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Kripke Semantics of Modal Logic

W4

W1

- The “universe” seen as a collection of worlds.
- Truth defined “in each world”.
- Say U is the universe.
- I.e. each w e U is a prepositional or predicate model.

W2

W3

Duminda Wijesekera

Temporal Logic

- Special kind of modal logic to reason about time.
- There are many kinds of Temporal Logics
- Linear and Branching Time
- Future and Past times
- Discrete and Continuous time
- Operators in Temporal Logics (MacMillan’s Notation)
- O = next time F
- [] = always G
- = some times X
- = until U

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Prepositional Syntax

- Atomic Proposition letters p, q etc.
- If p, q are propositions then so are.
- MeaningLogical NotationModel Checking
- Next Time p: Op Xp
- All ways p: []p Gp
- In the future p: p Fp
- p until q: p q pUq

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Prepositional Semantics

- A collection of Kripke Worlds including the current one.
- Accessibility relation is evolution of time.

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Prepositional Semantics II

- |= Xp if some world accessible from the current satisfies p.
- |= Gp if every world accessible from the current satisfies p.
- |= Fp if some world in the future from the current satisfies p.

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PTL Axioms and Rules I

- Axioms
- G(A ->B) ->(GA -> GB)
- X(A ->B) -> (XA -> XB)
- (X A) <-> (XA)
- GA -> (A /\ XGA)
- G(A -> OA) -> (A -> []A)
- A U B -> XB
- A U B <-> B \/ (A /\ X(A U B ))

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