Temporal logics

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Temporal logics
Hybrid Systems PhD School Aalborg University
January 2007

• Anders P. Ravn
• Department of Computer Science,
• Aalborg University, Denmark

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Symbolic Analysis

• Q - states
• Q0 initial states,
• A - labels,
• ? - transition relation, A ? Q?Q

a
Theory T p1, pn , p is a predicate,
e.g. pred(X ? V) Meaning of p p ? Q
q1 ? q2 iff p(q1) r(q2) for all p, r ? T
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A Logic

• Syntax p ? L
• Model Theory (semantics) M p
• - meaning, mathematical objects p M M
  p
• Proof Theory (axioms, deduction rules)
• - - p (axiom)
• - p1 ? p2 (rewriting rule)
• - If -p1,,-pn then - pc
• A deduction p1,,pn - pc

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Properties of a Logic

• Sound
• If p is deductible then M p for any M
• - p implies p
• Complete
• If p holds for any model M then - p
• p implies - p
• Decidable
• If there is an algorithm to check - p

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CTL - Syntax

• State formula
• Path formula

O.Kupferman, M. Y. Vardi, and P. Wolper An
Automata-Theoretic Approach to Branching-Time
Model Checking Journal of the ACM (JACM) , 47, pp
312 360, 2000.
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CTL Model Theory 1

• w is a configuration (element of a trace (path)
  ?)
• L(w) is the set of Atomic Propositions (AP) that
  are true in w.
• State formulas

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CTL Model Theory 2

• w is a configuration (element of a trace (path)
  ?)
• ?n is the trace with the n first elements
  removed.
• Path formulas
• ?1 Û ?2 as long as ?1 is false ?2 must be
  true
• ( ) ?

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LTL Syntax

• Remove A and E from CTL

LTL
LTL
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CTL - Henzinger
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Mu-calculus
// fixpoint computation
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Summary

• Logics allow us to express properties
• Two ways of using them
• - Deduction proof theory
• - Specification and Model checking
• What is model checking?