Temporal Logics

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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

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

Duminda Wijesekera

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

Duminda Wijesekera

Prepositional Semantics
• A collection of Kripke Worlds including the current one.
• Accessibility relation is evolution of time.

Duminda Wijesekera

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.

Duminda Wijesekera

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 ))

Duminda Wijesekera

PTL Axioms and Rules II
• Rules
• modus ponens
• generalization

A

G A

A

X A

Duminda Wijesekera