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Unless you want to be more rigid in what you allow as a "description", every decidable language has a finite description - precisely the Turing machine that decides it. This Turing machine must exist …

I think I see where your confusion arises. Hopefully ;). Suppose you have a problem $\Pi \in NP$, and two algorithms $\mathcal{A}$ and $\mathcal{B}$ that both solve $\Pi$. Assume that $\mathcal{A}$ …

At the most basic level, the size of the input is the length of the string that encodes the input given to the Turing Machine - i.e. the number of cells it takes up on the tape under the chosen encodi …

Another hint: As $TQBF$ is PSPACE-complete, for every problem $L$ in PSPACE, there exists a reduction $f$ such that for every instance $x$ of $L$ we can construct an instance $f(x)$ of $TQBF$ in time …

A $\mathrm{No}$ answer means that there is no solution, not that there is a non-solution. So for Subset-Sum this means there is no subset of the input integers that sum to zero, or to put it another …

Unless I'm missing something obvious (it is getting late here), the second isn't possible - $BPP = coBPP$, so $NP=BPP \rightarrow NP = coBPP \rightarrow NP = coNP$. In the first case, some of the con …

Purely for clarity I'll lay out the definitions first, then give a reduction for your problem, so skipping a little ahead shouldn't cause a problem (if it does, start a bit earlier ;) ). The two probl …

We can get some sort of upper bound from some complexity inclusions. $TC^{0}$ is the class of polynomially sized, constant depth boolean circuits where we also have a $MAJORITY$ gate of unbounded fan …

Like Joe, I'm assuming that you want to find all maximal cliques. Assuming your representation can represent all graphs (and there's no reason to suspect otherwise), then there's certainly still poss …

The number of inputs to a gate is an independent restriction or allowance on the circuit. The number of gates in a circuit is simply the number of gates. Of course you can always measure the two sepa …

As Kaveh points out, any problem in $P$ by definition can be solved in polynomial time by a Turing Machine, so it in effect produces its own proof and checks it, hence $P = PCP[0,0]$ and the inclusion …

The formal name for a problem where you want to actually produce the solution as the output (not just say yes or no to whether one exists) is a Function Problem if every input has an associated output …

Asymptotic (/Big-Oh/Landau) notation is not special to polynomial time algorithms, or indeed any class of algorithm. It is the standard way of communicating (asymptotic) running times, amongst other r …

You might find Greenlaw, Hoover and Ruzzo's "Limits to Parallel Computation: P-Completeness Theory" interesting. It obviously goes a lot further than the limits of this question, but most pertinently …

Expanding on Pål GD's hint, you can reduced from the normal, unweighted version of VERTEX COVER. VERTEX COVER Input: A graph $G$ and an integer $k$. Question: Does $G$ have a vertex cover of s …