Formal Languages, Automata, Computation 1 Administrivia Klaus Sutner Carnegie Mellon University FLAC Fall 2019 Web & Piazza 3 Piazza 4 . is great It’s a good way to share information and get answers without http://www.cs.cmu.edu/~flac terrible delays. sucks http://piazza.com/ 15-453 It can be used to avoid work by asking lots of silly questions, and ⇒ relying 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. Course Staff 5 Course Material 6 Prof: Klaus Sutner, [email protected], No textbook, but lots of material on the web. http://www.cs.cmu.edu/~sutner If you absolutely want to have a textbook: TAs: Paul G¨olz, [email protected] M. Sipser Introduction to the Theory of Computation Cengage India, 2014 Course secretary: Rosie Battenfelder, [email protected] Sipser 7 Learning Style 8 The book is based on a great idea: for a complicated proof, first present The topics covered in this course translate into quite a bit of a fairly informal sketch, outlining the main ideas. material. Allocate enough time to deal with this material. Then get down into the weeds and produce a formal argument. Read the notes, search the net, go to the library, talk to each other, talk to us. Again, great idea, but in reality the difference between the informal and Post on piazza (but make sure to read previous posts first). the formal argument is a bit anemic. You really get a sketch, and then a detailed sketch. I know this is an obnoxious thing to say at this point, One of the desired outcomes of this course is that you know where but in the back of your mind always think about a theorem prover. to find more information should you ever need it (the lifelong learning meme). Samuel Beckett 9 Bureaucracy 10 Ever tried. Ever failed. The usual testing: homeworks 50% • midterm (in-house) 20% • No matter. Try Again. Fail again. Fail better. final 30% • Worstward Ho! There are no makeups; if you miss some assessment you can sit for an oral exam. Bureaucracy II 11 Preserving TA Sanity 12 Typeset your solutions to the homework and submit pdf on afs Midterm is in-house, 80 minutes on Oct 19. (more on this on Piazza). Final will be a project (think 8 pages), more later. If you have extensive conversations with other students about a HW, mention them as “collaborators” in your submission. Homework is obviously critically important. It’s 50% of your grade, so I trust you to do the work yourself. If you use a computer program in your homework, make sure to I hate having to play policeman. reference it properly (but do not hand in 50 pages of code). Lateness 13 Cooperation 14 You have a total of 6 (six) late days at your disposal; use prudently. Lectures will be warm and friendly. Make sure to be an active participant – FLAC is not a spectator sport. A late day is a discrete atom, with no smaller parts. You are strongly encouraged to talk about the course material to Mention lateness in the header of your HW. each other, the course staff and other students. I reserve the right to give you no credit at all for work submitted This includes discussions of homework problems. beyond the allotted time. Limits to Cooperation 15 More on Limits 16 Think of this as being able to copyright your solution: you may have However, even after ample consultation, the work you submit must had conversations about it with other during the problem solving be written entirely by yourself. phase, but the actual work is solely yours. No one should be in the room when you start writing things up. List all your “consultants” on the first page of your homework. We will provide a convenient template. Needless to say, you have to be able to explain all the details of your solution at any time. To avoid problems with originality, do not take notes when discussing homework problems. Don’t even think about copy & paste (from each other or the web), file sharing, clairvoyance, telepathy, . If you write on a board, erase everything in the end. If you have any questions about policy issues talk to the course staff, preferably some time before you get into trouble. Yet More on Limits 17 Email 18 Some of you may still remember email (a medieval communication tool, And, of course, all the official university policies apply. predating social networks). If you decide to get in touch via email, use Subject line: [FLAC] will miss midterm http://www.cmu.edu/academic-integrity/index.html or some such. I filter rather aggressively, make sure to have the [FLAC] tag. Wellness 19 Wellness II 20 Take care of yourself. Do your best to maintain a healthy lifestyle this semester by eating well, exercising, avoiding drugs and alcohol, getting enough sleep and taking some time to relax. This will help you achieve your goals and cope with stress. If you or someone you know is feeling suicidal or in danger of self-harm, All of us benefit from support during times of struggle. There are many call someone immediately, day or night: helpful resources available on campus and an important part of the college experience is learning how to ask for help. Asking for support CaPS: 412-268-2922 sooner rather than later is almost always helpful. Re:solve Crisis Network: 888-796-8226 If the situation is life threatening, call the police If you or anyone you know experiences any academic stress, difficult life On campus: CMU Police: 412-268-2323 events, or feelings like anxiety or depression, we strongly encourage you Off campus: 911 to seek support. Counseling and Psychological Services (CaPS) is here to help: call 412-268-2922 and visit their website at http://www.cmu.edu/counseling/. Consider reaching out to a friend, faculty or family member you trust for help getting connected to the support that can help. Formal Languages, Automata, Computation 22 This is the official course title for 15-453. Administrivia Mostly a historical artifact, a better title would be CAFL: Computation, Automata, Formal Languages 2 FLAC We’ll start with the general theory of computation, then dive all the way down to finite state machines, and then talk a bit about the Chomsky hierarchy and complexity theory. The Computational Universe 23 Theory First 24 This may sound like just a whole bunch of math. A (ironic?) fact of history: It is, but this is a CS course: we don’t live in the set-theory universe of The theory of computation predates computers. math, we live in the computational universe. In fact, computability theory was closely connected to problems in the What matters to us is not just the pure mathematical theory, but its foundations of mathematics in the 1930s, there simply was no computer computational meaning. In particular we want to emphasize algorithms. science at the time. Bear this in mind, otherwise things may occasionally seem a bit bizarre. As you will see, this makes life much more interesting, but also quite challenging: we need our arguments to be constructive and As it turns out, computability theory is highly relevant to computer computational, not just logically correct. science, but one needs a bit of background to see why. Early History 25 Eratosthenes 26 Around 240 BCE, error may have been as low as 1%. Also calculated diameter of sun, not as accurate (27 times Earth, in reality 109). Early mathematics was very much focused on computation (Plimpton 322, about 1800 BCE). Beginnings of Abstraction 27 Full Abstraction 28 Things changed when the field matured. For example, the main accomplishments in mathematics in the 19th century can be summarized like so: complex variables abstract groups set theory While such “top ten” lists are often contentious, this one is fairly uncontroversial. Note that computation is essentially absent, the level of abstraction is rather high (logical depth). Even More Abstraction 29 What Could Go Wrong? 30 Despite all the progress, in the last third of the 19th century a number of annoying problems came to light. If Gauss says he has proved something, it seems very probable to me; if Cauchy says so, it is about as likely as not; if Dirichlet says so, it is certain. Carl Gustav Jacob Jacobi For example, seemingly intuitive notions in analysis such as continuity and differentiability are much more subtle than they might seem. It is very, very easy to trip up. Levels of Trouble 31 Weierstrass Monster 32 f(x) = bn cos(anπx) n 0 X≥ paradox para doxa; outside of the doctrine, a misalignment with common sense 2 1.5 antinomy anti nomos; against the law, not just a misalignment, but 1 something offensive, in need of correction 0.5 contradiction contra dicere; to speak against, a direct counterargument, 0.5 1 1.5 2 2.5 3 all hell breaks loose (aka inconsistency) -0.5 -1 -1.5 This hierarchy is non-standard and entirely informal, don’t over-interpret. For 0 < b < 1 and ab > 1 + 3/2π, this function is continuous but nowhere differentiable. A mild paradox. Cantor’s Work 33 More Trouble in Paradise 34 Burali-Forti The unit square has the same size as the unit interval. The set of all ordinal numbers. There are more reals than rationals. Russell There is an infinite hierarchy of increasing infinities. S = x x / x { | ∈ } K¨onig,Richard Paradoxical, no more.