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Fundamental AlgorithmsFundamental Algorithms for System Modeling, Analysis, and Optimization

Stavros Tripakis, Edward A. LeeUC B k lUC BerkeleyEECS 144/244Fall 2013

Copyright © 2013 E A Lee J Roydhowdhury S A Seshia S TripakisCopyright © 2013, E. A. Lee, J. Roydhowdhury, S. A. Seshia, S. TripakisAll rights reserved

Discrete Event Simulation

Stavros Tripakis: EECS 144/244 – Fall 2013 Discrete Event Simulation 1 / 66

Timed Systems

In embedded / cyber-physical systems, timing is key:

I proper timing = an issue of correctness

I the right values, at the right time (not too late, not too early).

C.f.: real-time controllers (automotive, robotic surgeon, ...).

Contrast this to personal computers: “best-effort” systems –timing an issue of performance.


(Timed) Discrete-Event (DE) Systemsvs. Continuous Control Systems

Continuous control systems:

I Coming from continuous system theory.

I Typically implemented by periodic sampling controllers (alsoevent-driven controllers): these are discrete, but try toapproximate the continuous ones.

Discrete-event systems:

I More “sparse” events (typically).

I Discrete control: e.g., mode switches.

I Typically higher level: e.g., supervisory control (DE) vs. cruisecontrol (continuous).

I Other application domains:I Queueing theory.I Circuits (VHDL, Verilog, SystemC).


Example: controller and monitor for an avionic electric power systemp y


Example: simple light controller modeled as a timedautomaton

off medium brighttouch

c := 0

touch

c < 2

touch

c ≥ 2

touch

Input event: “touch”.

Output: light (“off”, “medium”, “bright”).


Example: circuit timing modeled with timed automata

fi v�� vk� � �Ci ��

fi v�� vk� � ��Ci ��

vi � vi � �

fi v�� vk� � fi v�� vk� � �

vi � � �Ci � li

vi � �

vi �

vi � � �Ci � ui

vi � �Ci � ui

vi � �Ci � li

Fig� �� The two automata Ai and Ai associated with each equation

�

Sketch of Proof� By combining what claim � tells us about Ci and vi� whatclaim � tells us about their in�uence on vi� with the compositionality of ��claim � we obtain a characterization equivalent to de�nition � ut

Theorem� �Main Result�� Every k�wire circuit can be transformed into anequivalent timed automaton with �k variables and k clocks�

Proof� Given a system of delay inclusions� we create the corresponding systemof timed automata and compose them� obtaining

A �kO

i �

�Ai �Ai� � A � �V �V� C�R�O�

as described above� whose set of observable behaviors L�A��V� is exactly the setof solutions� utExample� Consider the simple circuit in �gure � We model x� as an oscillatorwith parameters l� and u�� Initially we obtain the four ��state automataA�� A��A� and A� whose R and O are written in table � After composing A� with A�

and A� with A� �and removing states and transitions whose R is false � theseindicate unstable states in which the automaton cannot stay for a non�zero

From [Maler and Pnueli(1995)].


Example: car wash

Taken from [Misra(1986)]:

I Source generates car arrivals at some arbitrary times.

I Attendant directs cars to car wash stations CW1 or CW2:

I if both CW1 and CW2 are free, then to CW1;I if only one is free, then to this one;I otherwise car waits until a station becomes free.I Cars are served by attendant in FIFO order.

I CW1 spends 8 mins to wash a car.

I CW2 spends 10 mins to wash a car.


This week: discrete-event systems

Fundamental models and algorithms:

I Discrete-event simulation.

I Model-checking (reachability) for timed automata: basicprinciples.


Recap: different ways to look at systems

operational denotational

monolithic automata, machines,transition systems, ...

functions (on values,streams, signals, ...),equations

compositional products of automata,etc.

function composition,sets of equations,fixpoints


Continuous vs. Discrete Event Systems

I Continuous systems: functions on continuous signals.Continuous signal x = continuous function of dense time (R+)

x : R+ → V

x(t): value of x at time t; belongs to some set of values V (e.g., R)

I Timed Discrete Event Systems: deal with timed discrete-eventsignals.Timed discrete-event signal: sequence of timed events.

continuoussystem

time

e6 e7 e8

t6 t7 t8

· · ·

e1 e2 e3 e4 e5

t1 t2 t3 t4 t5 time

· · ·systemevent

discrete

timetime

· · ·

· · ·


Timed Events

Main notion: event

I something occurring at some point in time

I may also carry a value

I event = (timestamp, value)

I systems are viewed as consumers/producers of event streams

-

6 6 6 6 6 - --

6 6 6

e1 e2 e3 e4 e5

t1 t2 t3 t4 t5 time

· · · system

time

e6 e7 e8

t6 t7 t8

· · ·


DIGRESSION:

EVENTS vs. STATES


From States to Events

If my formalism only has the notion of state, can I define events?

Event = change of state

Example: Lustre program

node UpwardEdge (X : bool) returns (E : bool);

let

E = false -> X and not pre X ;

tel


From Events to States

If my formalism only has events as primitives, can I define state?

State = history of events observed so farFormally:

I Σ: set of events

I Σ∗: set of finite event sequences = histories

I Every s ∈ Σ∗ can be seen as a state

C.f. a famous theorem:

Theorem (Myhill-Nerode theorem)A language L ⊆ Σ∗ is regular iff the equivalence relation over words

s ∼L s′ =̂ ∀s′′ ∈ Σ∗ : s · s′′ ∈ L⇔ s′ · s′′ ∈ L

has a finite set of equivalence classes. The number of equivalence classesof ∼L is the number of states in the smallest DFA recognizing L.




State = history of events observed so far

Formally:

I Σ: set of events





s ∼L s′ =̂ ∀s′′ ∈ Σ∗ : s · s′′ ∈ L⇔ s′ · s′′ ∈ L





State = history of events observed so farFormally:

I Σ: set of events





s ∼L s′ =̂ ∀s′′ ∈ Σ∗ : s · s′′ ∈ L⇔ s′ · s′′ ∈ L



END DIGRESSION


MODELING CONSIDERATIONS


Tagged signals

Tagged signal model (TSM)[Lee and Sangiovanni-Vincentelli(1998)]:

signal s = a set of events with timestamps:

s ⊆ T × V

where:

I T : set of tags (= timestamps)

I V : set of values.

Time T could be:

I discrete/“logical” (N),

I continuous/dense (R+),

I or even “superdense” (R+ × N)


Example Motivating the Need for Simultaneous E t Withi Si l > S D TiEvents Within a Signal => Super-Dense Time

Newton’s Cradle:Newton s Cradle:Steel balls on strings.

Collisions are events.

Momentum of the middle ball has three values atMomentum of the middle ball has three values at the time of collision.

Other examples: Batch arrivals at a queue. Software sequences abstracted as instantaneous. Transient states.


Super-Dense Timep

Lexicographic orderLexicographic order


Continuous vs. discrete signals

Intuition:

Discrete: only finite number of events can happen in a finiteamount of time.

Continuous: not discrete.

Let’s try to formalize this intuition.



Let T = R+, the set of non-negative reals.

Lets ⊆ R+ × V

be a signal.

Let t(s) be the set of all tags of signal s:

t(s) = {t | ∃(t, v) ∈ s}

I s is discrete if t(s) is order-isomorphic to a subset of N, whereN = {0, 1, 2, ...} is the set of natural numbers.

I Otherwise, s is continuous.


Order-isomorphisms

A order-isomorphic B means there is an order-preserving bijectionf : A→ B.

In our case the order is the usual < on R+ and N. So:

I f must be a bijection: (1) f(a) must be defined for all a ∈ A,(2) for all b ∈ B there must exist a ∈ A such that f(a) = b,and (3) a 6= a′ ⇒ f(a) 6= f(a′).

I f must be order-preserving: a < a′ ⇒ f(a) < f(a′).

Is N× N (with lexicographic order) order-isomorphic to N ?


Continous and discrete signals

Are the signals below discrete or continuous?

s1 = {(t, t) | t ∈ R+}?

Continuous.

s2 = {(n, v) | n ∈ N, v ∈ V }? Discrete.

s3 = {(0, 0), (1, 1, ), (2, 2)}? Discrete.

s4 = {(0.3, 0), (1.27, 1, ), (2π, 2)}? Discrete.

s5 = {}? Discrete.

s6 = {(t, t) | t ∈ [0, 1]}? Continuous.

s7 = {(t, btc) | t ∈ R+}? Continuous, according to ourdefinition. Could also be considered discrete since it’s piecewiseconstant. This is how discrete signals are modeled in Simulink.




s1 = {(t, t) | t ∈ R+}? Continuous.


s3 = {(0, 0), (1, 1, ), (2, 2)}? Discrete.

s4 = {(0.3, 0), (1.27, 1, ), (2π, 2)}? Discrete.

s5 = {}? Discrete.

s6 = {(t, t) | t ∈ [0, 1]}? Continuous.






s2 = {(n, v) | n ∈ N, v ∈ V }?

Discrete.

s3 = {(0, 0), (1, 1, ), (2, 2)}? Discrete.

s4 = {(0.3, 0), (1.27, 1, ), (2π, 2)}? Discrete.

s5 = {}? Discrete.

s6 = {(t, t) | t ∈ [0, 1]}? Continuous.

