Questions about general methods and techniques for proving multiple theorems. When asking about the proof of a single statement, use tags relating to what the proof is about instead.
Questions tagged [proof-techniques]
709 questions
197
votes
3 answers
Is there a system behind the magic of algorithm analysis?
There are lots of questions about how to analyze the running time of algorithms (see, e.g., runtime-analysis and algorithm-analysis). Many are similar, for instance those asking for a cost analysis of nested loops or divide & conquer algorithms,…
Raphael
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100
votes
5 answers
How to prove that a language is not context-free?
We learned about the class of context-free languages $\mathrm{CFL}$. It is characterised by both context-free grammars and pushdown automata so it is easy to show that a given language is context-free.
How do I show the opposite, though? My TA has…
Raphael
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100
votes
10 answers
How to prove that a language is not regular?
We learned about the class of regular languages $\mathrm{REG}$. It is characterised by any one concept among regular expressions, finite automata and left-linear grammars, so it is easy to show that a given language is regular.
How do I show the…
Raphael
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98
votes
11 answers
Solving or approximating recurrence relations for sequences of numbers
In computer science, we have often have to solve recurrence relations, that is find a closed form for a recursively defined sequence of numbers. When considering runtimes, we are often interested mainly in the sequence's asymptotic growth.
Examples…
Raphael
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79
votes
2 answers
What is coinduction?
I've heard of (structural) induction. It allows you to build up finite structures from smaller ones and gives you proof principles for reasoning about such structures. The idea is clear enough.
But what about coinduction? How does it work? How can…
Dave Clarke
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55
votes
9 answers
How to prove a language is regular?
There are many methods to prove that a language is not regular, but what do I need to do to prove that some language is regular?
For instance, if I am given that $L$ is regular,
how can I prove that the following $L'$ is regular, too?
$\qquad…
corium
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50
votes
6 answers
How to prove greedy algorithm is correct
I have a greedy algorithm that I suspect might be correct, but I'm not sure. How do I check whether it is correct? What are the techniques to use for proving a greedy algorithm correct? Are there common patterns or techniques?
I'm hoping this…
D.W.
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48
votes
4 answers
What are common techniques for reducing problems to each other?
In computability and complexity theory (and maybe other fields), reductions are ubiquitous. There are many kinds, but the principle remains the same: show that one problem $L_1$ is at least as hard as some other problem $L_2$ by mapping instances…
Raphael
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47
votes
2 answers
How to show that a function is not computable? How to show a language is not computably enumerable?
I know that there exists a Turing Machine, if a function is computable. Then how to show that the function is not computable or there aren't any Turing Machine for that. Is there anything like a Pumping lemma?
Similarly, how can we show a language…
user5507
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41
votes
2 answers
How do I construct reductions between problems to prove a problem is NP-complete?
I am taking a complexity course and I am having trouble with coming up with reductions between NPC problems. How can I find reductions between problems? Is there a general trick that I can use? How should I approach a problem that asks me to prove a…
Anonymous
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37
votes
3 answers
Why is Relativization a barrier?
When I was explaining the Baker-Gill-Solovay proof that there exists an oracle with which we can have, $\mathsf{P} = \mathsf{NP}$, and an oracle with which we can have $\mathsf{P} \neq \mathsf{NP}$ to a friend, a question came up as to why such…
Nikhil
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34
votes
4 answers
How to prove that a grammar is unambiguous?
My problem is how can I prove that a grammar is unambiguous?
I have the following grammar:
$$S
→ statement
∣ \mbox{if } expression \mbox{ then } S
∣ \mbox{if } expression \mbox{ then } S \mbox{ else } S$$
and make this to an unambiguous grammar, I…
user1594
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33
votes
7 answers
Is there a more intuitive proof of the halting problem's undecidability than diagonalization?
I understand the proof of the undecidability of the halting problem (given for example in Papadimitriou's textbook), based on diagonalization.
While the proof is convincing (I understand each step of it), it is not intuitive to me in the sense that…
user118967
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32
votes
5 answers
Proof that dead code cannot be detected by compilers
I'm planning to teach a winter course on a varying number of topics, one of which is going to be compilers. Now, I came across this problem while thinking of assignments to give throughout the quarter, but it has me stumped so I might use it as an…
thomas
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29
votes
3 answers
Are there any specific problems known to be undecidable for reasons other than diagonalization, self-reference, or reducibility?
Every undecidable problem that I know of falls into one of the following categories:
Problems that are undecidable because of diagonalization (indirect self-reference). These problems, like the halting problem, are undecidable because you could use…
templatetypedef
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