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Clock equivalence and regions

Clock equivalence and regions. Andres Lips. Clock equivalence (1). Definition: Satisfiability for a timed automaton A   if and only if M(A), (s 0 ,w 0 )   where w 0 (y) = 0 for all formula clocks y. M(A) is basis for model checking State space L x V(C) is infinite.

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Clock equivalence and regions

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  1. Clock equivalence and regions Andres Lips

  2. Clock equivalence (1) • Definition: Satisfiability for a timed automaton A  if and only if M(A), (s0,w0)  where w0(y) = 0 for all formula clocks y. • M(A) is basis for model checking • State space L x V(C) is infinite.

  3. Clock equivalence (2) • Introducing appropriate equivalence relation,  • Correctness • Finitness • M(A), ((l, v), w)  if and only if M(A), ((l, v’ ), w’ ) for vw  v’ w’ .

  4. Clock equivalence (3) • Paths starting at states which • agree on integer parts of all clock values • agree on ordering of the fractional parts of all clocks • are very similar. • Exceeding maximal constant -> value does not matter.

  5. Clock equivalence (4) • Finite state model -> region automaton • Equivalence classes -> regions • Model checking a timed automaton against TCTL-formula amounts to model checking its regions automaton against CTL-formula.

  6. Clock equivalence (5) • Basic recipe for model checking • Determine regions • Construct the region automaton R(A) • Apply CTL-model checking • A  if and only if [s0,w0]  SatR()

  7. Clock equivalence (6) • Definition: Clock equivalence. v  v’ if and only if • v(x)=v’(x) or v(x)>cx and v’(x)>cx • frac(v(x))frac(v(y)) iff frac(v’(x))frac(v’(y)) where v(x)cx and v(y)cy • frac(v(x))=0 iff frac(v’(x))=0 where v(x)cx • Example begins...

  8. Region automata (1) • Definition: Region • A region r is a pair (l, [v]) with location lL and valuation vV(C) • Abbreviations: • [s]=(l,[v]) • [s,w]=(l,[v  w])

  9. Region automata (2) • Let s,s’S such that s,w  s’,w’For any TCTL-formula , we have:M(A), (s,w)  iff M(A), (s’,w’)  • Using regions as states we construct finite-state automaton • Two types of transitions • due to the passage of time (solid arrows) • transitions of the timed automaton (dotted arrows) • Example continues...

  10. Region automata (3) • Definittion: Delay-successor region • Let r,r’ be two distinct regions. r’ is the delay successor of r, if there exists a dR+such that for each r=[s] we have and r’=[s+d] and for all 0d’<d:[s+d’]rr’

  11. Region automata (4) • Definition: Unbounded region • Region r is an unbounded region if for all clock valuations v such that r=[v] we have v(x)>cx for all xC.

  12. Region automata (5) • Region automaton • r=S/={[s]|sS} • r0=[s0] • rr’ iff • r is an unbounded region and r=r’, or • rr’ and r’=delsucc(r).

  13. Thank you andres.lips@mail.ee

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