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Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017
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Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

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Page 1: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Formal Languages, Automata,

Computation

Klaus Sutner

Carnegie Mellon University

Fall 2017

Page 2: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

1 Administrivia

� FLAC

Page 3: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Web & Piazza 3

http://www.cs.cmu.edu/~flac

http://piazza.com/ ⇒ 15-453

Page 4: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Piazza 4

. . . is great

It’s a good way to share information and get answers withoutterrible delays.

. . . sucks

It can be used to avoid work by asking lots of silly questions, andrelying on others to do all the heavy lifting.

This is an upper lever class, don’t play games.

Also, don’t repost the same question a dozen times.

Page 5: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Course Staff 5

Prof:Klaus Sutner, [email protected],http://www.cs.cmu.edu/~sutner

TAs:Xiaorong Zhang [email protected]

Yue Niu [email protected]

Course secretary:Rosie Battenfelder, [email protected]

Page 6: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Course Material 6

No textbook, but lots of handouts.

Additional material will be posted on the web.

If you absolutely want to have a textbook:

M. SipserIntroduction to the Theory of Computation

Page 7: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Learning Style 7

The topics covered in this course translate into quite a bit ofmaterial. Allocate enough time to deal with this material.

Read the notes, search the net, go to the library, talk to each other,talk to us.

Post on piazza (but make sure to read previous posts first).

One of the desired outcomes of this course is that you know whereto find more information should you ever need it (the lifelonglearning meme).

Page 8: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Samuel Beckett 8

Ever tried. Ever failed.

No matter. Try Again. Fail again. Fail better.

Page 9: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Bureaucracy 9

The usual testing:

• homeworks 50%• midterm (in-house) 20%• final 30%

There are no makeups; if you miss some assessment you can sit foran oral exam.

Page 10: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Bureaucracy II 10

Midterm is in-house, 80 minutes on Oct 19.

Final will be a project (think 8 pages), more later.

Homework is obviously critically important.

Page 11: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Preserving TA Sanity 11

Typeset your solutions to the homework and submit pdf on afs(more on this on Piazza).

If you have extensive conversations with other students about a HW,mention them as “collaborators” in your submission.

If you use a computer program in your homework, make sure toreference it properly (but do not hand in 50 pages of code).

Page 12: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Lateness 12

You have a total of 5 (five) late days at your disposal; use prudently.

A late day is a discrete atom, with no smaller parts.

Mention lateness in the header of your HW.

Page 13: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Cooperation 13

Lectures will be warm and friendly. Make sure to be an activeparticipant – FLAC is not a spectator sport.

You are strongly encouraged to talk about the course material toeach other, the course staff and other students.

This includes discussions of homework problems.

Page 14: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Limits to Cooperation 14

However, even after ample consultation, the work you submit mustbe written entirely by yourself.

List all your “consultants” on the first page of your homework. Wewill provide a convenient template.

To avoid problems with originality, do not take notes whendiscussing homework problems.

If you write on a board, erase everything in the end.

Page 15: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

More on Limits 15

Think of this as being able to copyright your solution: you may havehad conversations about it with other during the problem solvingphase, but the actual work is solely yours. No one should be in theroom when you start writing things up.

Needless to say, you have to be able to explain all the details of yoursolution at any time.

Don’t even think about copy & paste (from each other or the web),file sharing, clairvoyance, telepathy, . . .

If you have any questions about policy issues talk to the course staff,preferably some time before you get into trouble.

Page 16: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Yet More on Limits 16

And, of course, all the official university policies apply.

http://www.cmu.edu/academic-integrity/index.html

Page 17: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Email 17

Some of you may still remember email (a medieval communication tool,predating social networks). If you decide to get in touch via email, use

Subject line:

[FLAC] will miss midterm

or some such. I filter rather aggressively, make sure to have the [CDM]

tag.

Page 18: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Wellness 18

Take care of yourself. Do your best to maintain a healthy lifestyle thissemester by eating well, exercising, avoiding drugs and alcohol, gettingenough sleep and taking some time to relax. This will help you achieveyour goals and cope with stress.

All of us benefit from support during times of struggle. There are manyhelpful resources available on campus and an important part of thecollege experience is learning how to ask for help. Asking for supportsooner rather than later is almost always helpful.

If you or anyone you know experiences any academic stress, difficult lifeevents, or feelings like anxiety or depression, we strongly encourage youto seek support. Counseling and Psychological Services (CaPS) is here tohelp: call 412-268-2922 and visit their website athttp://www.cmu.edu/counseling/. Consider reaching out to a friend,faculty or family member you trust for help getting connected to thesupport that can help.

Page 19: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Wellness II 19

If you or someone you know is feeling suicidal or in danger of self-harm,call someone immediately, day or night:

CaPS: 412-268-2922Re:solve Crisis Network: 888-796-8226If the situation is life threatening, call the policeOn campus: CMU Police: 412-268-2323Off campus: 911

Page 20: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

� Administrivia

2 FLAC

Page 21: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Formal Languages, Automata, Computation 21

This is the official course title for 15-453.

Mostly a historical artifact, a better title would be

CAFL: Computation, Automata, Formal Languages

We’ll start with the general theory of computation, then dive all the waydown to finite state machines, and then talk a bit about the Chomskyhierarchy and complexity theory.

Page 22: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

The Computational Universe 22

This may sound like just a whole bunch of math.

It is, but this is a CS course: we don’t live in the set-theory universe ofmath, we live in the computational universe.

What matters to us is not just the pure mathematical theory, but itscomputational meaning. In particular we want to emphasize algorithms.

As you will see, this makes life much more interesting, but also quitechallenging.

Page 23: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Theory First 23

A (ironic?) fact of history:

The theory of computation predates computers.

In fact, computability theory was closely connected to problems in thefoundations of mathematics in the 1930s, there simply was no computerscience at the time. Bear this in mind, otherwise things may occasionallyseem a bit bizarre.

As it turns out, computability theory is highly relevant to computerscience, but one needs a bit of background to see why.

Page 24: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Early History 24

Early mathematics was very much focused on computation (Plimpton322, about 1800 BCE).

Page 25: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Eratosthenes 25

Around 240 BCE, error may have been as low as 1%. Also calculateddiameter of sun, not as accurate (27 times Earth, in reality 109).

Page 26: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Beginnings of Abstraction 26

Page 27: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Full Abstraction 27

Things changed when the field matured. For example, the mainaccomplishments in mathematics in the 19th century can be summarizedlike so:

complex variables

abstract groups

set theory

While such “top ten” lists are often contentious, this one is fairlyuncontroversial.

Note that computation is essentially absent, the level of abstraction israther high (logical depth).

Page 28: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Even More Abstraction 28

Page 29: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

What Could Go Wrong? 29

Despite all the progress, in the last third of the 19th century a number ofannoying problems came to light.

If Gauss says he has proved something, it seems very probableto me; if Cauchy says so, it is about as likely as not; if Dirichletsays so, it is certain.

Carl Gustav Jacob Jacobi

For example, seemingly intuitive notions in analysis such as continuityand differentiability are much more subtle than they might seem. It isvery, very easy to trip up.

Page 30: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Levels of Trouble 30

paradox para doxa; outside of the doctrine, a misalignment withcommon sense

antinomy anti nomos; against the law, not just a misalignment, butsomething offensive, in need of correction

contradiction contra dicere; to speak against, a direct counterargument,all hell breaks loose (aka inconsistency)

This hierarchy is entirely informal, don’t over-interpret.

Page 31: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Weierstrass Monster 31

f(x) =∑n≥0

bn cos(anπx)

0.5 1 1.5 2 2.5 3

-1.5

-1

-0.5

0.5

1

1.5

2

For 0 < b < 1 and ab > 1 + 3/2π, this function is continuous butnowhere differentiable. A mild paradox.

Page 32: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Cantor’s Work 32

The unit square has the same size as the unit interval.

There are more reals than rationals.

There is an infinite hierarchy of increasing infinities.

Paradoxical, no more.

Remarkably, Cantor was lead to his discoveries by studying Fourieranalysis, a relentlessly applied part of mathematics.

Page 33: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

More Trouble in Paradise 33

Burali-Forti

The set of all ordinal numbers.

Russell

S = {x | x /∈ x }

Konig, Richard

The least natural number not definable by at most 100 words.

Some actual antinomies, easily leading to biting contradictions.

Page 34: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Bad Proofs 34

The Four Color Theorem was first conjectured in 1852 by F. Guthrie: anymap planar can be colored with just four colors.

A. Kempe published a “proof” in 1879, but an error was found by P.Heawood in 1890. Heawood gave a correct proof of the weaker FiveColor Theorem.

An alternative “proof” was given in 1880 by P. G. Tait, but his argumenttoo had a bug, discovered in 1891 by J. Petersen.

The first major step forward came in 1969 when H. Heesch managed toshow that the proof could be reduced to checking finitely many cases.This opened the door for a computer-assisted proof: Appel and Haken1976, Gonthier 2006.

Page 35: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Grundlagenkrise 35

Adding up all these little insults and injuries to mathematical thoughtultimately added up to a major crisis: it was no longer clear thatmathematics was the ultimate standard for reliability and precision, thequeen of the sciences that dwarfed all other efforts.

To be sure, most mathematicians blithely ignored these issues, they weremore than happy to apply a highly developed machinery that couldapparently dismember problems of any complexity. Math worked, andwho cares about philosophical objections?

Page 36: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Hilbert and Poincare 36

Page 37: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Blowback 37

Formerly, when one invented a new function, it was to fur-ther some practical purpose; today one invents them in orderto make incorrect the reasoning of our fathers, and nothingmore will ever be accomplished by these inventions.

H. Poincare

Page 38: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Hilbert’s Program 38

. . . dispose of the foundational questions in mathematics assuch once and for all.

Formalize mathematics and concoct a finite set of axiomsthat are strong enough to prove all theorems of mathematics(completeness) and show that the system is consistent; bystrictly finitary means. Also show that statements about“ideal objects” can be proven in the system, without usingideal objects.

D. Hilbert, 1929

Poincare and Hilbert were arguably the two top mathematicians around1900; it is remarkable that their attitude towards foundational issues wasso diametrically opposed.

Page 39: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Hilbert’s Entscheidungsproblem 39

The Entscheidungsproblem is solved when one knows a pro-cedure by which one can decide in a finite number of oper-ations whether a given logical expression is generally validor is satisfiable. The solution of the Entscheidungsproblemis of fundamental importance for the theory of all fields, thetheorems of which are at all capable of logical developmentfrom finitely many axioms.

D. Hilbert, W. AckermannGrundzuge der theoretischen Logik, 1928

In modern terminology: Hilbert wanted a decision algorithm, more or lessfor all of mathematics.

Page 40: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Rebuilding Foundations 40

In a nutshell, Hilbert’s project seems to require at least the followingingredients:

Formalization of mathematics. This requires in particular a formallogical system: a formal language, rules of inference, notion of proof,axiom systems that describe our domains of discourse.

A completeness proof that shows that the system can prove all trueassertions.

A consistency proof that shows that the system cannot prove acontradiction (soundness of the system).

A decision algorithm that determines whether assertions in thesystem are true or false.

Page 41: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

The Trouble with Formal Systems 41

They don’t fit very well with the way actual humans conjecture, argue,think and reason.

Every little detail of an argument has to be spelled out withmind-numbing precision. As a result, proofs get to be very, very long,and are more or less incomprehensible (to humans, at least).

As a consequence, most mathematicians happily ignore logic and formalsystems. But note that writing a computer program is not much differentfrom writing a formal proof: the same, utterly annoying level of precisionand detail is required.

Page 42: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Gottlob Frege (1848-1925) 42

Page 43: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Frege’s Books 43

Begriffsschrift 1879 Introduces a surprsingly modern and careful systemof logic (except for notation).

Grundlagen 1884 Critique of previous foundational efforts, outline of hisown project.

Grundgesetze 1893/1903 Develops theory of arithmetic and analysis inhis system. Ruined by Russel’s discovery.

Page 44: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Frege in Attack Mode 44

He criticizes others (Dedekind, Peano) for the lack of precision in theirwork.

It cannot be demanded that everything be proved, becausethat is impossible; but we can require that all propositionsused without proof be expressly declared as such, so thatwe can see distinctly what the whole structure rests upon.After that we must try to diminish the number of primitivelaws as far as possible, by proving everything that can beproved. Furthermore, I demand–and in this I go beyondEuclid–that all methods of inference employed be specifiedin advance.

This is from “Grundgesetze der Arithmetik I,” published in 1893.

Page 45: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Alas . . . 45

Q(y)

P (x, y)

Q(x)

b a Q(a)

P (b, a)

Q(b)

Q(y)

P (x, y)

a Q(a)

P (x, a)

From Frege’s 1879 “Begriffsschrift” (concept script).

Page 46: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Principia Mathematica 46

The next large-scale attempt to provide a formal foundation for a largefragment of mathematics was undertaken by by Russell and Whitehead,leading to publication of three volumes 1910–1913.

PM was a remarkable effort and absolutely groundbreaking. Alas, boththeir notation system and their ontological assumptions were notsustainable. Russell later complained that his intellect

“. . . never quite recovered from the strain of writing it.”

Page 47: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Principia Mathematica 47

Page 48: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Progress Surrounding Hilbert’s Program 48

1889 Peano gives an axiomatization of basic number theory.

1899 Hilbert gives an axiomatization of basic geometry.

1910s Russell and Whitehead build a fairly comprehensive formalsystem of mathematics based on types.

1918 Bernays shows that propositional logics is sound andcomplete.

1929 Godel shows that first-order logic is sound and complete.

Page 49: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

A Catastrophe 49

The Incompleteness Theorem was announced by Godel on October 7,1930, at a conference in Konigsberg, the “First International Conferenceon the Philosophy of Mathematics”:

There is a formula of number theory such that Peano arith-metic proves neither the formula nor its negation.

Since either the formula or its negation must be true, we are missing outon a true statement of arithmetic.

It seems that the only person in the audience who understood what wasgoing on was von Neumann. Hilbert was in Konigsberg, he gave his“werden wissen” speech the next day, but apparently he did not attendGodel’s lecture.

Page 50: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Computation 50

One key idea in Godel’s proof is arithmetization: one can expressformulae and proofs, plus all necessary manipulations, in terms ofarithmetic. (The other is the Liar’s Paradox.)

The operations on numbers representing formulae are all easilycomputable.

Since they are easily computable, they can be handled by any formalsystem that is is expressive enough to cover some basic aspects ofarithmetic.

Note the irony: computability helps to demolish Hilbert’s dream, whichbasically says everything is computable.

Page 51: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Computer Science 51

“The standard of correctness and completeness necessary toget a computer program to work at all is a couple of ordersof magnitude higher than the mathematical community’sstandard of valid proofs.”

W. Thurston

To a programmer, all this may sound oddly familiar: mind-numbingprecision is also required in writing programs. In fact, much of thecomputability/complexity/automata machinery that is relevant for theGrundlagenkrise is also directly relevant to CS.

Page 52: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Sequencing 52

We will follow the historical chain of events:

1930s Computability theory

1950s Automata theory

1970s Complexity theory

Why? There are good reasons things happened in this particular order.

Page 53: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Computability 53

There are several problems we need to tackle:

Give a precise and robust definition of computability.

Provide evidence that our definition adequately reflects the intuitivenotion of computability.

Study the basic properties of computability and prove the mainresults in the area.

Develop tools to show that some problems are computationallysolvable or fail to be so solvable.

Do the same for a theory of tractable problems and practicalcomputation.

Page 54: Formal Languages, Automata, [1ex] Computationflac/pdf/lect-01.pdf · Formal Languages, Automata, Computation Klaus Sutner Carnegie Mellon University Fall 2017

Turing’s Machines 54