Modal logic and databases

1
Modal logic and databases
2
Terms

• Object terms
• Concept terms
• ?t object denoted by concept t in some context
• Type designations o (object) and c (concept)

3
Syntax
4
Semantics
5
Valuation
6
Truth and modal models
7
Relational databases

• Record basic unit of information in rdb
• Cant return it directly as answer to query
• Each one is a possible world
• Accessibility (i.e. proximity of possible
  worlds) the S5 logic
• db field attributes individual concepts
• db field entries individual objects

8
Sample database
9
Axioms
10
The worlds and mappings
Validity true in every world
11
Rigidity

• t is rigid if it always designates the same
  object, no matter which world
• FWIW in linguistics, proper names are rigid
• Rigidity can be relative w/rt subsets of all
  possible worlds
• Databases functional dependencies (e.g. between
  attributes)

12
Designation

• Designation is only possible when interpretation
  is grounded.

13
A query

• returns 2 and 5

14
Another query
returns t
15
Additional relation(ship)s
now add...
16
Higher-order relations
relation of type ltgtPERSON
relation of type ltgtLOCATION
17
The new (relational) constraint axioms
18
The new instance axioms
19
A sample derivation
Prove
Strategy Prove X Prove X is false
20
A shorthand derivation
21
The overall derivation
22
The strategy
Prove X is false
23
Reduce query to disjuncts
24
Introduce Axiom 5
25
Apply the shorthand derivation
F
26
Instantiate with query objects
27
?-reduce
28
Apply and reduce Axiom 7
29
Apply shorthand rule to 11
30
Apply disjunction rule
31
Instantiate
32
Contradiction! (lhs)
33
Instantiate
34
Contradiction!
35
Another example
36
Attributes and relations
37
Sample query 1
Which items have 2 cylinders?
38
Checking query 1 (for instance 3)
check the relevant world(s) with appropriate
mappings
and resolve each conjunct...
39
Sample query (2)

• What choices does a customer have when purchasing
  a 4-cylinder car?

40
Sample query (3)

• What features can a customer choose that are
  available for more than one product?