96
votes
Is legislation NP-complete?
It's undecidable because a law book can include arbitrary logic. A silly example censorship law would be "it is illegal to publicize any computer program that does not halt".
The reason results for ...
- 12.6k
36
votes
Accepted
Why there are no approximation algorithms for SAT and other decision problems?
Approximation algorithms are only for optimization problems, not for decision problems.
Why don't we define the approximation ratio to be the fraction of mistakes an algorithm makes, when trying to ...
D.W.♦
- 154k
36
votes
Accepted
How is the traveling salesman problem verifiable in polynomial time?
NP is the class of problems where you can verify "yes" instances. No guarantee is given that you can verify "no" instances.
The class of problems where you can verify "no" instances in polynomial ...
- 81.4k
34
votes
Accepted
Is legislation NP-complete?
Laws can include arbitrary language, and arbitrary language is able to express NP-complete logic. So in theory it would be possible to create an NP-complete or even an undecidable law. However, in ...
- 456
26
votes
Why are computability problems always written in full caps?
After the Cook-Levin Theorem Richard Karp realized that the complexity of computational problems could be compared.
His paper was prepared in a type-writer font, and used underlining and all-caps for ...
- 29.6k
23
votes
Accepted
Is the equality of two DFAs a decidable problem?
In order to decide whether the languages generated by two DFAs $A_1,A_2$ by the same, construct a DFA $A_\Delta$ for the symmetric difference $L(A_1) \Delta L(A_2) := (L(A_1) \setminus L(A_2)) \cup (L(...
- 275k
22
votes
Accepted
Is SAT an existential question?
The SAT problem is a decision problem. It means that an algorithm that solves SAT must answer true or false, and not necessarily ...
- 12.4k
19
votes
Accepted
Is Post Correspondence Problem in NP?
The Post correspondence problem is undecidable, and in particular it is not in NP. The reason that your idea doesn't work is that the witness is not necessarily of polynomial size (in fact, you just ...
- 275k
17
votes
Accepted
Why rectangle packing is NP-hard but maybe not in NP?
In order for a language $L$ to be in NP, there needs to be a way to certify that instance $x$ belongs to $L$. This "way" is a polynomial size witness which can be verified in polynomial time....
- 275k
16
votes
Accepted
is FIND WORDS in P?
Your language is in P. Suppose that the matrix is $n\times n$ and that the words have total length $\ell$. Each word can start at at most $n^2$ positions and be written in $O(1)$ many orientations, ...
- 275k
15
votes
How many decidable decision problems are there?
There are a countable infinity of decidable problems. There must be at least a countable infinity, because all of the languages $\{a\}$, $\{aa\}$, $\{aaa\}$, ... are decidable. There cannot be more ...
- 81.4k
15
votes
Why there are no approximation algorithms for SAT and other decision problems?
The reason you don't see things like approximation ratios in decision making problems is that they generally do not make sense in the context of the questions one typically asks about decision making ...
- 3,243
14
votes
Accepted
Do any decision problems exist outside NP and NP-Hard?
If $P = NP$, then any non-trivial language is NP-hard, and any trivial language belongs to NP. Hence, we do not get anything which is neither NP or NP-hard in this case.
If, however, $P \neq NP$, ...
- 2,649
14
votes
Accepted
Reconciling NP and the decision problem
By definition, all NP-complete problems are decision problems. In fact, the category of NP-completeness only applies to decision problems. Any other kind of problem cannot be NP-complete any more than ...
- 275k
14
votes
Why are computability problems always written in full caps?
In the area of discrete mathematics, sets are usually typeset in capital letters. The above problem classes are sets of problems, e.g. SAT is the set of all boolean satisfiability problems.
Thus, the ...
- 1,264
13
votes
Show that there are infinitely more problems than we will ever be able to compute
Reformulating in a more mathematically precise way, what the lecturer is trying to say is this: Any algorithm can be (uniquely) encoded as a finite string of bits, and any finite string of bits (...
- 4,979
12
votes
Is SAT an existential question?
If you have variables x1 to xn, and you decide in polynomial time whether the problem can be satisfied or not, then if it can be satisfied, you set x1 = true and check if it can still be satisfied. If ...
- 27.8k
11
votes
Is Post Correspondence Problem in NP?
Your witness is polynomial in the size of the solution not in the size of the input. You have no way of bounding the length of potential solutions. Your proof shows that PCP is recursively enumerable.
- 851
11
votes
Accepted
Why we cannot prove that a computable function is total?
You misunderstand soundness and completeness of logic. You think that it says:
A statement is true if, and only if, it is provable.
But it really says:
A statement is true in every model if, and ...
- 29.9k
11
votes
Accepted
Why are Chess, Mario, and Go not NP-complete?
It's a common misconception that chess is NP-hard. Generalized chess may be NP-hard. Chess has an 8x8 board, generalized chess has an nxn board with many pieces.
The question then becomes if ...
- 2,469
11
votes
Accepted
Show that there are infinitely more problems than we will ever be able to compute
If I understand you correctly, your question is —
why a solution can be encoded by a natural number and a problem with a real number.
(I assume that you understand the next phase of the proof ...
- 576
10
votes
Why are Chess, Mario, and Go not NP-complete?
Your understanding of what makes chess NP-Hard is slightly flawed. Yes, a nondeterministic machine is able to "play perfectly". But the language of chess is,
$$Chess = \{Pos \quad | \quad \text{White ...
- 4,282
10
votes
Is legislation NP-complete?
This is a very interesting question.
Law is somewhere between everyday language with its arbitrary, constantly changing and often soft rules, and programming language with its very specific, defined ...
- 209
9
votes
Accepted
Undecidability of telling if a program returns true or false
Assume you have a function like the one you described:
...
- 5,802
9
votes
Accepted
Prove that Hitting Set is NP-Complete
3SAT is reduced to the Hitting Set problem. Given a 3SAT $\phi$ having $m$ clauses and $n$ variables, define
$$S = \{ x_1, \dots x_n, \overline{x_1}, \dots , \overline{x_n}\}$$
$$S_i=\{y_1, y_2, y_3\...
- 9,727
9
votes
Accepted
Are there any proofs of exponential lower bound time complexity
The Time Hierarchy Theorem states that for any (reasonable) function $f$, there exists a problem that cannot be solved in time substantially faster than $f(n)$. The proof of this is similar to the ...
- 13.1k
8
votes
Accepted
The importance of the membership problem
The importance is that any computational problem where the answer is yes or no can be phrased as a membership problem in a language. The language is the set of strings for which the answer is yes.
- 81.4k
8
votes
Accepted
Do problems in P have a minimum number of YES and NO instances?
If a problem has only "YES" instances (resp. only "NO" instances), then the associated language, which is our formalization of a "problem" contains every word in $\Sigma^*$ (resp. no words), with $\...
- 17k
8
votes
Why there are no approximation algorithms for SAT and other decision problems?
In addition to the existing answers, let me point out that there are situations where it makes sense to have an approximate solution for a decision problem, but it works different than you might think....
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