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# Temporal Logics - PowerPoint PPT Presentation

Temporal Logics. SWE 623. Kripke Semantics of Modal Logic. W4. W 1. 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. Temporal Logic.

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### Temporal Logics

SWE 623

Duminda Wijesekera

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

• 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

Duminda Wijesekera

• 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

Duminda Wijesekera

• A collection of Kripke Worlds including the current one.

• Accessibility relation is evolution of time.

Duminda Wijesekera

• |= 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.

Duminda Wijesekera

• 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 ))

Duminda Wijesekera

• Rules

• modus ponens

• generalization

A

G A

A

X A

Duminda Wijesekera