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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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