Event-Clock%20%20Visibly%20Pushdown%20Automata

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Event-Clock Visibly Pushdown Automata

• Mizuhito Ogawa (JAIST)
• with Nguyen Van Tang
• SOFSEM 2009.1.27

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Model checking as an inclusion problem

• Paths generated by a model M is those that
  allowed by a specification S
• L(M) ? L(S) ? L(M) n L(S)C f
• Possible combinations
• M , S Finite Automata
• M Pushdown Automaton, S Finite Automaton
• M , S Pushdown Automata
• Possible extensions
• Timed constraints ?
• S beyond finite automata ?

3
Timed automata (Alur, et.al. 94)
press
next
Off
On
Menu
press
press
press

• Press quickly twice, it will enter to menu.
• Add time constraints e.g., quickly less-than
  1
• It sleeps (Off) when left more-than 5.
• Remark Time constraints contains integers only.

Accepts (press,2) (press, 2.5) (next,3)
(next,4.4) (press,8)
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Decidable properties of Timed Automata

• Boolean operations
• Decidable Union, intersection, emptiness
• Undecidable Complement, universality (2 clocks)
• Inclusion L(M) ? L(S)
• S has ?1 clock decidable (Ouaknine, et.al. 04)
• S has gt1 clocks undecidable (Alur, et.al. 94)

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Event-clock automata (Alur, et.al. 94)

• Def. The event-clock for a?S is a pair of clocks
  xa, ya
• xa event-recording ? record time since last a
• ya event-predicting ? record time until next a

a
b
e.g.,
b
b
a
a
q0
q1
q2
q0
q1
q2
yblt1
xa1
L1 (a,t1)(b,t2)(b,tn) tnt11
L2(a,t1)(a,tn-1)(b,tn) tn t1lt1

• e.g., Spec. like ack must come in 1 can be
  described.

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Properties of ECA

• Boolean operations
• Decidable all boolean operations.
• Determinizable (subset construction works)
• Language class relation
• ECA ? TA (An ECA can be encoded as a TA.)
• The class of ECA is incomparable to the class of
  deterministic TA.

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Encoding ECA to TA (Alur, et.al. 94)

• Event-recording clocks CR xa a?S
• Reset xa?CR when a is read.
• Event-predicting clocks CP ya a?S
• Let FP be the set of all event-predicting
  constraints.
• The set Q of states enlarged to QFP.
• Add fresh clocks z(yac) for each yac ?FP.

ya
c
0
yac made
a read
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Interval alphabet (DSouza 03)

• Def. Interval alphabet ? S IntvCS where
• CS xa, ya a ?S clocks
• Intv ri,ri, (ri,ri1), (rn,8) 0 ? r1 lt
  lt rn

All integers appearing in event-clock constraints

• Notation. Let ?(ai,ti) be a vector of clock
  values at ti
• uw((a1,t1)(an,tn)) (a1,I1)(an,In) with
  ?(ai,ti) ? Ii
• tw((a1,I1)(an,In)) (a1,t1)(an,tn)
  ?(ai,ti) ? Ii
• Lemma. If ?(ti) depends only on an input timed
  word
• For v??, tw(v) ?f implies uw(tw(v)) v
• For a timed word w, w?tw(uw(w))

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Untimed Translation
Translate event-clock constraints to interval
alphabet
e.g., Intv 0,0, (0,5), 5,5, (5,10),
10,10, (10,8) C xa, ya
(a,(0,0,))
(a,((0,5),))
q0
q1
q1
a, xalt10
(a,(5,5,))
q0
(a,((5,10),))
q2
b, ya gt5
(b,(,(5,10)))
q0
q2
(b,(,10,10))
M
(b,(,(10,8)))
ut(M)
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Timed Translations
Translate Interval alphabet to event-clock
constraints
a, (5ltxa?xalt10)?
(a,(5,10))
q0
q2
q0
q2
ec(ut(M))
ut(M)
Lemma. L(ec(ut(M))) L(M) for an ECA M.
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Model checking as an inclusion problem (again)

• Paths generated by a model M is those that
  allowed by a specification S
• L(M) ? L(S) ? L(M) n L(S)C f
• Possible combinations
• M , S Finite Automata
• M Pushdown Automaton, S Finite Automaton
• M , S Pushdown Automata
• Possible extensions
• Timed constraints ? ? Event-clock constraints
• S beyond finite automata ?

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Visibly Pushdown Automata (Alur, et.al. 04)
X
q
Y
ac / X
ac?Sc (call)
Z
Classification is universal (visibility)
q
q
Y
Y
ai
Z
ai?Si (local)
Z
ar / Y
q
Z
ar?Sr (return)

• Visibility implies height-deterministic and
  synchronous. (Only an input word decides the
  stack height.)
• ? Product construction (intersection) works!

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Visibly pushdown languages (VPL)

• VPL examples
• an bn (with a?Sc, b?Sr )
• Dyck language (well-balanced parantheses)
• (with left/right parantheses as in Sc / Sr )
• e.g. ( .. .. (..) .. .. ) OK, ( ..
  .. .. .. .. ) no.
• VPL is a proper subclass of DPDA
• an b an is not a VPL.
• words with equal number of a and b is not a VPL,
  e.g., abab, abba, baab,

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Properties of VPA and TVPA (Timed VPA)

• Boolean operations (of VPA)
• Decidable all boolean operations
• Determinizable
• Boolean operations (of TVPA)
• Decidable union, intersection, emptiness
• Undecidable Complement, universality (1 clock)
• Inclusion L(M) ? L(S)
• M,S (untimed) VPA decidable
• M TVPA, S TVPA undecidable (Emmi, et.al. 06)

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Event-Clock Visibly Pushdown Automata
Def. ECVPA VPA event clock constraints
Th 1. The inclusion problem for ECVPAs is
decidable.

• Proof (idea) L(M)?L(S) ? L(M)n L(ec(ut(S)c)) f
• S ECVPA (untimed translation)
• ut(S) VPA (complement)
• ut(S)c VPA (timed translation)
• ec(ut(S)c) ECVPA with L(S)c L(ec(ut(S)c))

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Inclusion between TVPA and ECVPA
Th 2. For a TVPA M and an ECVPA S, the inclusion
problem L(M) ? L(S) is decidable.

• Proof (idea) L(M) ? L(S) ? L(M) n L(S) f
• S ECVPA (untimed translation)
• ut(S) VPA (complement)
• ut(S)c VPA (timed translation)
• ec(ut(S)c) ECVPA (encoding EC-constraints)
• S TVPA with L(S)c L(S)

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Conclusion

• We showed that L(M) ? L(S)
• M, S ECVPA decidable
• M TVPA, S ECVPA decidable
• Compare L(M) ? L(S) when S has 1 clock
• M, S TA decidable (Oukline,
  et.al.04)
• M, S TVPA undecidable (Emmi, et.al. 06)
• M, S Buchi TA undecidable (Abdulla, et.al.
  05)
• Simple untimed / timed translations avoid complex
  subset construction argument of VPA.
• Buchi extensions of ECVPA are straight forward.