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 strategy Prove ¬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?