introduction to temporal logic troy reilly justin miller

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Introduction to Temporal Logic Troy Reilly Justin Miller

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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. - PowerPoint PPT Presentation

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Page 1: Introduction to Temporal Logic Troy Reilly Justin Miller

Introduction to Temporal LogicTroy ReillyJustin Miller

Page 2: Introduction to Temporal Logic Troy Reilly Justin Miller

Agenda

Historical background The four basic rules Temporal Frames Advanced Interpretation Advanced Rules Proofs Extensions

Page 3: Introduction to Temporal Logic Troy Reilly Justin Miller

Historical Background

Temporal Information Tense Logic Uses

Page 4: Introduction to Temporal Logic Troy Reilly Justin Miller

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”

Page 5: Introduction to Temporal Logic Troy Reilly Justin Miller

The Core Rules In Image

Page 6: Introduction to Temporal Logic Troy Reilly Justin Miller

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

Page 7: Introduction to Temporal Logic Troy Reilly Justin Miller

Translation Examples

G(p) → F(p) G(p → q) → (G(p) → G(q)) F(p) → F(F(p))

Page 8: Introduction to Temporal Logic Troy Reilly Justin Miller

Extensions to Core Temporal Logic

S(q, p) U(q, p) O(p)

Page 9: Introduction to Temporal Logic Troy Reilly Justin Miller

Proof Example

From ∃x(P(Killed(x, tuna))) → Dead(tuna) P(Killed(Curiosity, Tuna))

Prove

F(F(Dead(Tuna))

Page 10: Introduction to Temporal Logic Troy Reilly Justin Miller

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)))∃

Page 11: Introduction to Temporal Logic Troy Reilly Justin Miller

Temporal Logic in Software Verification

Normal Logic pre-conditions post-conditions invariant assertions

Temporal Logic Safety Assertion Liveness Assertion

Page 12: Introduction to Temporal Logic Troy Reilly Justin Miller

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