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Temporal Logics

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

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  1. Temporal Logics SWE 623 Duminda Wijesekera

  2. 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

  3. 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

  4. 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

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

  6. 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

  7. 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

  8. PTL Axioms and Rules II • Rules • modus ponens • generalization A G A A X A Duminda Wijesekera

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