CSC280/480, Spring 2023: Computer Models and Limitations
Instructor:
Prof. Lane A. Hemaspaandra.
Grad TAs:
Hanjia (Bruce) Lyu
and
Hecong (Alan) Wang.
Workshop (though called Recitations in CDCS) Leaders:
Ethan Ferland,
Ian Harding,
and Henry Welles.
Office Hour/Class TAs:
Vuong Ho, Zihao (Ziggy) Lin, and Quan Luu.
The Course Catalog Description:
This course studies
fundamental computer models and their computational
limitations. Finite-state machines and pumping lemmas, the
context-free languages, Turing machines, decidable and
Turing-recognizable languages, undecidability, and NP-completeness.
Textbook:
Introduction to the Theory of Computation, Michael Sipser,
third edition, 2013.
Be careful
not to accidentally get the first or second edition.
Note that we will be using
the (standard, USA-version)
third edition, not the "international" (third) edition;
in the international edition apparently
some problem numbering differs, so if you use the international
third edition (which you might get if you
buy a used one, so be careful), you
may
do the wrong problems!
(And no, I don't work for the publisher!)
The ISBN of the US edition (the international edition
has slightly different ISBNs) is listed by the publisher as:
ISBN-10: 113318779X and ISBN-13: 9781133187790.
You will be using
the book very, very often (you may find it
comforting to sleep with it---or your Kindle or whatever if you
get it in eTextbook form---under your pillow!---warning: don't
actually sleep with your Kindle under your pillow).
Prerequisites (very important and required): CSC173 and MTH 150.
Course Information Document/Syllabus:
course information document/syllabus (most current version, V. 1.1.2).
List of Our Coming Items (including all ROHs and also those OOHs
that are OOH/ROHs or that are Zoom-based pure OOHs)
If They Will Be Reached Via Zoom Meeting Links/Passwords:
Available
via a link in our Blackboard area, via, within "BB Parts
of Course Content," the link "Zoom Meeting Manager: Your springboard
to all our Zoom meetings."
Slides:
The main slide-set is
available in
the
"BB Parts of Course Content" part of our Blackboard
area.
Workshop Assignments (who is in which): These were set by
you as you registered via the registrar's system; each of you must
be registered for exactly one of the eight workshops (which are
called "recitations" in
registrar terminology).
ROH Assignments (who is in which):
The ROH assignments can be found inside our BB area, via an
item/link within our folder
BB Parts of Course Content.
Which Weeks Are ROH Weeks:
Whether a week will be an ROH
week will be posted here (sometimes very) shortly before the
given week, as ROH/nonROH-ness is dynamically decided based on pace/flow and
other factors.
-
Week 1 (Jan. 11-13): nonROH. (But, as always unless otherwise
explicitly
mentioned in this list, there WILL be your weekly recitations.)
But, as a very special, one-time case for this first week, you can
get attendance even for attending WS other than your home WS.
-
Week 2 (Jan. 16-20): nonROH. (Note: No class, office hours, etc. on 1/16, since it is MLK Day; except Hanjia and I will have a 615-715PM 1/16 Zoom office hour that you will be able to find the link to in our Zoom Meeting Manager on or before 1/16.)
-
Week 3 (Jan. 23-27): ROH. (Yes, the first ROH of the term!
Note:
And some TAs will likely be in class on Wednesday. Please make
sure to sit together in groups of between 3 and 5 people in class
on Wednesday, even from the very start of the class session, as
you'll be working in groups.)
-
Week 4 (Jan. 30-Feb. 3): ROH.
Note:
And some TAs will likely be in class on Monday. Please make
sure to sit together in groups of between 3 and 5 people in class
on Monday, even from the very start of the class session, as
you'll be working in groups.)
-
Week 5 (Feb. 6-10): nonROH.
-
Week 6 (Feb. 13-17): ROH.
-
Week 7 (Feb. 20-24): ROH.
-
Week 8 (Feb. 27-Mar. 3): Complex, special-case week:
- Wednesday, 3/1, 325PM (sharp!) Eastern
Standard Time is
Midterm 1, which is in-class (and there is
in general no makeup exam, see the syllabus for details).
-
No WSs and no WS attendance grades this week.
-
No ROHs and no ROH attendance grades this week. **But most or all ROHs
and OOH/ROHs
that are on Monday or Tuesday or earlier in the exam on Wednesday
will be offered this week as
to pure OOHs; the way you can tell which ones exist
is they will be listed in ZMM, but do be aware that within
ZMM, due to ZMM limitations,
they will still have their standard names (OOH, ROH, OOH/ROH), but
to be clear:
from 8AM 2/27 through 325PM 3/1, every OOH, ROH, and OOH/ROH in ZMM
that at its start time is still listed in ZMM will exist as a
special OOH (in fact, in concept by end-of-day 2/24
the TAs will have deleted
the nonoffered things in the after-exam part of Week 8,
and will have deleted any OOH, ROH, and OOH/ROHs in the first
part of the week that won't be given as OOHs, though as I write
this on the morning of 2/24 I know of no such drops yet during
the first half of that week).
-
No office hours this week from 325PM Wed. onward except Bruce
will have his Wed., March 1, 615-715PM Zoom OOH and Alan will
have his Wed., March 1, 740-840PM Wegmans 2215 OOH.
-
But on Mon., on Tues, and on Wed. before 325PM there will be
a pretty large list of pure office hours
(please
do use them if you need help in preparing!!!!); the list (plus
Bruce and Alan's Wed. evening ones mentioned above) is
as follows:
- See two items earlier for the slew of ROHs and OOH/ROHs
that this week will be given but will be available
(despite still being named ROHs or OOH/ROHs)
as OOHs on Monday,
and Tuesday, and a prefix of Wednesday. Again, see particularly
ZMM (any time starting the morning of *2/25*, but
then recheck closer to the day of the items you wish to
attend) to see the many things
available to attend that will function during that special
half-week as OOHs (even if named ROHs or OOH/ROHs in ZMM).
- Alan's (450PM WH 2215) and Bruce's (615PM zoom)
regular Monday OOHs will exist, and I will shift my
regular Monday office hour to be a bit earlier
namely it will be, by zoom via ZMM, 535PM-635PM; so between
those and the series of 730PM-10PM office hours of
Ziggy, Vuong, and Quan, there will be office hours
on Monday from 450PM to 10PM (except with a 15-minute
hole between 715PM and 730PM).
And
Alan's (615PM WH 2215) regular Tuesday OOH will exist.
- And as a reminder, as mentioned above, Alan and Bruce will
each have Wednesday evening office hours. But those are
the only office ours after Midterm 1 until the Monday
after Spring Break (so for example Ethan's Friday March 3 office
hour will not exist this week---nor will any office
hours on Thursday
or Friday of Week 8).
-
Week 9 (Mar. 13-17): ROH.
-
Week 10 (Mar. 20-24): ROH. (Warning: Week 11 will probably
NOT be an ROH week. So please do not miss the opportunity
THIS week---Week 10---gives you for the small-group-setting
feedback of an ROH, especially regarding proving things
undecidable! That is, attending ROH this week is even more
extremely important than usual.)
-
Week 11 (Mar. 27-March 31): ROH. (Yes, in a change from
what I was planning, this too is an ROH week. It is *that*
important that everyone master proving things undecidable!)
In fact, FYI, the main class lecture moves on to
complexity on Wed., March 29, but I've extended to an extra
week---this week---the focus of the WSs on undecidability,
as mastering that is *that* important, as well as
also having ROHs on that this week.)
In addition to the work on undecidability
at the WSs and the ROHs this week,
if after hearing our Monday 3/27 class lecture
and then reviewing the content of that
lecture that evening (it will review
the tools you have for classifying problems wrt
decidable/Turing-recognizable/co-Turing-recognizable and
you'll work a number of examples) you have not
yet become completely comfortable with proving things
undecidable via the template method, or with classifying
problems regarding whether they are decidable? Turing-recognizable?
co-Turing-recognizable?, then please use the week's
daily office hours
to come in and work on that!
-
Week 12 (Apr. 3-7): ROH.
-
Week 13 (Apr. 10-14): nonROH. IMPORTANT NOTE: As usual: (a) for
Zoom office hours this week, see
the long listing in ZMM
(most office hours this week
are via Zoom and there are A LOT of them since this is an nonROH
week, so please do use
them
to learn the material and prepare for Midterm 2; keep
in mind that since it is a nonROH week, each ZMM entry
marked EITHER OOH or OOH/ROH will exist this week!))
and (b) for in-person office
hours this week,
see the office hours listing in the syllabus.
(WARNING: So, as usual, if you look in
ZMM for this week, you will see all the Zoom hours,
but you won't see the in-person hours as they do not have a
ZMM entry, since they are in person; but they are an additional
resource to use, so please don't fail to, if you wish,
use them).
-
Week 14 (Apr. 17-21) Complex, special-case week but *IT IS AN ROH WEEK*:
- Wednesday, 4/19, 325PM (sharp!) Eastern
Standard Time is
Midterm 2, which is in-class (and there is
in general no makeup exam, see the syllabus for details).
-
No WSs and no WS attendance grades this week.
-
ALL OFFICE HOURS AND ROHS AND OOH/ROHs AND OOHs WILL BE AS IN EVERY TYPICAL
ROH WEEK EXCEPT ALAN'S IN-PERSON THURSDAY 450PM OFFICE HOUR IS
CANCELED.
So Week 14 is a ROH week. But it is a special one in the following
sense. As usual,
your ROH ATTENDANCE GRADE for the week comes ONLY from attending
your signed-up-for home workshop. But in
most weeks, students can attend extra tutorials only if the tutorial
leader allows them to, though they usually do;
but for this week, I have let the ROH leaders know to
allow any students
who show up to their tutorial to attend, so
no one should be turned away from a
tutorial (so if your home ROH is later in the week than Midterm 2,
note that you CAN in addition to your home ROH attend an earlier
one or ones if you wish; even if your home ROH is before Midterm 2
you can still attend multiple ROHS, of course, so
get in extra practicing, or
to ask questions). But again, as a reminder, your ROH attendance grade for the week comes
only from attending your home ROH.
-
Week 15 (Monday Apr. 24-Wednesday Apr. 26; last week of classes) Special-case week:
-
We have our final two class sessions: April 24 and April 26. Exciting
things will be covered (white rabbits, cheshire cats, counting,
elections, and
parallelism)!
-
No WSs and no WS attendance grades this week.
-
No ROHs and no ROH attendance grades this week.
-
For this week, as to what office hours exist,
the office hour listing in the syllabus does not apply, but,
rather, the following is the complete listing of what office
hours there are and how to get to them
(WARNINGS: (1) if you look in
ZMM for this week, you will see Bruce's and my Zoom hours listed below,
but you won't see the in-person hours of Alan as they do not have a
ZMM entry, since they are in person, but they---the ones listed
below---will indeed exist; (2) at the moment I'm writing this,
ZMM still shows entries for Week 15 for Quan, Vuong, and Ziggy,
but those sessions will not exist and the ZMM entries from them
should disappear soon, namely as soon as Quan/Vuong/Ziggy delete those
entries) (note: Alan's 4/25 office
hour is the course's final office hour of the term):
Monday 450PM--550PM, Alan, OOH, Wegmans 2215
Mondays 615PM--715PM Bruce OOH, see Zoom link in ZMM
Monday 615PM--715PM Lane OOH, see Zoom link in ZMM (note as per syllabus may end early if no one is there)
Tuesday 615PM--715PM, Alan, OOH, Wegmans 2215
Reading Assignments, starting with the date/time the reading is DUE:
- Due 2023/1/18/325pm (assigned on 2023/1/10):
Read the Course Information Document/Syllabus. (I know
it is long, but it is quite important. It contains many rules
regarding academic honesty as to what is and is not allowed
(and you are required to know and follow those rules), and
also it contains lots of logistic information about how the
course is run (and you are required to know and follow those
logistics), and also it contains detailed information (though
this probably falls also under logistics) on when the office
hours are, and more generally it tries to convey something of
the varying flavors that the workshops and ROHs will have
and of the natural and goals of the course. Even if most of that
leaves you cold (though I hope it does not),
the Academic Honesty issue should not; it is a
really, really bad, painful
topic on which to have problems, and the best
way to avoid problems is to---plus you are required to---read, know, and follow the
rules of the course... while of course also knowing the
general AS&E-wide Academic Honesty rules and framework---you
can find a link to the school-wide Academic Honesty site in the
"Other Important Links/Info/Resources" section of this page.)
- Due 2023/1/18/325PM (assigned 2023/1/12)]:
In SIP ("SIP" will be short for our Sipser, 3rd
edition, book) read Chapters 0, and
Chapters 1.1, 1.2, and 1.3.
Somewhat unusually, since I usually have the
reading trail the class/slides, here the reading
is leading the slides. Why? The reason is that
this reading is reviewing some
basics of discrete math/etc., and then is covering
work on finite-automata theory that was covered extremely
carefully in CSC 173. If you took CSC 173 very recently,
this reading may go relatively quickly and smoothly for you.
If you took CSC 173 long ago, your memory of CSC 173
may have faded, and so---although Sipser is
a beautifully written, remarkably readable book---the reading may
not be
as quick and smooth as if you did remember it very well, and so
to help you best
get you back to where you got to in CSC 173 on some
of these issues you might
want to partner with classmates to go over the
material. Also, the WSs in Week 2 (the Jan. 18-19 WSs)
will go over examples
of some of the more interesting constructions in that
reading (perhaps
the WS might go over turning NFAs to DFAs (aka the subset construction)
and turning DFAs into regular expressions (using GNFAs as a
key tool); in concept, you should remember well NFAs and regular
expressions (since you had them in 173), but if you
don't remember them well (or even if you do!), please do have read
in SIP the definitions of NFAs and regular expressions
before your WSs (fortunately, you will, given the due date above,
have done that!), and indeed
have done the whole reading before your WSs
(note: DFAs/NFAs/RegularExpressions are all
covered in our 173 prereq so with luck they are old friends to you... but
I realize that memories can fade with time... thus this reading to
bring back those warm memories of 173).
- Due 2023/1/23/325pm (assigned on 2023/1/18): (a) Please review
slides 1-123 of our first slide-set (side note: actually,
whenever we cover slides in class, you always should review them
after class, and so in general I won't list so as an explicit
assignment as it is globally an implicit assignment; but sometimes
I might make it explicit for extra emphasis, e.g., I am doing it
here as any worries you have during that review will, see later
in this sentence itself, indicate things you should re-read
in the SIP reading that was due today) (note: you can find the slide set
in our BB area), and review any of the
above-assigned Sipser reading that is on things you are not yet
fully comfortable with within those slides.
- Due 2023/1/25/325pm, but actually, reading it right away
would be a good idea (assigned on 2023/1/23): Read SIP section
1.4. This is generally what we covered in class today, so if
you were totally comfortable with and understand today's
lecture, you probably can read it quickly or just skim it, since
you already will (if the "if" part of this sentence holds)
understand it. But if some parts of today's lecture were not
clear or went by too quickly, you may wish to read this section
quite carefully.
Also, separately,
please review carefully (from earlier in SIP) pages 53-54, namely,
the notion/formalizing of an NFA, so know its tuple
definition, and know what each of the tuple's five parts is
(most esp. know the type of the transition function); in class
on 1/25, you'll be doing a (perhaps challenging) hands-on
exercise for which it will be important for you to come in with
a comfortable understanding of NFAs and their notation/formalism.
- Due 2023/2/6/325pm, but actually, reading it as soon after our 2/1
class as possible would be a
very good idea (assigned on 2023/2/1):
Read SIP
pages 101 through the middle of page 107
(which you
should generally know from CSC 173),
and Def. 2.8 and
the statement (you do not need to read the proof) of Theorem 2.9.
Read SIP Section 2.3!!!
If you don't remember SIP 2.2 from CSC 173 (where it typically
is outright assigned as a reading---not even from the Aho-Ull FOCS
book but
directly from SIP in fact), then
read SIP Section 2.2 (which covers what a PDA is
and why the set of languages accepted by them are exactly
the CFLs). You do NOT need to read SIP Section 2.4.
But please do carefully review our slides on SIP
Chapter 2, and you'll likely want to review especially
carefully the proof that the "a+b=c"-inspired set
is not a CFL.
- Due 2023/2/15/325pm. (Assigned 2023/2/13.) Read the
"Midterm 1 Scope Page" that is linked to
from the Other Important Links area below.
- Due 2023/3/13/325pm (assigned on 2023/2/22
but it is reasonable and fine to wait until after Midterm 1 if you are busy studying for that): Read SIP
Chapter 3 and Chapter 4.1.
(Note: This same reading---in fact indeed also
from SIP---is typically assigned in CSC 173 by George,
so with luck most or all of this reading will be familiar
to you.)
- Due 2023/3/20/325pm (assigned on 2023/3/15): (i) Read SIP Chapter
4.2 and 5.1. (ii) And,
when you do the reading of Chapter 5.1, also
read very carefully the document I've linked to below via an
"Other Important Links/Info/Resources" link called: "A likely very useful
template (regarding the task `prove a problem undecidable by
contradiction'): template and a worked example of using the
template," paying particularly close attention to (the whole
thing, definitely including the
worked example. It sets up a framework that may
make a bit easier the challenge of proving
languages undecidable.
- Due 2023/3/22/325pm
(assigned on
2023/3/20): Read SIP Chapter Chapter 5.3 except skip
reading Example 5.25 except do note (as is hidden in Example
5.25's pointer to Exercise 5.6) that \(\leq_m\) is
transitive, namely,
if \(A\leq_m B\) (say via f) and \(B
\leq_m C\) (say via g), then \(A \leq_m C\) via the
reduction \(\rm h(x) = g(f(x))\) (see the proof on page 242
of SIP for more details).
Re-reminder: If you have yet not read the "Other Important
Links" document "A likely very useful template (regarding the task `prove a problem undecidable by contradiction'): template and a worked example of using the template" (or if you read the template but did not yet
read carefully the
and the worked example it provides,
please, please make sure to do so right away; that
template/framework is central to our SIP-world approach to
proving things undecidable.
- Due 2023/4/3/325pm (assigned on 2023/3/29):
In SIP read (a) from the start of section 7.2 through
3/5 of the way down page 286 (though and including the words
"practical utility"), and (b) sections 7.3,
7.4, and from the start of 7.5 through and
including the end of page 313
(WARNING: SIP in Def. 7.18 gives as the definition
of NP something that is typically taken as a characterization
of NP that follows from the standard definition, and then he as
Theorem 7.20 turns what is typically the definition of NP into
something that is framed as a characterization of NP; in this
course, though, we will take, as our definition of NP the
standard one, namely, the class of languages accepted by
nondeterministic polynomial-time Turing machines)
(NOTE: in section 7.4 you'll read the proof
of the Cook-Levin Theorem, Theorem 7.37;
it is hard and it is worth having read, but I won't
be testing you, on Midterm 2, on the details
of the proof of that particular theorem, though
of course the theorem itself you should know).
- Due [various times, see the statement of
what is being assigned] (assigned on 2023/4/10):
As we go through our Power of Self-Reducibility unit, if in a given
class we go over challenge problem FOO's solution,
then please between that class and the next (unless you
utterly, totally understood the solution reached in
class) read the in-writing solution to challenge
problem FOO that you can find as the appropriate appendix
of the technical-report
https://arxiv.org/pdf/1902.08299.pdf. But do not read ahead to future
challenges or their solutions, as that would cheat you
of the chance to work on those and learn from trying
to solve them. So between the 4/10 class and the 4/12
class you should read Appendix A, and after the 4/12 class
you should read Appendix B (and perhaps---though only if we completed
Challenge Problem 3 in class on 4/12 or on 4/12 I ask you
to read Appendix---even Appendix C).
- Due 2023/4/14/1159pm. (Assigned 2023/4/12/823pm.)
Read the Midterm 2 Scope page that is linked to
from the Other Important Links area below.
- Due in part 2023/4/17/1159PM (if you see it in time
or else as soon thereafter as possible), and in part the morning of
April 19th (assigned on 2023/4/17/926pm). Read the posted-below
version of Midterm 2's
instructions (they are linked to from later on this same page,
in the "Other Important Links" section).
Read them now (April 17th) if possible (or if you see this
too late then as soon as possible thereafter) and then (so that you catch any
updates, and so it is fresh in your mind while you are taking
the exam), read them again either the evening before Midterm 2
or the morning of Midterm 2 (i.e., the morning of April 19th)
Exercises/Problems in cases when you are expected to work on
them before the workshop or ROHs (or class sessions)
at which you'll be going
over their solutions (all exercise and problem numbers are from
SIP unless otherwise stated) (note: for some or many workshops
(resp., ROHs), all the problems will be given and worked on
at the workshops (resp., ROHs), but THIS listing is cases
where you need to, before the workshop (resp., ROHs), do some
exercises or so on---which often some WS members will then be
asked to put up the answers to (resp., some ROH members will
be asked to present the answers to)):
- Due 2023/1/30/325pm (assigned 2023/1/25):
This is a (hard but easy---more on that later)
home exercise. As with all such, it won't be collected
or graded. (In the somewhat unlikely case that some WSs this
week feel that the pumping lemma is totally nailed down, they
might move on to doing this. But probably, all or almost all WSs
will not get to this... and so it truly will be a home exercise.)
But this is a type of problem that you saw one example of
today in class. But I think one example is not enough, and so
I think
people much need more practice on this flavor of
problem, and this exercise is such practice.
It is important you put time in (either alone or in groups)
on this exercise over the weekend, since just seeing me
present a solution in class on 1/30 won't improve your skills
on this type of problem nearly as much as if you work on
the problem yourself this weekend; even if you try but fail to solve
it, you'll be far, far more likely to understand the solution
I put up than if you did not try to solve it.
We will not be working in groups or individually on this
problem in class on Monday; you are expected to have already
worked on it over the weekend. And in class, I'll present a
solution. (If you feel already completely confident on
how to solve this type of problem, and can flawlessly write
down a 5-tuple that solves this with the approach that
is analogous to what we did in class on 1/25 for first
halves of regular sets, then challenge yourself by trying
to solve this without any "moving backwards" being used
as part of your solution; hint: to achieve that, you may have to
change from having your state set be \(\{q_0'\}\cup Q^2\) to
instead being \(\{q_0'\}\cup Q^3\).)
As always, try to first come up with a big-picture
idea for the strategy/approach of your solution (think fingers!). And
THEN go about implementing that, via building the
appropriate 5-tuple defining your NFA for the first-fifths language
of regular set L.
(It is not at
all unlikely that problem(s)/issue(s) related to this problem
could appear, for example, on Midterm 1.)
Here is the problem.
Show that for each regular set L,
the language FirstFifth(L) is regular.
FirstFifth(L) is defined as
\(\{ x ~ \mid ~ (\exists y) [ |y| = 4|x| \land xy \in L]\}\).
To solve this, you should note that if L is regular, there
is a 5-tuple giving a DFA accepting L. And then you should
show how to, in terms of the components of that 5-tuple,
define (as a new 5-tuple!) an NFA L' that accepts
FirstFifth(L).
As a hint (this hint is specifically about the solution that as its
state set uses \(\{q_0' \cup Q^2\}\)): Please review (via the video
for example) the in-class construction I did on
1/25 for first halves... because (HINT!!!!!)... you can use basically
that entire construction... you only have to adjust, and not even
very much at all (!!), one of the five components of the
5-tuple that is defining the NFA. (This hint is why I say that
this hard problem is also easy; it is not far from something
you know... you just have to try to understand things enough to
realize what change needs to be made.)
- Due 2023/2/6/325pm, but this may take you a while, and
working on it will likely really help you understand this
type of construction, so please start it a few days before
that... working on it joint with others is totally fine,
by the way (assigned on 2023/2/1):
Show that for each regular set L,
the language MiddleThird(L) is regular.
MiddleThird(L) is defined as
\(\{ x ~ \mid ~ (\exists w) (\exists y) [ |w| = |x| = |y|
\land wxy \in L]\}\).
To solve this, you should note that if L is regular, there is
a 5-tuple giving a DFA accepting L. And then you should show
how to, in terms of the components of that 5-tuple, define (as
a new 5-tuple!) an NFA L' that accepts MiddleThird(L). You of
course will want to first develop a strategy for your proof:
how many "fingers" will you use and what will each be doing?
But also, by now, you really need to be getting comfortable
with moving from a mental model of an NFA to a carefully
written and fully correct 5-tuple that defines the NFA. If
you're having trouble with this type of problem, or writing
5-tuples, please stop by (note: each of these is in Wegmans 2215)
Bruce's 2/1 615PM, or Alan's 2/2 450PM,
or Ethan's 2/3 noon office hours!
- Due by 2023/2/13/325pm (assigned 2023/2/3), but you'll be tackling some
of them, including your WSs one-to-put-up-in-class problem,
at our 2/8 and 2/9 WSs; but you already know everything you
need to know to do all of these, so to get maximum
benefit/improvement/challenge/skill-boosting, I'd suggest
you do these---as many as you can (and it is FINE to do them
jointly with a few CSC280 classmates)---before your 2/8 or
2/9 workshop (assigned 2023/2/3):
Problems for the 2/8 and 2/9 WSs and we will go
over your workshop-found solutions at the 2/13 and 2/15 class
sessions.
- Due by 2023/2/8/325pm. (Assigned 2023/2/6.) Both are
difficult.
-
Show
that the concatenation of 2 nonregular sets can be
regular (even if both are over the alphabet {0}).
-
Prove that if \(L\subseteq
\Sigma^*\) is a regular set and \(B\subseteq \Sigma^*\), then
\(L_{yxy} = \{x\in\Sigma^*~\mid~(\exists y \in B)[yxy\in L]\}\)
is a regular set. In doing this, you may find you will use
aspects of the "xy" case we covered today (2/6) in class, but
unlike that, might need to use a richer state space (as we often
have been doing in tackling a different type of challenge,
namely, fractional-language problems). If you find the yxy case
easy, then make your own string of x's and y's,
e.g. xxxyxyyyyxyyxyxxyxyxy (or perhaps something a bit shorter,
e.g., xyx or xxyx), and see if you can prove the same result for that
case (it does hold!).
- Due by 2023/2/15/325pm. (Assigned 2023/2/13.)
As a followup to the 2/8 and 2/13 classes which did (well, in some of
these
cases quite briefly,
e.g., for xxyxy it gave the strategy and then
embedded in the video the detailed construction)
the yxy, xyx, and xxyxy
cases,
and as practice on such problems and as preparation for
Midterm 1,
you may well want
more examples to work on---because you really should want to be able
to do basically any problem of this sort quickly, confidently, and
correctly. Any string of x's and y's makes a reasonable problem. But
for concreteness, let me suggest you do the yxx case. And then do the
(I kid you not) yxyyyyxxx case. If there is time on 2/15 in class
after we do the remaining 3 problems from the set-of-9-problems,
we'll go over the yxyyyxxx case. *Start by drawing
a strategy diagram for that case!* Then write, as a 5-tuple, an NFA
that realizes your strategy.
If you do other strings (e.g., xxxyxyxyxxyy), we won't go over them
in class,
but rather please go over
your answers with a classmate or few, helping each other. And if after
that and the Wed. class
you still are not confident about
how to do any such string, then definitely please
go to an OOH for help.
(After all, you might well be asked, on midterm 1, to do a
problem of that general sort.)
- Due 2023/3/22/325pm. (Assigned 2023/3/20.) Prove, using
the template method, that \(L_2 = \{\langle M\rangle \mid L(M)\)
is a decider\(\}\) is undecidable.
- Due 2023/4/5/325pm. (Assigned 2022/4/5.) Prove that SAT polynomial-time many-one reduces to the set {f | boolean formula f has at least two satisfying assignments}. NOTE: we will go over this problem in class on 4/5 or 4/10.
- Due 2023/4/12/325pm. (Assigned 2022/4/10.) Prove (or at least put in time trying to prove) that if SAT polynomial-time many-one reduces to (i.e., in Sipser-speak, polynomial-time mapping reduces to) a tally set, then P=NP.
Other
Important
Links/Info/Resources:
Some of My Favorite Bits of Science Wisdom:
-
After solving a challenging problem, I solve it again from scratch,
retracing only the *insight* of the earlier solution. I repeat this
until the solution is as clear and direct as I can hope for. Then I
look for a general rule for attacking similar problems, that *would*
have led me to approach the given problem in the most efficient way
the first time. -- Robert Floyd
-
In computer science, elegance is not a dispensable luxury, but a
matter of life and death. -- Edsger Dijkstra
-
Imagination is more important than knowledge. -- Albert Einstein
-
They are ill discoverers that think there is no land, when they can see
nothing but sea. -- Francis Bacon
Other Odds and Ends (Mostly Quotations):
-
I don't believe it. Prove it, and I still won't believe it.
--Life, the Universe and Everything
-
It was mentioned on CNN that the new prime number discovered
recently is four times bigger then the previous record.
-
Sooner or later society will realize that certain kinds of hard work
are in fact admirable even though they are more fun than just about
anything else. -- Donald E. Knuth
-
Getting tenure doesn't really change anything.
However, not getting tenure changes everything.
-
My late friend Stan Ulam used to remark that his life was sharply
divided into two halves. In the first half, he was always the
youngest person in the group; in the second half, he was always the
oldest. There was no transitional period.
-- Gian-Carlo Rota, "Indiscrete Thoughts"
-
Fools ignore complexity. Pragmatists suffer it. Some
can avoid it. Geniuses remove it.
-- Alan Perlis, Epigrams in Programming
-
More Quotes from Dijkstra:
* None of the programs in this monograph, needless to say, has been tested on a machine. (From "A Discipline of Programming.")
* Computer science is not about computers, any more than astronomy is about telescopes.
* The question of whether computers can think is just like the question of whether submarines can swim.
-
"Supposing a tree fell down, Pooh, and we were underneath it?" [said Piglet.]
"Supposing it didn't," said Pooh after careful thought.
Page Maintained by: Lane A. Hemaspaandra