Post on 01-Apr-2015
transcript
Modal Logicwith Variable Modalities
&its Applications to
Querying Knowledge Bases
Evgeny ZolinThe University of Manchester
zolin@cs.man.ac.uk
2/17
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%
3/17
Standard Modal Logic• (Multi-)modal language:
– propositional variables: p0 , p1 , …
– boolean connectives: ?, !
– modal operators (“modalities”): ¤1 , … , ¤m
• Modal formulas:
• Other connectives are definable:
4/17
Kripke Semantics
• Frame: F = hW ,R1, … ,Rmi, where Ri µ W£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,e ² iff M,e ² for any model M based on F
F ² iff F,e ² for all points e in the frame F
5/17
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…
6/17
Introducing Variable Modalities• The language is extended in two ways:
• Modal formulas:
• The dual variable modalities are defined as:
propositional variables: p0 , p1 , …
variable modalities: ¡0, ¡1, …
propositional constants: A1 ,…,An constant modalities: ¤1 ,…, ¤m
7/17
Semantics for Variable Modalities• Frame: F =hW ; V1 ,…,Vn; R1,…,Rm i, Vi µ W, Ri µ W£W
• Model: M =hF ,; S0,S1 ,…i, (pi)µ W ; 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,e ² 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
8/17
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 = hW i.
Ex.3:
Question: complexity of reasoning for the new language?
9/17
Complexity and further examplesTheorem. 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”
10/17
Part 2. Querying KBs using MLTask 1: Find all individuals a such that KB ² a:C ,
i.e. answer the query q(x) Ã x:C over a given KB.
Solution: KB ² a:C , KB [ { a::C } is unsatisfiable
Task 2: Find all individuals a such that KB ² aRa , i.e. answer the query q(x) Ã xRx over a given KB.
Solution a: KB ² aRa , KB ² a:9R.{a}
Recall that q(x) (reflexivity) is expressed by p ! ◊p
Solution b: KB ² aRa , KB ² a:(:P t 9R.P ) (P fresh)
11/17
Answering unary queriesTask 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:
¤R p ! §S (p Æ A) (where p is a variable, A a constant)
Solution: KB ² q(a) , KB ² a: :8R.P t 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!
xR
yS A
12/17
50%+25%+10%, for unary queriesDefinition. 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 ² a:C
Theorem (50%)
Theorem (25%) If then for any F and e,
Theorem (10%) If (and no ¡ in ), then for finitely branching frames:
13/17
Answering boolean queriesTask 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 ² a: :P t 9R.P (a,P are fresh)
Task 2. How to check whether KB ² Transitive(R) ?
Solution: KB ² a: :9R.P t 9R.9R.P (a,P are fresh)
Task 3. How to check whether KB ² R v S ?
Solution: KB ² a: :9R.P t 9S.P (a,P are fresh)
And so on: R1±R2 v R3±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 ◊R p ! ◊S p
14/17
50%+25%+10%, for boolean queriesDefinition. q globally corresponds to :
if for any frame F , we have:
Definition. A concept C answers a boolean query q :
q ¼ C, if for any KB, KB ² q , KB ² a:C (a – fresh)
Theorem (50%)
Theorem (25%) If then for any F,
Theorem (10%) If then for any finite frame F,
15/17
Mary Likes All CatsTask: KB ² “Mary likes all cats”
Mary (individual), Likes (role), Cat (concept)
Solution 1: KB ² Cat v 9 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.P t 8S.(:Cat t P )
16/17
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,b ² :◊> (b has no R-successors)
F,c ² ◊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 }
a
c
b
d
17/17
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!