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Modal logic and databases

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

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Modal logic and databases

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  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 • 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

  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 <>:PERSON relation of type <>:LOCATION

  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 Φ

  26. Instantiate with query objects

  27. λ-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?

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