<latexit sha1_base64="P4jUJHo6g1yopyZBD74hiv3LdI=">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</latexit> TCTL TCTL C 3 COMP 1 5 9 3 Algorithmic Verification Timed CTL and TCTL C Dr. Liam O’Connor CSE, UNSW (and LFCS, University of Edinburgh) Term 1 2020 1
TCTL TCTL C Timed Logic Timed CTL TCTL is CTL with clock constraints (as in TA) attached to U (and derived operators). 2
TCTL TCTL C Timed Logic Timed CTL TCTL is CTL with clock constraints (as in TA) attached to U (and derived operators). Note: The next-state operator X has no meaning in dense time. 3
TCTL TCTL C Timed Logic Timed CTL TCTL is CTL with clock constraints (as in TA) attached to U (and derived operators). Note: The next-state operator X has no meaning in dense time. Syntax ϕ ::= p | ¬ ϕ | ϕ ∧ ϕ | 4
TCTL TCTL C Timed Logic Timed CTL TCTL is CTL with clock constraints (as in TA) attached to U (and derived operators). Note: The next-state operator X has no meaning in dense time. Syntax ϕ ::= p | ¬ ϕ | ϕ ∧ ϕ | E ϕ U ∼ c ϕ | A ϕ U ∼ c ϕ Where p ∈ P is an atomic proposition and ( ∼ ) ∈ { <, ≤ , = , ≥ , > } . 5
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). 6
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). Our modelling relation is defined on a configuration (state). 7
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). Our modelling relation is defined on a configuration (state). Let Exec( s ) be the set of executions from configuration s , 8
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). Our modelling relation is defined on a configuration (state). Let Exec( s ) be the set of executions from configuration s , and Dur( p | ≤ k ) be the sum of delays along the prefix ρ | ≤ k of the execution ρ . 9
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). Our modelling relation is defined on a configuration (state). Let Exec( s ) be the set of executions from configuration s , and Dur( p | ≤ k ) be the sum of delays along the prefix ρ | ≤ k of the execution ρ . s | = p ⇔ p ∈ L ( s ) 10
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). Our modelling relation is defined on a configuration (state). Let Exec( s ) be the set of executions from configuration s , and Dur( p | ≤ k ) be the sum of delays along the prefix ρ | ≤ k of the execution ρ . s | = p ⇔ p ∈ L ( s ) s | = A ϕ U ∼ k ψ ⇔ ∀ ρ ∈ Exec( s ) . ρ | = ϕ U ∼ k ψ 11
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). Our modelling relation is defined on a configuration (state). Let Exec( s ) be the set of executions from configuration s , and Dur( p | ≤ k ) be the sum of delays along the prefix ρ | ≤ k of the execution ρ . s | = p ⇔ p ∈ L ( s ) s | = A ϕ U ∼ k ψ ⇔ ∀ ρ ∈ Exec( s ) . ρ | = ϕ U ∼ k ψ s | = E ϕ U ∼ k ψ ⇔ ∃ ρ ∈ Exec( s ) . ρ | = ϕ U ∼ k ψ 12
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). Our modelling relation is defined on a configuration (state). Let Exec( s ) be the set of executions from configuration s , and Dur( p | ≤ k ) be the sum of delays along the prefix ρ | ≤ k of the execution ρ . s | = p ⇔ p ∈ L ( s ) s | = A ϕ U ∼ k ψ ⇔ ∀ ρ ∈ Exec( s ) . ρ | = ϕ U ∼ k ψ s | = E ϕ U ∼ k ψ ⇔ ∃ ρ ∈ Exec( s ) . ρ | = ϕ U ∼ k ψ 13
TCTL TCTL C TCTL Semantics Semantics are defined on a timed transition system . Timed Transition Systems A TTS is a timed automaton with a labelling function L associating sets of atomic propositions to states (analogous to Kripke structures). Our modelling relation is defined on a configuration (state). Let Exec( s ) be the set of executions from configuration s , and Dur( p | ≤ k ) be the sum of delays along the prefix ρ | ≤ k of the execution ρ . s | = p ⇔ p ∈ L ( s ) s | = A ϕ U ∼ k ψ ⇔ ∀ ρ ∈ Exec( s ) . ρ | = ϕ U ∼ k ψ s | = E ϕ U ∼ k ψ ⇔ ∃ ρ ∈ Exec( s ) . ρ | = ϕ U ∼ k ψ ρ | = ϕ U ∼ k ψ ⇔ ∃ i . Dur( ρ | ≤ i ) ∼ k ∧ ρ i | = ψ ∧ ∀ j < i . ρ j | = ϕ 14
TCTL TCTL C Derived Operators Standard U is just U ≥ 0 . 15
TCTL TCTL C Derived Operators Standard U is just U ≥ 0 . Path formulae F ∼ k and G ∼ k are similar to normal CTL: 16
TCTL TCTL C Derived Operators Standard U is just U ≥ 0 . Path formulae F ∼ k and G ∼ k are similar to normal CTL: F ∼ k ϕ ≡ True U ∼ k ϕ G ∼ k ϕ ≡ ¬ F ∼ k ¬ ϕ 17
TCTL TCTL C Derived Operators Standard U is just U ≥ 0 . Path formulae F ∼ k and G ∼ k are similar to normal CTL: F ∼ k ϕ ≡ True U ∼ k ϕ G ∼ k ϕ ≡ ¬ F ∼ k ¬ ϕ Definition A timed automaton A satisfies a formula ϕ , written A | = ϕ iff its initial configuration ( q 0 , 0) satisfies ϕ i.e. ( q 0 , 0) | = ϕ . 18
TCTL TCTL C Derived Operators Standard U is just U ≥ 0 . Path formulae F ∼ k and G ∼ k are similar to normal CTL: F ∼ k ϕ ≡ True U ∼ k ϕ G ∼ k ϕ ≡ ¬ F ∼ k ¬ ϕ Definition A timed automaton A satisfies a formula ϕ , written A | = ϕ iff its initial configuration ( q 0 , 0) satisfies ϕ i.e. ( q 0 , 0) | = ϕ . Example The alarm is activated at most 10 time units after a problem occurs. 19
TCTL TCTL C Derived Operators Standard U is just U ≥ 0 . Path formulae F ∼ k and G ∼ k are similar to normal CTL: F ∼ k ϕ ≡ True U ∼ k ϕ G ∼ k ϕ ≡ ¬ F ∼ k ¬ ϕ Definition A timed automaton A satisfies a formula ϕ , written A | = ϕ iff its initial configuration ( q 0 , 0) satisfies ϕ i.e. ( q 0 , 0) | = ϕ . Example The alarm is activated at most 10 time units after a problem occurs. AG ( problem ⇒ AF ≤ 10 alarm ) 20
TCTL TCTL C Converting to Automata Let’s try to construct a timed (B¨ uchi) automaton that accepts all timed words that satisfy this property: AG ( problem ⇒ AF ≤ 10 alarm ) How do we know where to introduce clocks? 21
TCTL TCTL C TCTL C TCTL is CTL with explicit clock constraints and reset. Syntax ϕ ::= x ∼ k | x .ϕ | p | ¬ ϕ | ϕ ∧ ϕ | E ϕ U ϕ | A ϕ U ϕ Where x ∈ X is a clock variable and ( ∼ ) ∈ { <, ≤ , = , ≥ , > } . x .ϕ is a clock reset. 22
TCTL TCTL C TCTL C TCTL is CTL with explicit clock constraints and reset. Syntax ϕ ::= x ∼ k | x .ϕ | p | ¬ ϕ | ϕ ∧ ϕ | E ϕ U ϕ | A ϕ U ϕ Where x ∈ X is a clock variable and ( ∼ ) ∈ { <, ≤ , = , ≥ , > } . x .ϕ is a clock reset. Example (Alarm) How do we express: AG ( problem ⇒ AF ≤ 10 alarm ) in TCTL C ? 23
TCTL TCTL C Expressivity Result All TCTL formulae are expressive in TCTL by introducing a fresh clock for each constrained operator: E ϕ U ∼ k ψ ≡ ( x . E ϕ U ( ψ ∧ x ∼ k )) 24
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