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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

Temporal Logics

SWE 623

Duminda Wijesekera

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

Duminda Wijesekera

prepositional semantics ii
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
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
PTL Axioms and Rules II
  • Rules
    • modus ponens
    • generalization

A

G A

A

X A

Duminda Wijesekera

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