Agenda coming lectures ... Part I: Linear time temporal logic (LTL) - - PowerPoint PPT Presentation

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Agenda coming lectures ... Part I: Linear time temporal logic (LTL) - - PowerPoint PPT Presentation

Linear and Branching Temporal Logics 1 Frits Vaandrager Institute for Computing and Information Sciences Radboud University Nijmegen fvaan@cs.ru.nl June 25, 2015 1 Based on slides Julien Schmaltz F. Vaandrager Temporal Logics Agenda coming


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Linear and Branching Temporal Logics1

Frits Vaandrager

Institute for Computing and Information Sciences Radboud University Nijmegen fvaan@cs.ru.nl

June 25, 2015

1Based on slides Julien Schmaltz

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Temporal Logics

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Agenda coming lectures ...

Part I: Linear time temporal logic (LTL) Part II: Model checking LTL Part III: Branching time temporal logic (CTL) Part IV: Expressiveness of CTL vs LTL Part V: Model checking CTL Part VI: Binary decision diagrams and symbolic model checking Part VII: Partial order reduction

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Agenda for today

Course intro Linear time temporal logic

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Principles Syntax Semantics

Part I Linear Time Logic

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Principles Syntax Semantics

1

Principles

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Syntax Syntax Derived operators

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Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

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Principles Syntax Semantics

Principles: next time or until ...

Temporal logic = logic about time Abstract notion of (discrete) time = sequence of events Two principal operators

next A: at the next ”time” A holds A until B: A holds until B holds

Application to software/hardware specification

At the next clock cycle, the request signal must be high The request signal must be high until the acknowledge is high Eventually the request signal must become low again The arbiter always grants at most one request The elevator should never travel when the doors are open

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Principles Syntax Semantics Syntax Derived operators

Syntax

modal logic over infinite sequences [Pnueli 1977] Propositional logic

Atomic propositions: a ∈ AP Boolean connectives: ¬a and ϕ ∧ ψ

Temporal operators

”Next” noted X ϕ or ϕ ”Until” noted ϕ U ψ or ϕ ∪ ψ

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Principles Syntax Semantics Syntax Derived operators

Derived operators

ϕ ∨ ψ ≡ ¬(¬ϕ ∧ ¬ψ) ϕ ⇒ ψ ≡ ¬ϕ ∨ ψ ϕ ⇔ ψ ≡ (ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ) True (or ⊤) ≡ ϕ ∨ ¬ϕ False (or ⊥) ≡ ¬⊤ Fϕ (also noted ♦ϕ) ≡ ⊤ U ϕ ”eventually ϕ” Gϕ (also noted ϕ) ≡ ¬F¬ϕ ”globally ϕ”

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Intuitive semantics

... ... ... ... ... a a a a Xa a U b b a ∧ ¬b a ∧ ¬b Fa ¬a ¬a a Ga a a a

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Example: traffic lights

Whenever the light is red, it cannot become green immediately G(red ⇒ ¬Xgreen) The traffic light eventually becomes green Fgreen Once red, the light eventually becomes green G(red ⇒ Fgreen) After being red, the light goes yellow and then eventually becomes green G(red ⇒ X(redU(yellow ∧ X(yellowUgreen))))

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Classification of LTL Properties

Reachability

negated reachability: F¬ψ conditional reachability: ϕUψ reachability from any state: not expressible

Safety

simple safety: G¬ψ conditional safety (weak until): (ϕUψ) ∨ Gϕ

Liveness: G(ϕ ⇒ Fψ) and others Fairness: GFψ and others

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Semantics over words

A word σ is an infinite sequence of sets of atomic propositions. LTL property φ defines set of words for which the property is true. Words(ϕ) = {σ ∈ (2AP)ω | σ | = ϕ} σ | = a iff a ∈ A0 (or A0 | = a) σ | = ϕ ∧ ψ iff σ | = ϕ and σ | = ψ σ | = ¬ϕ iff σ | = ϕ σ | = Xϕ iff σ[1..] = A1A2A3... | = ϕ σ | = ϕUψ iff ∃j ≥ 0 : σ[j..] | = ψ and σ[i..] | = ϕ, 0 ≤ i < j for σ = A0A1A2..., σ[i..] = AiAi+1Ai+2... is suffix of σ from index i

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More semantics ...

σ | = Fψ iff

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More semantics ...

σ | = Fψ iff ∃j ≥ 0 : σ[j..] | = ψ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More semantics ...

σ | = Fψ iff ∃j ≥ 0 : σ[j..] | = ψ σ | = Gψ iff

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More semantics ...

σ | = Fψ iff ∃j ≥ 0 : σ[j..] | = ψ σ | = Gψ iff ∀j ≥ 0 : σ[j..] | = ψ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More semantics ...

σ | = Fψ iff ∃j ≥ 0 : σ[j..] | = ψ σ | = Gψ iff ∀j ≥ 0 : σ[j..] | = ψ σ | = GFψ iff

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More semantics ...

σ | = Fψ iff ∃j ≥ 0 : σ[j..] | = ψ σ | = Gψ iff ∀j ≥ 0 : σ[j..] | = ψ σ | = GFψ iff ∀j ≥ 0, ∃i ≥ j : σ[i..] | = ψ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More semantics ...

σ | = Fψ iff ∃j ≥ 0 : σ[j..] | = ψ σ | = Gψ iff ∀j ≥ 0 : σ[j..] | = ψ σ | = GFψ iff ∀j ≥ 0, ∃i ≥ j : σ[i..] | = ψ σ | = FGψ iff

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More semantics ...

σ | = Fψ iff ∃j ≥ 0 : σ[j..] | = ψ σ | = Gψ iff ∀j ≥ 0 : σ[j..] | = ψ σ | = GFψ iff ∀j ≥ 0, ∃i ≥ j : σ[i..] | = ψ σ | = FGψ iff ∃j ≥ 0, ∀i ≥ j : σ[i..] | = ψ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Duality

From the semantics, we have ¬F¬ϕ = Gϕ. Proof. σ | = ¬F¬ϕ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Duality

From the semantics, we have ¬F¬ϕ = Gϕ. Proof. σ | = ¬F¬ϕ σ | = ¬∃j ≥ 0 : σ[j..] | = ¬ϕ (Def. of F)

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Duality

From the semantics, we have ¬F¬ϕ = Gϕ. Proof. σ | = ¬F¬ϕ σ | = ¬∃j ≥ 0 : σ[j..] | = ¬ϕ (Def. of F) σ | = ∀j ≥ 0 : σ[j..] | = ϕ (Def. of ¬)

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Duality

From the semantics, we have ¬F¬ϕ = Gϕ. Proof. σ | = ¬F¬ϕ σ | = ¬∃j ≥ 0 : σ[j..] | = ¬ϕ (Def. of F) σ | = ∀j ≥ 0 : σ[j..] | = ϕ (Def. of ¬) σ | = Gϕ (Def. of G)

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Semantics over paths, states, and transition systems

Let TS = (S, Σ, T, I, AP, L) be a transition system and let ϕ be an LTL formula over AP. An infinite path π of TS satisfies ϕ iff the trace of π satisfies ϕ: π | = ϕ iff trace(π) | = ϕ A state s ∈ S satisfies ϕ iff all paths from s satisfy ϕ: s | = ϕ iff ∀π ∈ Paths(s) : π | = ϕ A transition system satisfies ϕ iff ϕ holds from all initial states: TS | = ϕ iff Traces(TS) ⊆ Words(ϕ) iff ∀s0 ∈ I : s0 | = ϕ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Example

{a, b} {a} {a, b}

TS | = Ga TS | = X(a ∧ b) TS | = G(¬b ⇒ G(a ∧ ¬b)) TS | = bU(a ∧ ¬b)

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Semantics of negation

For paths, it holds π | = ϕ iff π | = ¬ϕ since: Words(¬ϕ) = (2AP)ω \ Words(ϕ) But: TS | = ϕ and TS | = ¬ϕ are not equivalent in general We have: TS | = ¬ϕ implies TS | = ϕ. TS neither satisfies ϕ or ¬ϕ if there are paths π1 and π2 such that π1 | = ϕ and π2 | = ¬ϕ.

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Example

A transition system for which TS | = Fa and TS | = ¬Fa.

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

More dualities and idempotent laws

Duality ¬Gϕ ≡ F¬ϕ ¬Fϕ ≡ G¬ϕ ¬Xϕ ≡ X¬ϕ Idempotency GGϕ ≡ Gϕ FFϕ ≡ Fϕ ϕU(ϕUψ) ≡ ϕUψ (ϕUψ)Uψ ≡ ϕUψ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Absorption and distributive laws

Absorption FGFϕ ≡ GFϕ GFGϕ ≡ FGϕ Distribution X(ϕUψ) ≡ (Xϕ)U(Xψ) F(ϕ ∨ ψ) ≡ Fϕ ∨ Fψ G(ϕ ∧ ψ) ≡ Gϕ ∧ Gψ But we have: F(ϕ ∧ ψ) ≡ Fϕ ∧ Fψ G(ϕ ∨ ψ) ≡ Gϕ ∨ Gψ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Absorption Laws(1)

FGFϕ ≡ GFϕ More formally: GFϕ means ∀i ≥ 0, ∃j ≥ i : σ[j..] | = ϕ FGFϕ means ∃k ≥ 0, ∀i ≥ k, ∃j ≥ i : σ[j..] | = ϕ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Absorption Laws(2)

GFGϕ ≡ FGϕ More formally: FGϕ means ∃i ≥ 0, ∀j ≥ i : σ[j..] | = ϕ GFGϕ means ∀k ≥ 0, ∃i ≥ k, ∀j ≥ i : σ[j..] | = ϕ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Distributive Laws (1)

X(ϕUψ) ≡ (Xϕ)U(Xψ) F(ϕ ∨ ψ) ≡ Fϕ ∨ Fψ

ϕ ∨ ψ

G(ϕ ∧ ψ) ≡ Gϕ ∧ Gψ

ϕ ∧ ψ ϕ ∧ ψ ϕ ∧ ψ

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Distributive Laws (2)

F(a ∧ b) ≡ Fa ∧ Fb TS | = F(a ∧ b) and TS | = Fa ∧ Fb

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Principles Syntax Semantics Intuitive semantics Semantics over words Semantics over paths and states Laws

Distributive Laws (3)

G(a ∨ b) ≡ Ga ∨ Gb TS | = G(a ∨ b) and TS | = Ga ∨ Gb

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Temporal Logics