, … , ¤m • Modal formulas: • Other connectives are definable:
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
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…
Introducing Variable Modalities • The language is extended in two ways: • Modal formulas: • The dual variable modalities are defined as:
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
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?
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”
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)
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 Cwill answer your query!
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:
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
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,
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)
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 }
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!
