Chapter 7: Verification using Timed Automata

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Chapter 7 Verification using Timed Automata

• Albert M. K. Cheng

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Verification Strategy

• Given
• Implementation automaton A_I
• Specification automaton A_S
• Verify
• A_I satisfies A_S by showing that the language
  accepted by A_I is a subset of the language
  accepted by A_S, that is,
• L(A_I) is subset of L(A_S)

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Example 1 Untimed automaton representing climate
control unit
S_4
S_1
cold
turn_on_ac
hot
turn_on_heater
S_0
S_5
S_2
turn_off_heater
comfort
comfort
turn_off_ac
S_6
S_3
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Untimed Process

• Pair (E,S), where E is the event set and S is the
  set of possible traces
• Example Traffic light
• event set green, yellow, red
• trace greenyellowredgreenyellow
• red

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Untimed Trace

• Linear sequence of observable events of a
  process
• p bar p_1 p_2 p_3
• infinite word over nonempty subsets of event
  set E

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Example 1t Timed version of automaton
representing climate control unit
Two clocks c_1, c_2
c_1 0
S_4
(c_1 lt 10)?
S_1
cold
turn_on_ac
hot
turn_on_heater
(c_2 lt 5)?
c_2 0
S_0
S_5
S_2
(c_1 lt 2)?
turn_off_heater
comfort
comfort
turn_off_ac
S_6
S_3
c_1 0
c_2 0
(c_2 lt 2)?
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Timed word over alphabet Sigma

• (rho bar, tau bar)
• rho bar is infinite word, tau bar is time
  sequence
• Timed language is set of timed words over Sigma
  
• Process p, e.g., (green,yellow,red,(green
  yellow red)w)

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Timed Traces

• Time sequence tau bar tau_1, tau_2, tau_3,
• Timed trace is a time trace over E (set of
  events) is a pair (rho bar, tau bar)
• Example rho_p (cold,0), (turn_on_heater,10),
  (comfort,15), (turn_off_heter,20), (cold,50),