Welcome toModeling time in computing

A PhD (not only) DEIB courseTeachers:

Carlo FuriaDino MandrioliAngelo MorzentiMatteo Rossi

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Outline:• Introduction

• A few formal preliminaries• The various “dimensions” of time modeling• Traditional approaches to time modeling (in engineering)

revised• Operational (abstract machine-based) time dependent

models• Synchronous• Asynchronous

• Algebraic approaches• Logic-based approaches• Model-checking and other “dual-language” approaches to

model and analyze time dependent systems• Epilogue and conclusions

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(Flexible) Schedule and organization (1)

Monday, 8th : 9.00 --- 11.30 (Mandrioli)Wednesday, 10th: 9.00 --- 12.30 (Mandrioli)Thursday, 11th: 9.00 --- 13.00 (Rossi)Friday, 12th: 9.00 --- 13.00 (Furia)Monday, 15th : 9.00 --- 12.30 (Morzenti)Tuesday, 16th: 14.00 – 16.30 (Morzenti)Thursday, 18th: 9.00 --- 12.30 (Rossi-Mandrioli)Thursday, 18th: 14.00 --- 15.30 (Mandrioli)

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(Flexible) Schedule and organization (2)• Time slots redundant (on purpose)

• To allow for plenty of (informal and free) discussion• (For “official” PhD students):

• Each teacher will propose a few exercises (rather heterogeneous in style!) during his presentation

• Students are invited to select a few of them (their choice) and to develop and supply them to the corresponding proponent(s), within 15 days (flexible)

• Their work will be evaluated • Further deepening of specific issues, possibly towards real research welcome

References:Furia C., Mandrioli D., Morzenti A., Rossi M., “Modeling Time in Computing: A Taxonomy and a Comparative Survey”, ACM Computing Surveys (CSUR) Volume 42 , Issue 2 , February 2010, 59 pages.Furia C., Mandrioli D., Morzenti A., Rossi M., Modeling Time in Computing. Springer, EATCS monograph series, 2012(go to http://link.springer.com/book/10.1007/978-3-642-32332-4/page/1 and download it ASAP!)

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(Flexible) Schedule and organization (3)

• Necessary background:• A little bit of:• Automata theory• Mathematical logic • Propositional and predicate calculus

• (if not … let me know!)

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Introduction

• What is time?•

• If an alarm is raised at time t0, the system must be shut down within time t1, with t1 ≤ t0 + k

• Action B can occur only after action A• Switching a transistor’s state takes a few nanoseconds• Sorting an array of size n by means of a merge-sort algorithm

takes a time poportional to n times the logarithm of n• The age of the universe is approximately 13.7 billion years

(what happened/existed before the birth of the universe?)• …

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1

0

t

t0 dt)t(vs)t(s

• … but also• “You are always late”• Time flies• In pleasant events time is faster, unpleasant

ones never end• 1/10 sec is the time that elapses between …• To achieve good coordination we must

synchronize our clocks (there is no time without clock)

• Time can only advance, never go back• Zeno’s paradox: Achille, though much faster than

the turtle will never reach itModeling time 7

• For a long “time”, time in engineering: • A mathematical variable t• But:• Always an independent variable? (state(t))• Continuous or discrete?• Time unit: nanoscond or century?• …

• Nowadays “traditional” ways of modeling time are not always adequate: a more general framework is needed: this is our goal here

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• Next steps: • Some basic mathematic terminology• “Dimensions”to evaluate and compare time-

related models

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Some basic terminology(apologies to those who already know it)

• Language, in the most general meaning of the term:• An expression/communication means• A collection of sentences (strings –not

necessarily linear) of characters in an alphabet• Natural• Programming• Musical• Graphic• A (communication) protocol• …

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Syntax and semantics (1)(apologies to those who already know it)

• Syntax: how to build language sentences:• A sentence consists of a noun phrase followed by a verb

phrase. The noun phrase is a noun (possibly preceded by a determiner such as an article) or a pronoun. …

• <Conditional-statement> ----> • If (<Cond> ) <Statement> [else <Statement>]

• But also:

• …Modeling time 11

Syntax and semantics (2)(apologies to those who already know it)

• Semantics: assigning meaning to language sentences:• if (x > 3) { x = x + 1; } else { x = x – 1; }:

• f(1) = 0; f(2) = 1; f(5) = 6; ….

• Ambiguous semantics • (different meanings for the same sentence):• if (C1) if (C2) S2 else S3• 10: ten, two or what?• …

• Conversely: • Multiple expressions for the same meaning:

• for (i = 0, i < n, i++) do …• i = 0; while (i < n) ….

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Language formality (1)

• Formal languages: • A ---> a A | b ===> L = { a+b}

• Informal (natural) languages:• To be or not to be• Io credo ch’ei credette ch’io credessi• ….

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a

b

Language formality (2)

• Partially formal (semiformal) languages: • Often:• Formal syntax an informal semantics

• (a) (b)

“three-way handshake”: (a) the communication is successfully established (b) a timeout occurs after the syn message. what if the syn-ack message is received after the timeout is triggered?

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(Languages for) system modeling

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Operational vs. descriptive languages

Operational formulation:“When the last digit of the correct security code is entered, the safe opens; then, if the safe remains open for three minutes, it automatically closes.”

Descriptive formulation:“The safe is open if and only if the correct security code has been entered no more than three minutes ago.”

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Operational vs. descriptive languages

Operational formulation:

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closed

1 correct

digit

2 correct digits

open

1 minute elapsed

2 minutes elapsed

incorrect digit

correct digit

incorrect digit

incorrect digit

correct digit

correct digit

1 minute

1 minute

1 minute

Operational vs. descriptive languages

Descriptive formulation:( Open Closed ) ( Open Correct_code_entered_within_three_minutes ) (*)

Open Correct_code_entered_within_three_minutes

(**)(Safe_opened_in_the_past Stay_open_issued_since_last_opening)

Mathematical logic(s): the principal descriptive formal language(s)

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Operational vs. descriptive languages

Descriptive formulation (in predicate calculus):

t ( (Open (t) Closed (t)) (*+)

(Open (t) u ( (t − 3 u < t) Correct_code_entered (u) ) )

(preliminary, self-testing) exercise:Provide a predicate logic extension of the propositional formula (**)

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Operational vs. descriptive languages:not that sharp:

Operational (style) formulation (in predicate calculus): (*++)

t ( (Open (t) Closed (t))

(Correct_code_entered (t) u ( (t + 1 u t + 3) Open (u)))

(u ( (t u t + 2) (Open (u) Correct_code_entered (u))) Closed (t + 3)) (Closed (t) Correct_code_entered (t) Closed (t + 1)) )

The flexibility and generality of mathematical logic(s) allows for adopting an operational as well as a descriptive style.

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The dimensions of time• Time is a physical quantity• Once we set the appropriate unit to measure we have

fully characterized it:• Seconds?• Too long or too short?• The real axis? The discrete axis? Semi-infinite or two-sided?• …• Not everything so obvious• Modeling time must be evaluated from different

perspectives: “dimensions”.• They will be our reference to evaluate and compare (time-

dependent system models)

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1. Discrete vs. Dense Time Domains

• A well known distinction:• N (natural numbers), Z (integers), vs. • Q (rational numbers), R (reals), Q+, R+, …

• But also:• Continuous vs. Non-Continuous Time Models

• Many “clocks” are uncomparable:• What’s the ratio between one year and one day, and one

second?• More technically:

• We will see that some algorithms to decide system properties work well with time domain Q but do not apply when time domain is R

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2. Bounded, Finite, and Periodic Time Models

• In principle time is infinite … but:• Is it always necessary to “inspect” system behavior in

[- … + ]?• If system behavior is periodic …• Perhaps to understand whether the ABS of a car behaves

properly it is suffcient to observe its behavior within a time window of a few seconds -minutes at most-

• If the (time) domain is bounded and discrete it is finite … and originally undecidable problems become decidable ===> we can “approximate” problems that are undecidable in general cases into decidable ones by suitably restricting the domain of interpretation• E.g., through discretization and sampling

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3. Hybrid systems and (time) models• Traditionally discrete and continuous (system)

models were two well-separated worlds:• (differential) equations, Laplace and Fourier

transforms, …• vs.

• Difference equations, Z transforms, automata, …• Nowadays, typically:

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Continuous process

Computer-basedcontroller

4. Ordering vs. metric

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

a b a b a b a

• Every a is followed by a b within 4 time units and every b is followed by an a after exactly 2 time units.

• Every a is followed by a b and every b is followed by an a.• If (after) an alarm is raised the system must be shut down,

then repaired and finally can be restarted• If an alarm is raised the system must be shut down within 10

seconds; it must be repaired within 3 minutes and must be restarted within 10 minutes.

4.1 Total vs. partial ordering

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• Time is totally ordered … isn’t it?• Yes but …

• In a car there is:• The anti-lock braking system • Another subsystem electronically controlled is responsible for moving the car

windows. • the braking system must release the brakes within, say, 1/10 second whenever the

wheels are blocked and the vehicle is moving, • the electric windows must shut completely within, say, 7 seconds whenever a

passenger clicks the button. • In an overall model of the car, the events “wheels become blocked” and “brakes

released” are strictly ordered, and so are the two other events “button clicked” and “windows become closed”.

• However, there is no reason to define an order between events of the braking subsystem and of the window control: the events in the overall system are only partially ordered, and so are the instants of time when they may occur.

5. Time granularity

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• We are designing a car engine:• We “reason” in terms of (k)r.p.m.

• We are designing an electronic circuit:• We “reason” in terms of nanoseconds

• We are designing a university curriculum:• We “reason” in terms of (credits per) year

• … but, what if• We are designing a hydroelectric, computer controlled, power system:

• Water dynamics evolves in the “scale” of minutes• Electricity generation and distribution too (approximately ….)• Control subsystem is responsible to take “decisions” (e.g. load shedding) in real-

time i.e., within seconds or less (and its internals have a dynamic paced by micro or nanoseconds)

• Water accumulation in the reservoirs takes weeks or months• …

• How should we model the global system?

5. Time granularity

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• How should we model the global system?• From a mathematical point of view the answer could seem

obvious:• Refer everything to the smallest time unit• OK, but it is at least “uncomfortable” to say that a

reservoir of xxx m3 takes yyyy nanoseconds to fill up.

• But in other cases it is not just a matter of “changing time unit”:

5. Time granularity

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• “every month, if an employee works, then she gets her salary”• “whenever an employee is assigned a job, this job should be

completed within three days”• We decide that the month is not the appropriate time unit to

describe the firm’s life and we move to the finer time unit “day”:

• “every day, if an employee works, then she gets her salary” ??• “this job has to be finished within 3 days from now” at 4 PM on

16 June 2011. • “This job has to be finished within 3.24.60.60 seconds counting from

now” ?• “this job has to be finished by 6 PM on 19 June 2011”?• “this job has to be finished by midnight on 19 June 2011”?

5. Time granularity

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• There is lot of implicit knowledge in the interpretation of sentences that refer to a coarse time scale at the level of a finer one.

• Exercise:Consider the following sentences:Tomorrow, I will eatTomorrow, I will workTomorrow, I will go to the bank to pay my monthly billsTomorrow, I will stay in the city.

• Based on your intuitive understanding thereof:• Build sample interpretations of the above sentences in a time axis with a finer time unit than

the day (say hours or -better-minutes)• By choosing the appropriate time unit (again, hours or minutes) build suitable first-order

formulas that formalize the semantics of the above sentences in agreement with their common sense interpretation.

6. Linear vs. Branching Time Models

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• (a) (b)

• Time model (b) can be useful e.g., to interpret the behavior of parallel systems

• Notice that in this figure time is still “metric”, but if we “erase” the metrics therefrom we obtain a partial order.

• Deterministic evolution:• x(t+1) = f(x(t), u(t))

• Nondeterministic evolution:• x(t+1) = f(x(t), u(t)) OR g(x(t), u(t)) OR …

• Nondetermism useful in various contexts:• (Advanced) programming languages• Search algorithms• Concurrent systems

7. Deterministic, Nondeterministic, and Probabilistic Models

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a

a

• Probabilistic (stochastic) evolution:

• A)

• B)

7. Deterministic, Nondeterministic, and Probabilistic Models

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a /30%

a /70%

0

1

t

Probability distribution

• Nondeterministic behavior could be naturally “paired with” branching time, but the two concepts are different

• Nondeterministic modeling not to be confused with probabilistic modeling:• Nondeterminism ---> lack of knowledge: anyone of the possible behaviors

may occur, but no indication on preference: e.g., an implementer to which a nondeterministic specification is given is totally free to choose any implementation compatible with it.

• Probability distribution is knowledge about possible behaviors: for large numbers of “experiments” 30% go according to (a) and 70% according to (b). Conversely, in a nondetermistic specification all of them could be of the same type.

• Mathematical models for specifying nondeterministic systems are of the same type as deterministic ones (in general, roughly speaking, we need quantification w.r.t. possible behaviors); stochastic modeling follows a sharply different mathematical approach, rooted in probability calculus.

7. Deterministic, Nondeterministic, and Probabilistic Models

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• Sometimes we refer explicitly to a time value in a given time axis:• During the year 1625, a dramatic famine struck Europe; the famine lasted until the

beginning of the year 1630.

• Other times we refer implicitly to the time occurrence of some events:• Every lie is eventually uncovered

• Engineering artifacts are often time-invariant systems, naturally described with an implicit “now”:• The speed of a braking car decreases proportionally to the relative time since when

braking starts.• The discharge time of a capacitor attached to a resistor only depends on the resistor’s

resistance, the capacitor’s capacity, and the initial charge accumulated, irrespective of the absolute time when discharging starts.

• Often there is a blend of implicit and explicit –possibly metric- time reference:• If I push the button (implicit now) the window will be open within 3 seconds (explicit

time bound)

8. Implicit vs. Explicit Time Reference

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• Starting from Zeno’s paradox (Achille and the turtle):• System’s behavior (model) can exhibit:

• An infinite sequence of steps that “never passes a time threshold” (just called “Zeno behavior”).

• A continuous behavior with a frequency that goes to (such as sin (1/x) )• An infinite sequence of steps that “does not converge” as it happens with Zeno’s

behaviors but whose relative distance gets smaller and smaller (e.g., as in the series (this behavior is labeled as “Berkeley behavior”).

• Are such behaviors realistic? Should they be ruled out a priori from the models? Should their (non) existence be verified a posteriori?

• (We will see that) Some models allow for 0-time transitions, i.e., changes of system state during which time does not progress: is this a reasonable time modeling approach? (remember that we pointed out that time evolution often occurs at very different time granularities).

9. The Time Advancement Problem

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n1

• Consider a system whose state s evolves according to the function of time s(t) = sin(t2). How would you classify such a behavior? A Zeno behavior? A Berkeley behavior? None of them?

• An unbounded sequence of events occur each at time t1, t2, …, ti, ti+1, …, where

t1 = 0tk+1 = tk + dk for k ≥ 1

1. Define, if possible, a sequence of values d1, d2, d3, … such that the resulting sequence of events is:

2. Zeno and all events but the first occur at irrational times3. Zeno and all events occur at integer times4. Non-Zeno and Berkeley5. Zeno and Non-Berkeley6. Non-Berkeley and all events but the first occur at irrational times7. Non-Berkeley and all events occur at integer times

A couple of (real-time) exercises

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• Systems are composed of various modules• When such modules “run” concurrently various aspects of their

composition impact on their time behavior• Typical risks:

• Deadlock: the whole system is blocked• Starvation: some system component never progresses (either because it can’t

–it is blocked or dead- or because it could run but never gets permission to proceed)

• Two basic approaches to compose concurrent modules:• Synchronous• Asynchronous

• Two basic approaches to let concurrent modules communicate between each other:• Accessing shared resources• Message passing

10. Concurrency and Composition

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• In this course our attention is focused on modeling rather than on (algorithms and methods to) analyze and verify model properties.

• However, analyzability (e.g., decidability) is an important property on whose basis to evaluate and compare the overall quality of a model.

• A few typical qualities on which to evaluate them:• Expressiveness

• Mathematically defined• Subjectively evaluated (better called “naturalness” to avoid misunderstandings)

• Decidability and complexity (nowadays much attention devoted to this (these) aspect(s))

• Analysis and verification techniques• Static versus dynamic• Exhaustive/brute force (+ heuristic) vs. “clever”

11. The analysis and verification issue

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1. Revisiting traditional time modeling and analysis techniques at the light of the above dimensions (and not only)

2. To introduce more recent and sophisticated ones on the basis of an already solid background

3. At the end: summing up

4. Throughout: discuss, clarify, comment!

What next?

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