models, languages, dynamics Eugene Asarin PIMS/EQINOCS Workshop on Automata Theory and Symbolic Dynamics LIAFA - University Paris Diderot and CNRS

Transcription

1 models, s, LIAFA - University Paris Diderot and CNRS PIMS/EQINOCS Workshop on Automata Theory and Symbolic Dynamics

2 Context A model for verification of real-time systems Invented by Alur and Dill in early 1990s Precursors: time Petri nets (Bethomieu) Now: an efficient model for verification, supported by tools (Uppaal) A popular researh topic ( 8000 citation for papers by Alur and Dill) modeling and verification decidability and algorithmics and very recent: Inspired by TA: hybrid, data, on nominal sets

3 Outline

4 Before we begin: timed words and s A word: u = abbabb represents a sequence of events in some Σ.

5 Before we begin: timed words and s A word: u = abbabb represents a sequence of events in some Σ. A timed word: w = 0.8a2.66b1.5b0a b b represents a sequence of events and delays.

6 Before we begin: timed words and s A word: u = abbabb represents a sequence of events in some Σ. A timed word: w = 0.8a2.66b1.5b0a b b represents a sequence of events and delays. It lives in a timed monoid Σ R +,

7 Before we begin: timed words and s A word: u = abbabb represents a sequence of events in some Σ. A timed word: w = 0.8a2.66b1.5b0a b b represents a sequence of events and delays. It lives in a timed monoid Σ R +,but forget about it For us it sits in (R + Σ) (words on an infinite alphabet), that is w = (0.8,a),(2.66,b),(1.5,b),(0,a),( ,b),( ,b).

8 Before we begin: timed words and s A word: u = abbabb represents a sequence of events in some Σ. A timed word: w = 0.8a2.66b1.5b0a b b represents a sequence of events and delays. It lives in a timed monoid Σ R +,but forget about it For us it sits in (R + Σ) (words on an infinite alphabet), that is w = (0.8,a),(2.66,b),(1.5,b),(0,a),( ,b),( ,b). Geometrically w is a point in several copies of R n : w = (0.8,2.66,1.5,0, , ) R 6 abbabb

9 Before we begin: timed words and s A word: u = abbabb represents a sequence of events in some Σ. A timed word: w = 0.8a2.66b1.5b0a b b represents a sequence of events and delays. It lives in a timed monoid Σ R +,but forget about it For us it sits in (R + Σ) (words on an infinite alphabet), that is w = (0.8,a),(2.66,b),(1.5,b),(0,a),( ,b),( ,b). Geometrically w is a point in several copies of R n : w = (0.8,2.66,1.5,0, , ) R 6 abbabb A timed is a set of timed words - examples below.

10 What are TA Recipe: how to make a timed automaton take a finite automaton put it into continuous time add some variables x 1,...,x n, called clocks. make all them run: ẋ i = 1 everywhere. add guards to some transitions (e.g. x 3 < 7) add resets to some transitions (e.g. x 2 := 0) serve and enjoy!

11 An example of a timed automaton automaton (we forget to write ẋ = 1): a,x [1;2]? q 1 q 2 b,x := 0

12 An example of a timed automaton automaton (we forget to write ẋ = 1): Its run a,x [1;2]? q 1 q 2 b,x := 0 (q 1,0) 1.83 (q 1,1.83) a (q 2,1.83) 4.1 (q 2,5.93) b (q 1,0) 1 (q 1,1)

13 An example of a timed automaton automaton (we forget to write ẋ = 1): Its run a,x [1;2]? q 1 q 2 b,x := 0 (q 1,0) 1.83 (q 1,1.83) a (q 2,1.83) 4.1 (q 2,5.93) b (q 1,0) 1 (q 1,1) Its trace 1.83a4.1b1a a timed word

14 An example of a timed automaton automaton (we forget to write ẋ = 1): Its run a,x [1;2]? q 1 q 2 b,x := 0 (q 1,0) 1.83 (q 1,1.83) a (q 2,1.83) 4.1 (q 2,5.93) b (q 1,0) 1 (q 1,1) Its trace 1.83a4.1b1a a timed word Its timed : set of all the traces starting in q 1, ending in q 2 : {t 1 as 1 bt 2 as 2 b...t n a i.t i [1;2]}

15 An example of a timed automaton automaton (we forget to write ẋ = 1): Its run a,x [1;2]? q 1 q 2 b,x := 0 (q 1,0) 1.83 (q 1,1.83) a (q 2,1.83) 4.1 (q 2,5.93) b (q 1,0) 1 (q 1,1) Its trace 1.83a4.1b1a a timed word Its timed : set of all the traces starting in q 1, ending in q 2 : {t 1 as 1 bt 2 as 2 b...t n a i.t i [1;2]} Observation Clock value of x: time since the last reset of x.

16 Some simple exercises Draw timed for specifications: Request a arrives every 5 minutes.

17 Some simple exercises Draw timed for specifications: Request a arrives every 5 minutes. Request a arrives every 5 to 7 minutes.

18 Some simple exercises Draw timed for specifications: Request a arrives every 5 minutes. Request a arrives every 5 to 7 minutes. a arrives every 5 to 7 minutes; and b arrives every 3 to 10 minutes.

19 Some simple exercises Draw timed for specifications: Request a arrives every 5 minutes. Request a arrives every 5 to 7 minutes. a arrives every 5 to 7 minutes; and b arrives every 3 to 10 minutes. Request a is serviced within 2 minutes by c or rejected within 1 minute by r.

20 Some simple exercises Draw timed for specifications: Request a arrives every 5 minutes. Request a arrives every 5 to 7 minutes. a arrives every 5 to 7 minutes; and b arrives every 3 to 10 minutes. Request a is serviced within 2 minutes by c or rejected within 1 minute by r. The same, but a arrives every 5 to 7 minutes.

21 Modeling exercise 2 Scheduling Schedule two jobs on one CPU and one printer with a total execution time up to 16 minutes. Job 1 : Compute (10 min); Print (5 min) Job 2 : Download (3 min); Compute (1 min); Print (2 min) Try it : 1 without preemption; 2 with preemptible computing.

22 Main theorem Theorem (Alur, Dill) Reachability is decidable for timed.

23 Main theorem Theorem (Alur, Dill) Reachability is decidable for timed. Classical formulation Empty problem is decidable for TA

24 Main theorem Theorem (Alur, Dill) Reachability is decidable for timed. Classical formulation Empty problem is decidable for TA Both are the same Non-empty Reach(Init,Fin)

25 Proof idea Split the state space Q R n into regions s.t. all the states in one region have the same behavior; there are finitely many regions;

26 Proof idea Split the state space Q R n into regions s.t. all the states in one region have the same behavior; there are finitely many regions; Build a region automaton (its states are regions)

27 Proof idea Split the state space Q R n into regions s.t. all the states in one region have the same behavior; there are finitely many regions; Build a finite region automaton (its states are regions)

28 Proof idea Split the state space Q R n into regions s.t. all the states in one region have the same behavior; there are finitely many regions; Build a finite region automaton (its states are regions) Test reachability in this region automaton.

29 Proof idea Split the state space Q R n into regions s.t. all the states in one region have the same behavior; there are finitely many regions; Build a finite region automaton (its states are regions) Test reachability in this region automaton. Two difficulties What does it mean: the same behavior? How to invent it?

30 Proof idea Split the state space Q R n into regions s.t. all the states in one region have the same behavior; there are finitely many regions; Build a finite region automaton (its states are regions) Test reachability in this region automaton. Two difficulties What does it mean: the same behavior? Bisimulation. How to invent it? A&D invented it using ideas of Berthomieu (Time Petri nets). In fact it is rather natural.

31 Region equivalence Definition Two states of a TA are region equivalent: (q,x) (p,y) if Same location: p = q Same integer parts of clocks: i ( x i = y i ) Same order of fractional parts of clocks i,j ({x i } < {x j } {y i } < {y j }) Look at the picture!

32 Region equivalence Definition Two states of a TA are region equivalent: (q,x) (p,y) if Same location: p = q Same integer parts of clocks: i ( x i = y i ) Same order of fractional parts of clocks i,j ({x i } < {x j } {y i } < {y j }) Look at the picture!

33 Region equivalence Definition Two states of a TA are region equivalent: (q,x) (p,y) if Same location: p = q Same integer parts of clocks: smalli ( x i = y i ) Same order of fractional parts of clocks smalli,j ({x i } < {x j } {y i } < {y j }) Or they are both big : i ((x i > M) (y i > M)) Look at the picture!

34 Region equivalence Definition Two states of a TA are region equivalent: (q,x) (p,y) if Same location: p = q Same integer parts of clocks: smalli ( x i = y i ) Same order of fractional parts of clocks smalli,j ({x i } < {x j } {y i } < {y j }) Or they are both big : i ((x i > M) (y i > M)) Look at the picture! Definition Equivalence classes of are called regions.

35 Region equivalence Definition Two states of a TA are region equivalent: (q,x) (p,y) if Same location: p = q Same integer parts of clocks: smalli ( x i = y i ) Same order of fractional parts of clocks smalli,j ({x i } < {x j } {y i } < {y j }) Or they are both big : i ((x i > M) (y i > M)) Look at the picture! Definition Equivalence classes of are called regions. Lemma (Region equivalence is a bisimulation) Equivalent states can make the same transitions, and arrive to equivalent states.

36 Decision algorithm Build a region automaton RA States are regions. There is a transition r 1 a r2 if some (all) element of r 1 can go to some element of r 2 on a. There is a transition r 1 τ r2 if some (all) element of r 1 can go to some element of r 2 on some t > 0

37 Decision algorithm Build a region automaton RA States are regions. There is a transition r 1 a r2 if some (all) element of r 1 can go to some element of r 2 on a. There is a transition r 1 τ r2 if some (all) element of r 1 can go to some element of r 2 on some t > 0 Check whether some final region in RA is reachable from initial region.

38 Outline

39 Closure properties Definition regular is a accepted by a TA

40 Closure properties Definition regular is a accepted by a TA Theorem regular s are closed under,, projection, but not complementation.

41 Closure properties Definition regular is a accepted by a TA Theorem regular s are closed under,, projection, but not complementation. Fact Determinization impossible for timed.

42 properties Definition regular (TRL) is a accepted by a TA

43 properties Definition regular (TRL) is a accepted by a TA Theorem Decidable for TRL (represented by TA): L =, w L, L M =.

44 properties Definition regular (TRL) is a accepted by a TA Theorem Decidable for TRL (represented by TA): L =, w L, L M =. Proof. Immediate from Alur&Dill s theorem.

45 properties Definition regular (TRL) is a accepted by a TA Theorem Decidable for TRL (represented by TA): L =, w L, L M =. Theorem Undecidable for TRL (represented by TA): L universal (contains all the timed words), L M, L = M.

46 properties Definition regular (TRL) is a accepted by a TA Theorem Decidable for TRL (represented by TA): L =, w L, L M =. Theorem Undecidable for TRL (represented by TA): L universal (contains all the timed words), L M, L = M. Proof. Encoding of runs of Minsky Machine as a timed s.

47 Reminder: regular expressions Definition Regular expressions: E ::= 0 ε a E +E E E E Theorem (Kleene) Finite and regular expression define the same class of s.

48 Reminder: regular expressions Definition Regular expressions: E ::= 0 ε a E +E E E E Theorem (Kleene) Finite and regular expression define the same class of s. Example a,b q p b r a ((a+b)a) (a+b)b

49 regular expressions A natural question How to define regular expressions for timed s?

50 regular expressions A natural question How to define regular expressions for timed s? E ::= 0 ε t a E +E E E E E I E E [a z]e

51 regular expressions A natural question How to define regular expressions for timed s? E ::= 0 ε t a E +E E E E E I E E [a z]e Semantics: t = R 0 a = {a} E 1 E 2 = E 1 E 2 E I = {σ E l(σ) I} E 1 E 2 = E 1 E 2 0 = ε = {ε} E 1 +E 2 = E 1 E 2 E = E [a z]e = [a z] E

52 A good example and a theorem a,x [1;2]? q 1 q 2 b,x := 0 {L = {t 1 as 1 bt 2 as 2 b...t n a i.t i [1;2]}

53 A good example and a theorem a,x [1;2]? q 1 q 2 b,x := 0 {L = {t 1 as 1 bt 2 as 2 b...t n a i.t i [1;2]} ( ) An expression for L : ta [1;2] tb Theorem (A., Caspi, Maler,95) Automata and regular expressions (with and [a z]) define the same class of timed s

54 A nasty example Intersection needed [ACM] a x 2 := 0 b x 1 = 1? c x 2 = 1? {t 1 at 2 bt 3 c t 1 +t 2 = 1,t 2 +t 3 = 1} = ta tbtc 1 tatb 1 tc

55 Another nasty example Renaming needed [Herrmann] a a a a a y := 0 x = 1? y = 1? [b a] ( (ta) tb(ta) 1 (ta) tb 1 (ta) ).

56 Outline

57 Preliminary considerations Aim: describe timed regular s as dynamical systems Solution: from Nicolas Basset s MSc thesis (2010) Limitations: deterministic, bounded intervals between events

58 Shifts Full timed shift Fix Σ an alphabet and M R. Let C = Σ [0;M] - a compact set (with a natural metrics). S = C Z : set of bi-infinite timed words (with a natural metrics). E.g....t 1 a 1 t 0 a 0 t 1 a 1 t 2... Shift σ : S S E.g. σ({c n }) = {c n+1 }.

59 Shifts Full timed shift Fix Σ an alphabet and M R. Let C = Σ [0;M] - a compact set (with a natural metrics). S = C Z : set of bi-infinite timed words (with a natural metrics). E.g....t 1 a 1 t 0 a 0 t 1 a 1 t 2... Shift σ : S S E.g. σ({c n }) = {c n+1 }. The simplest example: studied by Weiss and Lindenstrauss! Σ = {a};m = 1;C = [0;1] S = [0;1] Z = {t 1,t 0,t 1,...}

60 subshifts Definition A subshift X S with X closed and σ-invariant. Two standard ways to define subshifts by an open set of forbidden patterns; by a closed set of allowed patterns for some lengths. A timed automaton which is deterministic, w/o initial final states; with closed guards; corresponds to a timed subshift; we call it sofic!

61 an example p a,x [1;2]? b,x [5;10],x := 0 q exercice Compute forbidden patterns.

62 What about entropy? We try to compute it for [0;1] Z Define C(n,ǫ): size of ǫ-net in [0;1] n. Compute it: C(n,ǫ) = (1/ǫ) n.

63 What about entropy? We try to compute it for [0;1] Z Define C(n,ǫ): size of ǫ-net in [0;1] n. Compute it: C(n,ǫ) = (1/ǫ) n. Growth rate of C: h ǫ = lim n logc(n,ǫ) n = log 1 ǫ. Entropy h = lim ǫ 0 h ǫ =

64 What about entropy? We try to compute it for [0;1] Z Define C(n,ǫ): size of ǫ-net in [0;1] n. Compute it: C(n,ǫ) = (1/ǫ) n. Growth rate of C: h ǫ = lim n logc(n,ǫ) n Entropy h = lim ǫ 0 h ǫ = = log 1 ǫ.

65 What about entropy? We try to compute it for [0;1] Z Define C(n,ǫ): size of ǫ-net in [0;1] n. Compute it: C(n,ǫ) = (1/ǫ) n. Growth rate of C: h ǫ = lim n logc(n,ǫ) n Entropy h = lim ǫ 0 h ǫ = = log 1 ǫ. The same is true for almost all reasonable timed subshifts.

66 We still try! Given a timed subshift L Entropy renormalized Define L n -set of timed words of length n (btw L n R n Σ n ) Define C(n,ǫ): size of ǫ-net in L n. Compute it: C(n,ǫ) = (1/ǫ) n Vol(L n ).

67 We still try! Given a timed subshift L Entropy renormalized Define L n -set of timed words of length n (btw L n R n Σ n ) Define C(n,ǫ): size of ǫ-net in L n. Compute it: C(n,ǫ) = (1/ǫ) n Vol(L n ). Growth rate of C: h ǫ = lim n logc(n,ǫ) n = log 1 ǫ +lim n logvol(l n ). n We call the last term volumic entropy: H(L) = lim n logvol(l n ) n

68 We still try! Given a timed subshift L Entropy renormalized Define L n -set of timed words of length n (btw L n R n Σ n ) Define C(n,ǫ): size of ǫ-net in L n. Compute it: C(n,ǫ) = (1/ǫ) n Vol(L n ). Growth rate of C: h ǫ = lim n logc(n,ǫ) n = log 1 ǫ +lim n logvol(l n ). n We call the last term volumic entropy: H(L) = lim n logvol(l n ) n Information production in a timed subshift

69 Outline

70 It was in this talk : a beautiful variant of automat coming from practice Infinite-state, but many questions decidable A non-trivial of s Symbolic can be defined, many thing remain to study...in particular the flow

71 Advertisement Next talk (Aldric): how to characterize and compute the volumes and the volumic entropy. After that (Nicolas): MME for timed Merci EQINOCS, more results will follow