Introduction to Temporal Logic Troy Reilly Justin Miller

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