이론적 인 컴퓨터 과학에 반 직관적 인 결과가 있습니까?

일부 수학 및 논리 역설은 컴퓨터에 자동으로 적용될 수 있지만 컴퓨터 과학 자체에서 발견 된 역설이 있습니까?

역설적으로 나는 모순처럼 보이는 직관적 인 결과를 반대하는 것을 의미합니다.

네트워크 흐름이 다항식 시간 카운터 직관적이라는 사실을 알았습니다. 많은 NP-Hard 문제보다 첫눈에 훨씬 어렵습니다. 또는 다르게 말하면 CS에는 결과를 해결하는 데 필요한 실행 시간이 예상보다 훨씬 좋은 결과가 많이 있습니다.

A family of counter-intuitive results is the whole "prove an upper bound to prove a lower bound" family of results. The Meyer result that $\mathsf{P} = \mathsf{NP}$ implies $\mathsf{EXP} \not\subseteq \mathsf{P}/poly$ is one example of this, and this came to my mind from both Ketan Mulmuley's GCT work as well as Ryan Williams' recent result that again used an upper bound for CIRCUIT-SAT to prove a lower bound for $\mathsf{NEXP}$ in terms of $\mathsf{ACC}$.

SAT has a polynomial-time algorithm only if P=NP. We don't know whether P=NP. However, I can write down an algorithm for SAT which is polynomial-time if P=NP is true. I don't know the correct reference for this, but the wikipedia page gives such an algorithm and credits Levin.

Computability certainly screws most students. A beautiful example with high confusion rate is this:

$f(n) := \begin{cases}1, \quad \pi \text{ has } 0^n \text{ in its decimals} \\\\ 0, \quad else\end{cases}$

Is $f$ computable?

The answer is yes; see a discussion here. Most people immediately try constructing $f$ with present knowledge. That can not work and leads to a perceived paradox which is really just subtleness.

One surprising and counter intuitive result is that $IP = PSPACE$, proved using arithmetization around 1990.

As Arora & Barak put it (p. 157) "We know that interaction alone does not give us any languages outside NP. We also suspect that randomization alone does not add significant power to computation. So how much power could the combination of randomization and interaction provide?"

Apparently quite a bit!

As Philip said, Rice's theorem is a good example: one's intuition before studying computability is that there must surely be something we can compute about computations. It turns out that we can only compute something about some computations.

How about Martin Escardo's publications showing that there are infinite sets that can be exhaustively searched over in finite time? See Escardo's guest blog post on Andrej Bauer's blog, for instance, on "Seemingly impossible functional programs".

The Recursion Theorem certainly seems counter-intuitive the first time you see it. Essentially it says that when you are describing a Turing Machine, you can assume it has access to its own description. In other words, I can build Turing Machines like:

TM M accepts n iff n is a multiple of the number of times "1" appears in the string representation of M.

TM N takes in a number n and outputs n copies of itself.

Note that the "string representation" here is not referring to the informal text description, but rather an encoding.

Proving information-theoretic results based on complexity-theoretic assumptions is another counter-intuitive result. For instance, Bellare et al. in their paper The (True) Complexity of Statistical Zero Knowledge constructively proved that, under the certified discrete log assumption, any language that admits honest-verifier statistical zero knowledge also admits statistical zero knowledge.

The result was so odd that it surprise the authors. They pointed out this fact several times; for instance, in the introduction:

Given that statistical zero-knowledge is a computationally independent notion, it is somewhat strange that properties about it could be proved under a computational intractability assumption.

PS: A stronger result was later proved unconditionally by Okamoto (On Relationships between Statistical Zero-Knowledge Proofs).

Description of some terms

Since the above result includes a lot of cryptographic jargon, I try to informally define each term.

1. Certified discrete log assumption: It is hard (for poly-size circuits) to solve the discrete logarithm, even if the group prime ($p$) is certified; that is, the factorization of $p-1$ is given.
2. Zero knowledge: A protocol which yields no knowledge to polynomial-time bounded parties.
3. Statistical zero knowledge: A protocol which yields no information, even to computationally unbounded parties, except with negligible probability.
4. Honest-verifier zero knowledge: A protocol which yields no knowledge to polynomial-time bounded parties, if they act as specified by protocol.

How about the fact that computing permanent is #P-Complete but computing determinant - a way weirder operation happens to be in the class NC?

This seems rather strange - it did not have to be that way (or maybe it did ;-) )

The linear programming problem is solvable in (weakly) polynomial time. This seems very surprising: why would we be able to find one among an exponential number of vertices of a high-dimensional polytope? Why would we be able to solve a problem which is so ridiculously expressive?

Not to mention all the exponential-size linear programs which we can solved by using the ellipsoid method and separation oracles, and other methods (adding variables, etc.). For example, it's amazing that an LP with an exponential number of variables such as the Karmakar-Karp relaxation of Bin Packing can be efficiently approximated.

Whenever I teach automata, I always ask my students if they find it surprising that nondeterminism doesn't add any power to finite-state automata (i.e., that for every NFA is there is an equivalent -- possibly much larger -- DFA). About half the class reports being surprised, so there you go. [I myself have lost the "feel" for what is surprising at the intro level.]

Students definitely find it surprising at first that $R\neq RE$. I challenge them to produce an algorithm that determines whether a given java program will halt, and they typically try to search for endless while loops. As soon as I show them ways of constructing loops whose termination is far from obvious, the surprise factor goes away.

I have found the A simple public-key cryptosystem with a double trapdoor decryption mechanism and its applications paradoxical, because it is a adaptive chosen ciphertext secure scheme which is homomorphic.