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Introduction to Temporal Logic Troy Reilly Justin Miller

Introduction to Temporal Logic Troy Reilly Justin Miller. Agenda. Historical background The four basic rules Temporal Frames Advanced Interpretation Advanced Rules Proofs Extensions. Historical Background. Temporal Information Tense Logic Uses. The Core Rules.

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Introduction to Temporal Logic Troy Reilly Justin Miller

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  1. Introduction to Temporal Logic Troy Reilly Justin Miller

  2. Agenda • Historical background • The four basic rules • Temporal Frames • Advanced Interpretation • Advanced Rules • Proofs • Extensions

  3. Historical Background • Temporal Information • Tense Logic • Uses

  4. The Core Rules • P(x) “It has at some time been the case that x” • F(x) “It will at some time be the case that x” • H(x) “It has always been the case that x” • G(x) “It will always be the case that x”

  5. The Core Rules In Image

  6. Temporal Frames • Set T of time entities t, coupled with an ordering relation < on T • At each t, each atomic formula is assigned a truth value

  7. Translation Examples • G(p) → F(p) • G(p → q) → (G(p) → G(q)) • F(p) → F(F(p))

  8. Extensions to Core Temporal Logic • S(q, p) • U(q, p) • O(p)

  9. Proof Example • From • ∃x(P(Killed(x, tuna))) → Dead(tuna) • P(Killed(Curiosity, Tuna)) • Prove • F(F(Dead(Tuna))

  10. Advanced Translation Example • A philosopher will be a king • ∃x(Philosopher(x)&F(King(x))) • ∃xF(Philosopher(x) & King(x)) • F(∃x(Philosopher(x) & F(King(x)))) • F(∃x(Philosopher(x) & King(x)))

  11. Temporal Logic in Software Verification • Normal Logic • pre-conditions • post-conditions • invariant assertions • Temporal Logic • Safety Assertion • Liveness Assertion

  12. References • Temporal Logic • http://plato.stanford.edu/entries/logic-temporal/ • Temporal Logic • Yde Venema • Temporal Representation and Inference • Barry Richards, Inge Bethke, Jaap van der Does, Jon Oberlander • Temporal Logic Mathematical Foundations and Computational Aspects • Dov Gabbay, Ian Hodkinson, Mark Reynolds • Diagnosing Java code: Assertions and Temporal Logic in Java programming • Eric Allen

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