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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 Basics
Timed Run
Untimed Words
Timed Language
Untimed Language
Timed Constraints and Clock Interpretation
Timed Transition Systems
Timed Regular Language
2 Timed Automata
Deﬁnition
Checking Emptiness
Restriction to integer constants
Clock Region
The Region Automaton
The Untiming Construction
Deterministic Timed Automaton
Deﬁnition
Example

Outline
3 Distributed Timed Automaton
Deﬁnition
Example
Semantics
Timed Automaton with Independently Evolving Clocks
Construction of icTA from DTA
Universal Semantics
Existential Symantics
Weird Behaviour
4 Conclusion

Basics
Timed Run
Definition
A time sequence τ = τ1τ2 . . . is an infinite sequence of time values τi ∈ R
with τi > 0, satisfying the following constraints:
(1) Monotonicity: increases strictly monotonically;i.e., τi < τi+1 for all
i ≥ 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 an
infinite word over Σ and σ is a time sequence.

Basics
Timed 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.

Basics
Untimed 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 word
here is aab . . . .
Similarlly if we consider only non-inﬁnite timed words, then aab is an
untimed word.

Basics
Timed 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 is
b 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). . .
}

Basics
Untimed Language
Deﬁnition
For a timed language L over Σ, Untime(L) is the ω-language consisting of
σ ∈ Σω such that (σ, τ) ∈ L for some time sequence τ.
Example
For the timed language given in the above example, the untimed language
is (a + b)b(a + b)ω and the corresponding ﬁnite language is
(a + b)b(a + b)∗

Basics
Deﬁnition
For a set X of clock variables, the set Φ(x) of clock constraints δ is
deﬁned inductively by
δ := x ≤ c | c ≤ x | ¬δ | δ1 ∧ δ2
where x is a clock in X and c is constant in Q.

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

Basics
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 input
symbol a. The set λ ⊆ C gives the clocks to be reset with this
transition, and δ is a clock constraint over C.

Basics
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.

Basics
Definition
A run r, denoted by (´s, ´v) of a timed transition table Σ, S, S0, C, E over
a 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 following
requirements:
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 vi
equals [λi → 0](vi−1 + τi − τi−1)
The set inf(r) consists of those states s ∈ S such that s = si for infinitely
many i ≥ 0.

Basics
Example
In the transition system given in above example, the only inﬁnite run
possible is
(q0, [0, 0]) −→
1
(q0, [1, 1])
a
−→
1
(q1, [0, 1]) −→
2
(q1, [1, 2])
b
−→
2
(q0, [1, 0])
−→
3
(q0, [2, 1])
a
−→
3
(q1, [0, 1]) . . .

Basics
Definition
A timed language L is a timed regular language iff L = L(A) for some
TBA A.
Example
The timed language defined in above example is a timed regular language.
As it is accepted by timed Büchi automata obtained from timed transition
system defined above with one extension of defining any state in that
automata as good state. In you consider non-infinite sense of TBA(TA),
we will timed regular language as a timed language accepted by timed
automata.

Timed Automata
Definition
Definition
The timed Büchi 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 of
accepting states.
A run r=(´s, ´v) of a TBA over a timed word (σ, τ) is called an accepting
run iff inf (r) F = φ.
For a TBA A, the language L(A) of timed words it accepts is defined to be
the set {(σ, τ) | A has an acceptiing run over (σ, τ)}.

Timed Automata
Deﬁnition
Figure: 1.TBA

Timed Automata
Deﬁnition
Example
The ﬁgure 1 represents a TBA. 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 = {q0}
The run
(q0, [0, 0]) −→
1
(q0, [1, 1])
a
−→
1
(q1, [0, 1]) −→
2
(q1, [1, 2])
b
−→
2
(q0, [1, 0])
−→
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 Automata
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 is
the timed transition table obtained by replacing each constant d in each
clock constraint lebelling edges of A by t.d.

Example
Lets take an example of run (σ, τ)
(s, [1.2, 2.2]) −−→
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]) −→
3
(s, [15, 25])
a
−→
3
(s , [15, 25])
with all transitions constraint rational constants multiplied by 10. The
corresponding transition would be
(s, s , a, {}, x ≤ 15 ∧ y ≤ 25) ∈ E .

Concept 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]) −→
1
(q0, [1, 1])
a
−→
1
(q1, [0, 1]) −→
2
(q1, [1, 2])
b
−→
2
(q0, [1, 0])
−→
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 cx
be the largest integer c such that (x ≤ c) or (c ≤ x) is a subformula of
some clock constraint appearing in E.
The equivalence relation ∼ is defined over the set of all clock
interpretations for C; v ∼ v iff all the following conditions hold:
(1) For all x ∈ C, either v(x) and v (x) 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 interpretations
induced 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 and
d − 1 < y < d appear in (1) for some c,d, whether fract(x), is less
than, equal to, or greater than fract(y).

Figure: 2.Region Types

Concept 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 are
called adjacent regions. And region 2 is called successor of clock region 1
as region 2 can be reached here just by passing the time from the
valuations in region 1.

Figure: 3.Adjacency of Clock Regions

Definition
For a timed transition table A = Σ, S, S0, C, E , the corresponding region
automaton 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 a
clock region.
The initial states are of the form s0, [v0] where s0 ∈ S0 and
v0(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 → 0]α .

Figure: 4. Another TBA

Figure: 5. Region Automaton

Example
For the TBA A = (S, S0, Σ, C, E, F) where
S = {S, P, Q, R}
S0 = S
Σ = {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]) −−→
0.35
(P, [0.35 0.35])
0.35
−−→
a
(Q, [0 0.35] −−→
0.75
(Q, [0.4 0.75])
b
−−→
0.75
(R, [0.4 0]) −−→
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, ˇα) 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 such
that αi satisfies [(x = 0) ∨ (x > cx )].

Example
The run in the above example is progressive as reset occurs in the states
for each variable.

Untiming Construction
Theorem
Given a TBA A = Σ, S, S0, C, E, F , there exists a Büchi automaton over
Σ which accepts Untime[L(A)].
Proof.
Given the timed automata we can construct corresponding TBA from that
by using the above procedure. And it is clear from the above that for every
accepting run on timed automata we can generate corresponding state
sequence of TBA from that ending in the same final state. For TBA
condition to satisfy we will consider only combination of final states as
final state which occur infinitely often. It is also clear that TBA can only
accept untimed words. So, the proof.

Deterministic 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 are
mutually exclusive (i.e. δ1 δ2 is unsatisfiable).
A timed automaton is deterministic iff its timed transition table is
deterministic.

Deterministic Timed Automata
Example
The above example is of deterministic timed automata as there is only one
possible move from a given state, with a given transition and given
valuation.

Distributed Timed Automaton
Deﬁnition
Deﬁnition
A distributed timed automaton (DTA) over the set of processes Proc is a
structure D = ((Ap)p∈Proc, π) where Ap = (Sp, Σp, Zp, δp, Ip, ip, Fp) are
timed automata such that the alphabets Σp are pairwise disjoint, and π is
a (total) mapping from p∈Proc Zp to Proc such that, for each p ∈ Proc,
we have Reset(Ap) ⊆ π−1(p) ⊆ Zp.

Example
Figure: 6.Distributed Timed Automata

Example
Example
Proc = {p, q}
Consider the DTA as shown in the ﬁgure 6. It consists of two timed
automatas Ap and Aq where
Ap = (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)}
and
δq = {(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 function
from clock variables to processes.

Semantics
τ = (τp)p∈Proc where τp : R≥0 → R≥0.

Semantics
Figure: 7.Behaviour of DTA with uniform linear rates

Semantics
Figure: 8.Behaviour of DTA with nonuniform rates

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

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

For s = (sp)p∈Proc ∈ p∈Proc Sp and A ⊆ Σ with A = φ, we deﬁne the
state 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), and
F = ( p∈Proc Fp) × {φ}. Then transition in BD are of two types-

(T1) The ﬁrst type in an -move, which guesses the set of processes of the
DTA that will move next and the transitions that each of them would
perform. In addition it checks the guard that each of them must
satisfy and resets the clocks as well. Thus,((s, φ), , ϕ, R, (s , A)) ∈ δ
if there are some non-empty set P ⊆ Proc and transitions
(¨sp, ap, ϕp, Rp,¨sp) ∈ δp, p ∈ P,such that sp = ¨sp and sp = ¨sp for all
p ∈ P, and sq = sq for all q ∈ ProcP, ϕ = ∧p∈Pϕp,
R = p∈P Rp ∪ {z}, and A = {ap|p ∈ P}{ }.

(T2) This move performs an action from its guessed set A and then
removes it.
((s, A), a, true, ∅, (s, A{a})) ∈ δ
for all s ∈ p∈Proc Sp, A ⊆ Σ and a ∈ A. This completes the
deﬁnition 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.

Universal Symantics
Deﬁnition
Let B = (S, Σ, Z, δ, I, i, F, π) be an icTA and τ ∈ Rates. The language of
B 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 deﬁne
L∃(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.

Universal Symantics
Deﬁnition
For a DTA D and τ ∈ Rates, we get L(D, τ) = L(BD, τ) to be the language
of D wrt. τ. and we deﬁne L∃(D) = L∃(BD) as well as L∀(D) = L∀(BD) to
obtain its existential and universal semantics, respectively.

Universal Symantics
Example
In the ﬁgure 9, the universal language i.e. the language accepted for all
possible dynamic rates is
L∀(A) = {a, ab}

Existential Symantics
Example
In the ﬁgure 9, the existential language i.e. union of the language
accepted for possible dynamic rates is
L∃(A) = {a, ab, b, c}

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

Conclusion
1 Timed Automata
2 Distributed Timed Automata
3 IcTA

Thank you

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