There is a puzzle in the book Discrete Mathematics and its applications Page 23 - Exercise 16 . Puzzle goes like -
"An Explorer captured by a group of two cannibals. There are two type of cannibals-those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular cannibal always lies or always tells the truth. He is allowed to ask the cannibal exactly one question.
Find the question that the explorer can use to determine whether the cannibal always lies or always tells the truth."
I know some solutions like you can ask cannibals "are you a cannibal", "is the colour of sky is blue?". But that is not how logic works right?
There is another solution which is perfect. Solution is, you ask cannibal "If I were to ask you if you are a liar, would you answer 'Yes'?"
Truther cannibal will answer "No" because he is not a liar.
But I don't know what Liar cannibal will answer. Will he also answer "No" because he always lies?
And I want to know how double Negation is working here.
Let's break it into parts.
The liar cannibal will always say the wrong answer. When you ask him "If I were to ask you if you are a liar, would you answer 'Yes'?" he answers the opposite.
The liar cannibal thinks for himself: "If I was asked if I am a liar, would I answer 'Yes'?" Because he is a liar, if he was asked, he would answer 'No, I am not a liar'. But that not you asked him. You asked "If I were to ask you". So the liar would lie about that too- instead of saying 'No, I am not al liar' he would say "Yes, If I was asked if I am a liar I would answer 'Yes'".
The double negation is something like this:
not (if I was asked: not (am I a liar?))
"am I a liar?" is true so (not (am I a liar?)) is false. Because of that the whole statement is true.
Hope it was clear :) This is indeed a tricky question.
If you asked the straightforward question "Are you a liar?" then a Truther cannibal would truthfully answer No, but a Liar would also answer No and you couldn't tell them apart.
Your "question within a question" query would work though. If you asked one of the cannibals "If I were to ask you if you are a liar, would you answer 'Yes'?", the Truther cannibal will still truthfully answer No, but a Liar would lie about lying and answer Yes.
Side note: It's good to see schools are still teaching from Rosen's Discrete Math. It's by far one of my favorite text books.
Related
I'm looking for an automatic theorem proving system, which can prove this:
Crocodile took mans child. Man asked crocodile not to eat his child. But Crocodile said: I'll return your child to you, if you will tell me, what am I going to do with him.
Analytical solution to his looks like this:
x - crocodile will eat child
y - men answers: crocodile will eat child
~ means equality, ! means not, -> implies, + OR;
((x~y)->!x) and ((x XOR y)->x) =
(x! and y +!x and y+!x)(!x!y+x and y+x) =
(!x+!y)(x+!y) = !y;
So, the answers is that men has to say: "You are not going to eat the child";
Now, there are plenty of systems listed here:
http://en.wikipedia.org/wiki/Automated_theorem_proving
I've tried 5-6 of them, but couldn't really understand what am I doing here. How to formalize this theorem inside them, so that I could enter this first part of it:
((x~y)->!x) and ((x XOR y)->x)
and get the answer
y as an output.
Can any once advice, which system at least capable of doing so automatically, and give me some more hints?
Regards,
Konstantin.
Well, your question target it's a lot higher than usual, I don't understand sufficiently your task to help...
I just goggled 'tableaux in Prolog', and would suggest to try first the simpler one, before delving into something more sophisticated (but I don't even know if that kind of logic is appropriate for your task).
After lots of reserach, I actually found out there are many programs like this, so my answer is : yes, such theorem can be proved automatically. Online example: http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-zentral.html
Checkout prolog. It's great for logical propositions and that sort of thing. Start with a read through the wiki and see if it sounds like what you want. It is a logical programming language - it will help you to build your own theorem proving algorithms.
Wiki:
http://en.wikipedia.org/wiki/Prolog
Tutorials:
http://cs.union.edu/~striegnk/courses/esslli04prolog/index.php
I was talking to someone in a bar not too long ago, and the inevitable question of "What do you do?" Came up. My response "Oh, I just recently graduated; I work in company X. I'm a programmer" didn't quite satisfy her.
I've had this conversation a hundred times and people are never quite happy with the answer. They always have more questions. I think this is because, for the mostpart, people don't really know what a programmer does. The answer to this question normally gives you an insight into a person (sort of): Accountants might be analytical and good with numbers, lawyers might be good with words and debate etc. 'programmer' doesn't.
so I tried to give an answer that would give that sort of insight.
I tried to illustrate my point of "Well It's basically fancy problem solving" with an example of a classic programming problem.
The first thing that jumped into my head (for some reason) was the smallest subset problem... but I'd have to explain arrays, time complexity, etc
I thought of n-queens problem, but she'd never seen a chessboard in her life (yeah, I know)
Towers of hanoi was too hard to envision in general...
So I Was stuck and the topic changed.
Can you guys think of an example of a problem; that a complete lay(wo)man can understand, which would give someone an idea of the kind of thinking programmers have to do? (and by extension save me answering awkward questions about how I make my living)
I get asked what I do too, and basically, all I say is that You wouldn't understand much about it, which will get you one of two responses: "Try me" and shes interested ->move to next section of comment, or she will just shut down and think that you think she's not intelligent. USE WITH DISCRETION.
Now, if she says try me, you have a challenge. Obviously what you do is going to be over her head, but you know you can explain a little to her. Basically, what I try to explain is that I am a logician, kinda like a magician. When the computer needs to be told to do something, and no one else knows how to tell it, they call me up. Sometimes it's simple, other times, its likes climbing Mt. Everest with your fingers on some impossibly difficult task. Imagine the worlds largest word search, and you don't know what words you're looking for. That's what I do. I find those words. So as you could imagine, I'm quick to notice little things and remember them. ;)
That should elicit the kind of response out of any women worth your time.
Good Luck!
AllDistinct(a1 , . . . , an )
if (n = 1)
return True
for i := n down to 2
begin
if (LinearSearch(a1 , . . . , ai−1 ; ai ) != 0)
return False
end
return True
Give a big-O bound on the running time of AllDistinct. For full credit, you must show
work or briefly explain your answer.
So the actual answer for this according to the solution for this problem is O(n^2). However, since BigO is the worst case running time, could I have answered O(n^100000) and gotten away with it? Theres no way they can take points off for that since its technically the correct answer right? Although the more useful O(n^2) is obvious in this algorithm, I ask because we might have a more difficult algorithm on the upcoming exam, and incase I cant figure out the 'tight' bound, I could make up some extremely large value and it should still be correct, right?
Yes, if a function is in O(n^2), it is also in O(n^1000).
Whether you'll get full (or any) credit for answering this way depends on the person grading your exam of course, so I can't tell you that (probably not though). But yes, it is technically correct.
If you do decide to go this way though, you should probably chose something like O(n^n) or O(Ackermann(n)), since for example exponential functions are not in O(n^1000).
Another problem is that you will probably be asked to proof the bound as well. This will be hard to do if you don't actually know the running time of the function. "n^n is really large, so the running time will probably be less than that" is not a proof. Though on the upside if you manage to correctly proof that the function is in O(n^n), you'll probably get at least partial credit.
That would be a trivial answer to the question. Although correct, it tells you nothing and is thus worthless. It's not about right or wrong, it's about bad and good. The better your answer, the more points you'll get for it. The question does not say you'll get credit for a terrible bad bound. Bad answers give bad marks?
(Asking for Big Theta would be a harder question. I would play nice :)
No.
It might be all clever and HA! I got you! but that's not the idea. (and you know that)
If the professor ask you for BigO of it you can answer whatever BigO you think but you must prove it as it say For full credit, you must show work or briefly explain your answer.
BigO is not useless. For problems it's easy to get Upper Bound (BigO) graeter; e.g.
Sorting problem: you have the simple bubble sort and you can proof that is n^2 (right?), so upper bound of Sorting problem is n^2 (because exists and algorithm that solve it in that time, but if you go on with maths, you see that the problem has a lower bound of log(n!) ). So n^2 was a good answer until you proof it's log(n!). There are many problems that we know just BigO but not the lower bound, so it's not useless.
If you can say that a program halts you can always compute is BigO with some math, but is not always easy (exists even ammortized complexity) but it's simpler than problems lowerBound. So BigO is not so important in algorithm, but it's not useless.
The important thing is that you understand what it means; then if you can get any BigO of that program you can write it on that exam paper that is a function from Student to number.. and good luck.
At a guess, you'd probably have to talk to the professor, and argue with him a bit to even get partial credit for an answer like that. Depending on how much he values theory vs. practicality, he might give you partial credit, or he might give no credit -- but I can hardly imagine a professor who'd give any credit without your explicitly pointing out how it's (semi-)correct, and some might not even then.
I was a prof. Profs make up exam questions, and those can have bugs. It's embarrasing when you have to throw out a question because it's got a bug and people can give trivial answers. In this case the bug is "a big-O bound". Making exam questions is tricky, because you don't want to err on the side of saying too much, like some kind of airtight lawyer statement, because that will confuse people even more.
After all, the reason for doing this is, hopefully, you'll learn something useful. If you see an ambiguous question like this, the prof will appreciate it if you say something like "I assume you mean a good big-O bound."
I am reading 'The Little Schemer' in an effort to better understand some of the core elements of programming (namely recursion) and to get more of an idea how to think like a programmer.
The book comes recommended as an entry-level book and the introduction states that all I need to know are English, numbers and counting (which I do).
I am kind of confused though as the first section and questions start off by asking "Is it true that this is an atom?"
Am I missing something? Am I supposed to know what an atom is? I am confused as I thought it was meant to be in more plain English.
Thanks in advance,
Tim
It can be a hard book to get into; it took me two tries separated by about a year. The way you to read it is that you are figuring out these concepts for yourself by listening in on a dialogue between two other people. The first question about a concept will lose you, but the hope is that you say, "Aha! I've figured out the concept they must be talking about" before the end of the questions on a given topic. By the end of the section you'll be answering the questions yourself before reading the answers in the book.
If you hit the end of a section and haven't gotten to that point, start over again but try to give the answers yourself without reading them. When you can supply the answers yourself, you've either figured out the concept in your own terms or memorized the answers in the book. Later sections will refer back to these concepts, though, and will reinforce your understanding.
Think of the student in the book as a proxy for you who seems to begin each section smarter than you, but who you outpace by the end of the section.
The book uses a sort of "constructivist" learning model. It asks you to figure things out before you know the formal definitions. The idea is to develop an intuition before formality (I believe, although that may not be the intention of the authors). You may find this annoying at first, but when you get to the higher-level concepts, you will find yourself understanding things way better than you would from reading R5RS, for example. Continuations had me completely baffled until I read all the way through this book. Stick with it and you'll get why the authors take this approach.
On the left of the page:
"Is it true that this is an atom?
atom"
On the right of the page, 2cm away:
"Yes, because atom is a string of
characters beginning with the letter
a".
And similar questions and answers about atoms in the same format for the remainder of the page. I don't think it takes a genius to work out what is going on here.
An atom in Scheme is like in english, something that you can't divide.
Here are some atoms:
'foo 'bar 'baz 123 '() '+
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Today there was a question on SO, where the author was given an NP-complete problem during an interview and he obviously hadn't been told that it was one.
What is the purpose of asking such questions? What behavior does the interviewer expect when asking such things? Proof? Useful heuristics? And is it even legitimate to ask one if it's not a well-known NP-complete problem everyone should know about? (there's a plenty of them)
Completely legitimate to me. If you are Computer Science professional there are good chances that you can either argument informally why the problem seems to be hard, or (even better) provide a sketch of reduction from a known NP-hard problem.
Many real world problems eventually turn out to be NP-hard, and stackoverflow also has now and then questions about the complexity of a problem which turns out to be a difficult one (NP-hard, for instance). It is an important part of a CS professionals toolbox to be able to recognize and to argue for problems which are known to be difficult to solve.
I don't see any problem with asking something like this. Also, programmers should NOT be expected to recognize NP-complete problems by rote. They should, however, be able to identify that their algorithm is potentially slow regardless of whether a given problem is NP-complete.
Sure, why not? NP-complete doesn't mean unsolvable, it just means your solution will be slow. You may be looking to see if the candidate will choose the brute-force solution, or try a dynamic programming solution. And this type of question can lead into questions about runtime and other useful theory.
There's a category of interview questions that are illegal in some countries, usually pertaining to personal details that are none of the employer's business. That aside, any question is fair game if the interviewer feels it'll help get an idea of the interviewee's capabilities!
If you're hiring for a position that calls for a thinker rather than just a code monkey, it may be useful to throw this kind of problem at the applicant. Who cares if a problem is "well known" to be NP? If the guy is good he'll come to that understanding in analyzing the problem. That may well be the result the interviewer wants to see, or the applicant can go on to do some more pre-analysis and describe how he'd brute-force the problem, or what optimizations he can think to apply to make it more manageable.
It's good to ask a question that is hard to answer, to see how a programmer reasons through a problem.
But it all depends on how the interviewer asks the question, and prompts the programmer towards a solution if they aren't a mathematical genius (i.e. to see how they reason, and how they react to questions like "that's a good start, but what if...") rather than to detect if they are autistic and can provide an optimal solution in 4.3 seconds).
It's worth remembering that interviews are highly stressful affairs in which many people find such questions very difficult to answer well - a much simpler question will usually suffice without putting the interviewee under undue stress/pressure.
If you do it to deliberately try to see how they deal with stress is just stupid - that isn't the sort of stress a programmer has to deal with in their job, so you're not testing anything worthwhile.
I think it's valid to ask a question you know the interviewee won't know the answer to.
Everyone encounters problems they don't know the answer to. This type of question will give you insight as to what the interviewee's internal process is. If they logically conclude things and start to formulate a correct answer, even if it's not the best dynamic programming algorithm for it, it shows that they can reason well and discover an answer.
Also, since they likely don't know everything about the problem, this sort of question lets you see how comfortable the interviewee is with asking for help or clarification.
I think the best way to answer this type of question is to ask for any clarifications if something is missing or not well known, and then postulate an answer, pointing out why you think it is correct, and why it likely isn't the best solution.
I don't see a problem with this, but I do somewhat question the usefulness of these sorts of questions in interviews in general.
The benefit of asking questions like this, as an interviewer, is see how the person approaches a problem, and how they think. If you tell them to talk it out, you can find out quite a bit about how they will approach a difficult problem.
That being said, during an interview, most people aren't at their best - so throwing something that's somewhat "tricky" like this is often overkill, IMO.
It's sort of mean to ask nigh-impossible questions without informing the interviewee of it, but in observed problem solving, the question is often asked so that you may demonstrate critical thinking skills, how you approach problem solving, and how you handle pressure or failure.
I've been asked interview questions I couldn't solve, and I don't think I've ever "failed" an interview because of it.
No, it's rude and a sign that the interviewer just likes being in a position of power. Haha, peon! I know the answer, and you don't! And boy, do I love to make you squirm trying to come up with it!
About the only way it could be even slightly valid as a useful interview question is if it were a well-known question or one that was somehow obviously NP-complete, and asked in a way that encouraged discussion of feasibility.
Is it fair to ask in an interview how to factorise numbers?
That's not known to be NP-C, but no polynomial-time solution[*] is known, so it is certainly not known to be in P.
I think the answer to both my question, and the original question, is "yes", and for the same reasons. Some problems have no solution which scales well, but do need to be solved anyway for certain inputs. If you need programmers who can handle such problems, there's a good way to let them prove it in interview, and that's to pitch them one and see whether they freak out.
If someone claims a CompSci background, then they should even be able to provide good solutions to certain NP-C problems on demand, such as solving the knapsack problem with dynamic programming. I would consider it pointless asking an applicant for a programming job to take a problem they've never seen before, and actually prove it NP-complete (for example by reducing knapsack to the specified problem). You don't need very many programmers per company who can do that (usually 0), and all you'll likely discover is how long the candidate keeps at it before attempting to change the subject and do something more valuable with the interview time...
[*] polynomial in the size of the input in bits, that is. You often see people discussing algorithmic complexity of integer problems like factorisation in terms of the size of the number represented by the input, e.g. "sqrt(N) trial divisions". But that's not how NP and NP-C are defined.
That is evil!
If the interviewer asks an NP-complete question in an interviewer the only response they can reasonably expect is that the interviewee respond with a proof that the problem is NP-complete. In a low-stress environment like a university homework question, this usually takes a bright student 2-3 or more hours to prove. The proof itself can take several pages to write out completely, perhaps several hours of work itself. In a high-stress environment like an interview you can expect that the interviewee may not even recognize that this is np-complete.
The only reasonable alternative is that the interviewee produce an approximation algorithm; however, in this case the interviewer should make it explicitly clear that they are fine with approximations.
Even so, most approximations only come with an order of 2 of the correct answer.
I guess there is 1 more alternative: the interviewee suggests that a search-type algorithm maybe the most suitable (take for example the integer-domain optimization problem which is NP-complete, most approximation algorithms use a branch and bound search spin on the simplex algorithm to produce decent results.)
There is nothing wrong with giving an NP-complete problem as a programming challenge during an interview. I only see something wrong with expecting to find a polynomial-time solution to the problem during the interview.
An interviewer should want to see how a candidate deals with a variety of situations -- including situations that the candidate can't find an easy solution to. "Impossible" questions show how the candidate reacts when there's no simple solution. Does the candidate just give up? How many different attempts does the candidate search? How far-reaching are the solutions tried? When does the candidate ask for help -- and how? Does the candidate complain that the problem "isn't fair"?
In short, such an interview question isn't about solving P=NP... it's a psychological answer.
I prefer asking them to prove that P != NP or P == NP. Someday a candidate will answer it, I'll steal their answer and be famous!
On a more serious note, though, I think it's completely fair. Most NP complete problems are easy to solve, they just run very slowly. Unless the job requires them to know a lot about complexity theory, though, all they need to demonstrate is that they understand the solution will be slow. Bonus points if they know it's non-polynomial time, gold star if they know it's NP complete.
If such a question was given before an interview(to be answered at an interview) I would say it's ok.. but to just solve such a difficult problem as that on the spot is definitely not going to be done well by any programmer, and if the programmer does do it well that just means they can act on the spot(which isn't always the best thing for programming as designing things needs to take time and check every possible flaw) or that they have seen a similar problem before.
Edit:
Or possibly discussion about the problem would be good, like say laying down a plan of action whether or not you completely solve it.. and discussing how feasible and if there is a fast(but difficult) way to do it and such. I would not say that the interviewee should have to write down over 50 lines of C code in an interview to solve it though