beamerpresentation

Distributed Real Time Systems

Rahul Wani

Indian Institute of Technology,Bombay

May 2, 2014

Rahul Wani (Indian Institute of Technology,Bombay)Distributed Real Time Systems May 2, 2014 1 / 59

Outline

1 BasicsTimed RunUntimed WordsTimed LanguageUntimed LanguageTimed Constraints and Clock InterpretationTimed Transition SystemsTimed Regular Language

2 Timed AutomataDefinitionChecking Emptiness

Restriction to integer constantsClock RegionThe Region Automaton

The Untiming ConstructionDeterministic Timed Automaton

DefinitionExample


Outline

3 Distributed Timed AutomatonDefinitionExampleSemanticsTimed Automaton with Independently Evolving ClocksConstruction of icTA from DTA

Universal SemanticsExistential SymanticsWeird Behaviour

4 Conclusion


BasicsTimed Run

Definition

A time sequence τ = τ1τ2 . . . is an infinite sequence of time values τi ∈ Rwith τi > 0, satisfying the following constraints:

(1) Monotonicity: increases strictly monotonically;i.e., τi < τi+1 for alli ≥ 1.

(2) Progress:For every t ∈ R, there is some i ≥ 1 such that τi > t.

A timed word over an alphabet Σ is a pair (σ, τ) where σ = σ1σ2 . . . is aninfinite word over Σ and σ is a time sequence.


BasicsTimed Run

Example

The sequence(a, 3.4)(a, 4)(b, 5) . . .

over Σ = {a, b} is a timed word as

3.4 < 4 < 5 < . . .

Here a, b are also called actions.Similarlly a sequence

(a, 3.4)(a, 4)(b, 5)

is timed word if we consider finite sequences.


BasicsUntimed Words

Example

Given a timed word(a, 3.4)(a, 4)(b, 5) . . .

untimed word is just a word without having timestamps i.e. untimed wordhere is aab . . . .Similarlly if we consider only non-infinite timed words, then aab is anuntimed word.


BasicsTimed Language

Definition

A timed language over Σ is a set of timed words over Σ.


Example

Consider the timed language over Σ = {a, b} where the second position isb and this b should only occur in between timestamp 1 and 2.L= {x | (a1, τ1)(a2, τ2)(a3, τ3)... ∈ x ∧ ∀i( i ≥ 1 ∧ τi < τ(i+1) ∧ ai ∈Σ) ∧ a2 = b ∧ 1 ≤ τ2 ∧ τ2 ≤ 2}L= {

(a,0)(b,1). . . ,. . . ,. . . ,. . . ,(a,1.999. . . )(b,2). . . ,(b,0)(b,1). . . ,. . . ,. . . ,. . . ,(b,1.999. . . )(b,2). . .

}


BasicsUntimed Language

Definition

For a timed language L over Σ, Untime(L) is the ω-language consisting ofσ ∈ Σω such that (σ, τ) ∈ L for some time sequence τ .

Example

For the timed language given in the above example, the untimed languageis (a + b)b(a + b)ω and the corresponding finite language is(a + b)b(a + b)∗


BasicsTimed Constraints and Clock Interpretation

Definition

For a set X of clock variables, the set Φ(x) of clock constraints δ isdefined inductively by

δ := x ≤ c | c ≤ x | ¬δ | δ1 ∧ δ2where x is a clock in X and c is constant in Q.


BasicsTimed Constraints and Clock Interpretation

Example

Let X = {x , y} is the set of clock variables x , y .The clock constraints overthese clock variables are defined as any valid combination of abovespecified atomic formulae. The atomic formulae are x ≤ c1 , x ≥ c2 ,y ≤ c3 , y ≥ c4 where c1, c2, c3, c4 are constants in Q. The timeconstraints may be any valid combination of above atomic formulae with¬ and ∧ operations. The valuation of clock is snapshot of the values ofthe clock variables at a certain point in time space. i.e. Lets say clockvaluation is [x = 3, y = 4] at say global time 4 given that clock x isreseted at global time 1.


BasicsTimed Transition Systems

Definition

A timed transition table A is a tuple 〈Σ, S , S0,C ,E 〉, where

Σ is a finite alphabet,

S is a finite set of states,

S0 ⊆ S is a set of start states,

C is a finite set of clocks, and

E ⊆ S × S × Σ× 2C × φ(C ) gives the set of transitions. An edge〈s, s ′

, a, λ, δ〉 represents a transition from state s to state s′

on inputsymbol a. The set λ ⊆ C gives the clocks to be reset with thistransition, and δ is a clock constraint over C.


BasicsTimed Transition Systems

Example

Lets take example of a transition system-A = 〈Σ, S ,S0,C ,E 〉 where Σ = {a, b}S = {q0, q1}S0 = {q0}C = {x , y}E = {(q0, q1, a, {x}, y ≥ 1 ∧ y ≤ 1), (q1, q0, b, {y}, x ≥ 1 ∧ x ≤ 1)}We will refer this example in further sections.


BasicsTimed Regular Language

Definition

A run r, denoted by (s, v) of a timed transition table 〈Σ, S ,S0,C ,E 〉 overa timed word (σ, τ) is an infinite sequence of the form

r : 〈s0, v0〉σ1−→τ1〈s1, v1〉

σ2−→τ2〈s2, v2〉

σ3−→τ3

. . .

with si ∈ S and vi ∈ [C −→ R], for all i ≥ 0, satisfying the followingrequirements:

Initialization: s0 ∈ S0, and v0(x) = 0 for all x ∈ C

Consecution: for all i ≥ 1, there is an edge in E of the form〈si−1, si , σi , λi , δi 〉 such that (vi−1 + τi − τi−1) satisfies δi and viequals [λi 7→ 0](vi−1 + τi − τi−1)

The set inf(r) consists of those states s ∈ S such that s = si for infinitelymany i ≥ 0.



Example

In the transition system given in above example, the only infinite runpossible is

(q0, [0, 0])ε−→1

(q0, [1, 1])a−→1

(q1, [0, 1])ε−→2

(q1, [1, 2])b−→2

(q0, [1, 0])

ε−→3

(q0, [2, 1])a−→3

(q1, [0, 1]) . . .



Definition

A timed language L is a timed regular language iff L = L(A) for someTBA A.

Example

The timed language defined in above example is a timed regular language.As it is accepted by timed Buchi automata obtained from timed transitionsystem defined above with one extension of defining any state in thatautomata as good state. In you consider non-infinite sense of TBA(TA),we will timed regular language as a timed language accepted by timedautomata.


Timed AutomataDefinition

Definition

The timed Buchi automaton (in short TBA) is a tuple 〈Σ,S ,S0,C ,E ,F 〉,where 〈Σ,S , S0,C ,E 〉 is a timed transition table, and F ⊆ S is a set ofaccepting states.A run r=(s, v) of a TBA over a timed word (σ, τ) is called an acceptingrun iff inf (r)

⋂F = φ.

For a TBA A, the language L(A) of timed words it accepts is defined to bethe set {(σ, τ) | A has an acceptiing run over (σ, τ)}.



Figure: 1.TBA



Example

The figure 1 represents a TBA. A = 〈Σ,S ,S0,C ,E ,F 〉 whereΣ = {a, b}S = {q0, q1}S0 = {q0}C = {x , y}E = {(q0, q1, a, {x}, y ≥ 1 ∧ y ≤ 1), (q1, q0, b, {y}, x ≥ 1 ∧ x ≤ 1)}F = {q0}The run(q0, [0, 0])

ε−→1

(q0, [1, 1])a−→1

(q1, [0, 1])ε−→2

(q1, [1, 2])b−→2

(q0, [1, 0])

ε−→3

(q0, [2, 1])a−→3

(q1, [0, 1]) . . .

is accepting run as inf (r) = q0 ∩ F . The language containing these words(here only one) is the language accepted by TBA.


Checking Emptiness of Timed AutomataRestriction to integer constants

Lemma

Consider a timed transition table A, a timed word (σ, τ), and t ∈ Q. (s, v)is a run of A over (σ, τ) iff (s, t.v) is a run of A, over (σ, t.τ), where At isthe timed transition table obtained by replacing each constant d in eachclock constraint lebelling edges of A by t.d.


Checking Emptiness of Timed AutomataRestriction to integer constants

Example

Lets take an example of run (σ, τ)(s, [1.2, 2.2])

ε−−→0.3

(s, [1.5, 2.5])a−−→0.3

(s′, [1.5, 2.5])

with transition (s, s′, a, , x ≤ 1.5 ∧ y ≤ 2.5) ∈ E .

Corresponding (σ, 10.τ) can be given as(s, [12, 22])

ε−→3

(s, [15, 25])a−→3

(s′, [15, 25])

with all transitions constraint rational constants multiplied by 10. Thecorresponding transition would be(s, s

′, a, {}, x ≤ 15 ∧ y ≤ 25) ∈ E

′.


Checking Emptiness of Timed AutomataConcept of Clock Region

Definition

For a timed transition table 〈Σ,S , S0,C ,E 〉, an extended state is a pair〈s, v〉 where s ∈ S and v is a clock interpretation for C.



Example

Lets take example of a TBA given in figure 1-A = 〈Σ, S ,S0,C ,E ,F 〉 where Σ = {a, b}S = {q0, q1}S0 = {q0}C = {x , y}E = {(q0, q1, a, x , y ≥ 1 ∧ y ≤ 1), (q1, q0, b, y , x ≥ 1 ∧ x ≤ 1)} F = {q1}And a run associated with this(q0, [0, 0])

ε−→1

(q0, [1, 1])a−→1

(q1, [0, 1])ε−→2

(q1, [1, 2])b−→2

(q0, [1, 0])

ε−→3

(q0, [2, 1])a−→3

(q1, [0, 1]) . . .

The extended states for this run are(q0, [0, 0]), . . . , (q0, [1, 1]), . . . , (q1, [0, 1]), . . . , (q1, [1, 2]), . . . ,(q0, [1, 0]), . . . , (q0, [2, 1]), . . . , (q1, [0, 1]), . . .



Definition

Let A = 〈Σ,S ,S0,C ,E 〉 be a timed transition table. For each x ∈ C ,let cxbe the largest integer c such that (x ≤ c) or (c ≤ x) is a subformula ofsome clock constraint appearing in E.The equivalence relation ∼ is defined over the set of all clockinterpretations for C; v ∼ v

′iff all the following conditions hold:

(1) For all x ∈ C , either bv(x)c and bv ′(x)c are the same, or both v(x)

and v′(x) are greater than cx

(2) For all x , y ∈ C with v(x) ≤ cx and v(y) ≤ cy ,fract(v(x)) ≤ fract(v(y)) iff fract(v

′(x)) ≤ fract(v

′(y))

(3) For all x ∈ C with v(x) ≤ cx , fract(v(x)) = 0 iff fract(v′(x)) = 0

A clock region for A is an equivalence class of clock interpretationsinduced by ∼.



A region can be specified by:-

1 for every clock x ,one clock constraint from the set

{x = c | c = 0, 1, . . . , cx}∪{c−1 < x < c | c = 1, . . . , cx}∪{x > cx}

2 For every pair of clocks x and y such that c − 1 < x < c andd − 1 < y < d appear in (1) for some c,d, whether fract(x), is lessthan, equal to, or greater than fract(y).



Figure: 2.Region Types


Checking Emptiness of Timed AutomataConcept of Region Automaton

Definition

A clock region α′

is a time-successor of a clock region α iff for each v ∈ α,there exists a positive t ∈ R such that v + t ∈ α′

.

Example

The triangular clock regions 1 and 2 defined in the above figure 3 arecalled adjacent regions. And region 2 is called successor of clock region 1as region 2 can be reached here just by passing the time from thevaluations in region 1.



Figure: 3.Adjacency of Clock Regions



Definition

For a timed transition table A = 〈Σ, S , S0,C ,E 〉, the corresponding regionautomaton R(A) is a transition table over the alphabet Σ.

The states of R(A) are of the form 〈s, α〉 where s ∈ S and α is aclock region.

The initial states are of the form 〈s0, [v0]〉 where s0 ∈ S0 andv0(x) = 0 for all x ∈ C .

R(A) has an edge 〈〈s, α〉, 〈s ′, α

′〉, a〉 iff there is an edge

〈s, s ′, a, λ, δ〉 ∈ E and a region α“ such that

(1) α′′

is a time-successor of α,(2) α“ satisfies δ, and(3) α

′= [λ 7→ 0]α

′′.



Figure: 4. Another TBA



Figure: 5. Region Automaton



Example

For the TBA A = (S , S0,Σ,C ,E ,F ) whereS = {S ,P,Q,R}S0 = SΣ = {a, b}C = {x , y}E = {(S ,P, a, {y}, φ), (P,Q, a, {x}, y ≤ 1), (Q,R, b, {y}, x ≤1), (R,P, a, {y}, φ)}F = {P}The TBA for this is given in the figure 4.The region automata for the same is given in figure 5.



Definition

For a run r = (s, v) of A of the form

r : 〈s0, v0〉σ1−→τ1〈s1, v1〉

σ2−→τ2〈s2, v2〉

σ3−→τ3

. . .

define its projection [r ] = (s, [v ]) to be a sequence

[r ] : 〈s0, [v0]〉 σ1−→ 〈s1, [v1]〉 σ2−→ 〈s2, [v2]〉 σ3−→ . . .



Example

For the run(S , [0 0])

a−→0

(P, [0 0])ε−−→

0.35(P, [0.35 0.35])

0.35−−→a

(Q, [0 0.35]ε−−→

0.75

(Q, [0.4 0.75])b−−→

0.75(R, [0.4 0])

ε−−→1.75

(R, [1.4 1])a−−→

1.75(P, [1.4 0]) . . . .

sequence of the projection on region automata is(S , {x = 0, y = 0}) a−→ (P, {x < 1, x > 0, y = 0}) a−→ (Q, {y < 1, y >

0, x = 0}) b−→ (R, {x < 1, y = 0}) a−→ (P, {x > 1, y = 0}) . . . .This example is also referenced further.



Definition

A run r = (s, α) of the region automaton R(A) of the form

r : 〈s0, α0〉σ1−→ 〈s1, α1〉

σ2−→ 〈s2, α2〉σ3−→ . . .

is progressive iff for each clock x ∈ C , there are infinitely many i ≥ 0 suchthat αi satisfies [(x = 0) ∨ (x > cx)].



Example

The run in the above example is progressive as reset occurs in the statesfor each variable.


Checking Emptiness of Timed AutomataUntiming Construction

Theorem

Given a TBA A = 〈Σ,S ,S0,C ,E ,F 〉, there exists a Buchi automaton overΣ which accepts Untime[L(A)].

Proof.

Given the timed automata we can construct corresponding TBA from thatby using the above procedure. And it is clear from the above that for everyaccepting run on timed automata we can generate corresponding statesequence of TBA from that ending in the same final state. For TBAcondition to satisfy we will consider only combination of final states asfinal state which occur infinitely often. It is also clear that TBA can onlyaccept untimed words. So, the proof.


Checking Emptiness of Timed AutomataDeterministic Timed Automata

Definition

A timed transition table 〈Σ,S , S0,C ,E 〉 is called deterministic iff

(1) it has only one start state, | S0 |= 1, and

(2) for all s ∈ S , for all a ∈ Σ, for every pair of edges of the form〈s,−, a,−, δ1〉 and 〈s,−, a,−, δ2〉, the clock constraints δ1 and δ2 aremutually exclusive (i.e. δ1

∧δ2 is unsatisfiable).

A timed automaton is deterministic iff its timed transition table isdeterministic.


Checking Emptiness of Timed AutomataDeterministic Timed Automata

Example

The above example is of deterministic timed automata as there is only onepossible move from a given state, with a given transition and givenvaluation.


Distributed Timed AutomatonDefinition

Definition

A distributed timed automaton (DTA) over the set of processes Proc is astructure D = ((Ap)p∈Proc , π) where Ap = (Sp,Σp,Zp, δp, Ip, ip,Fp) aretimed automata such that the alphabets Σp are pairwise disjoint, and π isa (total) mapping from

⋃p∈Proc Zp to Proc such that, for each p ∈ Proc,

we have Reset(Ap) ⊆ π−1(p) ⊆ Zp.


Distributed Timed AutomatonExample

Figure: 6.Distributed Timed Automata


Distributed Timed AutomatonExample

Example

Proc = {p, q}Consider the DTA as shown in the figure 6. It consists of two timedautomatas Ap and Aq whereAp = (Sp,Σp,Zp, δp, Ip, ip,Fp), Aq = (Sq,Σq,Zq, δq, Iq, iq,Fq)Sp = {q1, q2, q3} and Sq = {q1′ , q2′ , q3′}, Σp = {a, b} and Σq = {a, b}Zp = {x , y} and Zq = {x , y}δp = {(q1, q2, a, φ, y ≤ 1), (q2, q3, c , φ, x > 1), (q3, q3, a, {x}, φ)}

andδq = {(q1′ , q2′ , b, φ, x > 1), (q2′ , q3′ , b, φ, y ≤ 1), (q3′ , q2′ , d , φ, y ≥ 2)}Ip = φ and Iq = φ ,ip = {q1} and iq = {q1′}Fp = {q3} and Fq = {q3′}The DTA corresponding to these timed automatas is given by-

D = (Ap,Aq, π) and π(x) = p and π(y) = q is the mapping functionfrom clock variables to processes.


Distributed Timed AutomatonSemantics

τ = (τp)p∈Proc where τp : R≥0 → R≥0.



Figure: 7.Behaviour of DTA with uniform linear rates



Figure: 8.Behaviour of DTA with nonuniform rates


Distributed Timed AutomatonIcTA

Definition

A timed automton with independently evolving clocks(icTA) over Proc is atuple B = (S ,Σ,Z, δ, I, i ,F , π) where (S ,Σ,Z, δ, I, i ,F ) is a timedautomaton and π : Z→ Proc maps each clock to a process.


Figure: 9.An icTA with independent clocks x,y


Distributed Timed AutomatonConstruction of icTA from DTA

For s = (sp)p∈Proc ∈∏

p∈Proc Sp and A ⊆ Σ with A 6= φ, we define thestate invariants I (s, φ) =

∧p∈Proc Ip(sp) and

I (s,A) = z ≤ 0 ∧∧

p∈Proc Ip(sp). Morever we get i = ((ip)p∈Proc , φ), andF = (

∏p∈Proc Fp)× {φ}. Then transition in BD are of two types-



(T1) The first type in an ε-move, which guesses the set of processes of theDTA that will move next and the transitions that each of them wouldperform. In addition it checks the guard that each of them mustsatisfy and resets the clocks as well. Thus,((s, φ), ε, ϕ,R, (s

′,A)) ∈ δ

if there are some non-empty set P ⊆ Proc and transitions(sp, ap, ϕp,Rp, s

′p) ∈ δp, p ∈ P,such that sp = sp and s

′p = s

′p for all

p ∈ P, and sq = s′q for all q ∈ Proc\P, ϕ = ∧p∈Pϕp,

R =⋃

p∈P Rp ∪ {z}, and A = {ap|p ∈ P}\{ε}.



(T2) This move performs an action from its guessed set A and thenremoves it.

((s,A), a, true, ∅, (s,A\{a})) ∈ δ

for all s ∈∏

p∈Proc Sp,A ⊆ Σ and a ∈ A. This completes thedefinition of the icTA BD associated with DTA D.



Figure: 10.Part of icTA for DTA D



Example

The icTA for DTA in figure 6 is as shown in the figure 10.


Distributed Timed AutomatonUniversal Symantics

Definition

Let B = (S ,Σ,Z, δ, I, i ,F , π) be an icTA and τ ∈ Rates. The language ofB wrt. τ denoted by L(B, τ), is the set of words w ∈ Σ∗ such that(B, τ) : i

w−→ s for some s ∈ F . Morever, we defineL∃(B) =

⋃τ∈Rates L(B, τ) to be the existential semantics and

L∀(B) =⋂τ∈Rates L(B, τ) to be the universal semantics of B.



Definition

For a DTA D and τ ∈ Rates, we get L(D, τ) = L(BD, τ) to be the languageof D wrt. τ . and we define L∃(D) = L∃(BD) as well as L∀(D) = L∀(BD) toobtain its existential and universal semantics, respectively.



Example

In the figure 9, the universal language i.e. the language accepted for allpossible dynamic rates is

L∀(A) = {a, ab}


Distributed Timed AutomatonExistential Symantics

Example

In the figure 9, the existential language i.e. union of the languageaccepted for possible dynamic rates is

L∃(A) = {a, ab, b, c}


Distributed Timed AutomatonWeird Behaviour

Figure: 11.An icTA with independent clocks x,y


Conclusion

1 Timed Automata

2 Distributed Timed Automata

3 IcTA


Thank you


beamerpresentation

Education

timed worda

timed word as3

set of timed words

x c c x

set x of clock constraints

basicstimed constraints

noninfinite timed words

set x of clock variables