Modal logic and databases

Terms

Object terms Concept terms ↓t: object denoted by concept t in

some context Type designations: o (object) and c

(concept)

Syntax

Semantics

Valuation

Truth and modal models

Relational databases

Record: basic unit of information in rdb Can’t 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

Sample database

Axioms

The worlds and mappings

Validity: true in every world

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)

Designation

Designation is only possible when interpretation is grounded.

A query

returns: 2 and 5

Another query

returns: t

Additional relation(ship)s

now add...

Higher-order relations

relation of type <>:PERSON

relation of type <>:LOCATION

The new (relational) constraint axioms

The new instance axioms

A sample derivation

Prove:

Strategy:

Prove X

Prove ¬X is false

A shorthand derivation

The overall derivation

The strategyProve ¬X is false

Reduce query’ to disjuncts

Introduce Axiom 5

Apply the shorthand derivation

Φ

Instantiate with query objects

λ-reduce

Apply and reduce Axiom 7

Apply shorthand rule to 11

Apply disjunction rule

Instantiate

Contradiction! (lhs)

Instantiate

Contradiction!

Another example

Attributes and relations

Sample query 1

Which items have 2 cylinders?

Checking query 1 (for instance 3)

check the relevant world(s) with appropriate mappings:

and resolve each conjunct...

Sample query (2)

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

Sample query (3)

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