Linear and Branching Time Safety, Liveness, and Fairness

1. Linear and Branching TimeSafety, Liveness, and Fairness Lam Quoc Dang FAdCo T-79.5306 Reactive Systems

2. Contents • Linear and Branching Time • General Overview • Linear Time • Branching Time • Safety, Liveness, and Fairness • Safety Property • Liveness Property • Fairness Assumption

3. General Overview • An abstraction mechanism consisting of π and Ɛπ, Σ and ƐΣ, or both is used. • Properties of individual states, transitions, and their relations over time are specified.

4. Linear Time • Looking at each complete execution separately • ”K ||− φ” is defined as Ɛπ(K) ||− φ, or ƐΣ(K) ||− φ, or Ɛπ+Σ(K) ||− φ. • Property φ belongs to a system iff K ||− φ for every K ∈ CEx. • Such property is a linear-time property. • Eg: Reachability of a deadlock, 4-boundedness of a Petri net place p are linear-time property.

5. Linear Time • Checking: including a proposition is_deadlock in π, checking whether is_deadlock holds in any abstracted states in sequences of Ɛπ(CEx). • A structural transition t ∈ T is Petri-net-live iff ”t is enabled”. • Petri-net-liveness of t is not a linear-time property.

6. Branching Time • Petri-net-liveness is determined with t_enab ∈ π and all execution trees shown. • An execution tree represents all executions starting with s1 ∈ S1, and records all positions where two executions separate. • Execution tree of state space (S, T, Δ, S1) with s1 ∈ S1 is formally defined as rooted edge-labelled graph (V, E, S1).

7. Branching Time • Each node in execution tree contains only information provided by Ɛπ. • Edges of execution tree are defined in two different ways if structural transitions are abstracted with Σ or away totally. • Property whose validity is defined on Ɛ-abstracted execution trees is branching-time property. • Branching-time property is proper if it is not linear-time property, eg: Petri-net-liveness.

8. Safety Property • Safety in concurrent systems corresponds to partial correctness in sequential programs. • Linear-time safety is property of Ɛ-abstracted executions having finite counterexamples. • A safety property belongs to a system iff all its executions have such property. • No property is simultaneously a safety and liveness property.

9. Safety Property • Eg: ”the program will not terminate” is safety property, so as ”prefix” property of fifo queue and 4-boundedness of Petri net place. • Safety property can be defined in branching-time by replacing ”execution trees” for ”executions”. • Verification for safety is easier than for liveness properties (existing algorithms and not depending on fairness).

10. Liveness Property • Liveness (progress) in concurrent systems corresponds to termination in sequential programs. • Liveness is system-specific and more difficult to formulate than termination. • Linear-time liveness is property whose counterexamples are Ɛ-abstractions of ony complete executions.

11. Liveness Property • Eg: ”the program will eventually terminate” is liveness property, so as the ”length” property of fifo queues. • Counterexamples of linear-time liveness are infinite executions (finite ”prefix” + repeating finite cycle). • Counterexamples for proper branching-time properties are more complicated.

12. Fairness Assumption • Fairness is assumption often needed for ensuring liveness. • Two well-known notions: weak fairness (justice) and strong fairness (compassion). • If a structural transition t is enbled in every state from some point on (but never occurs), weak-fairness will eventually occur.

13. Fairness Assumption • Fairness is used to modelling process schedule to ensure each process gets processor time. • Fairness is used to ensure a serve serves its clients equally. • In strong fairness, if t is enabled infinitely many times, it should occur infinitely many times.

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