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数学代考|计算复杂性理论代写computational complexity theory代考|Interactive Proof Systems

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数学代考|计算复杂性理论代写computatiknal complexity theory代考|Examples and Definitions

The notion of interactive proof systems is most easily explained from a game-theoretic view of the complexity classes. In the general setting, each problem A is interpreted as a two-person game in which the first player, the prover, tries to convince the second player, the verifier, that a given instance x is in A. On a given instance x, each player takes turn sending a string $y_{i}$ to the other player, where the $i$ th string $y_{i}$ may depend on the input $x$ and the previous strings $y_{1}, \ldots, y_{i-1}$. After a number $m$ of moves, the prover wins the game if the verifier is able to verify that the strings $x$, $y_{1}, \ldots, y_{m}$ satisfy a predetermined condition; otherwise, the verifier wins. Depending on the computational power of the two players, and on the type of protocols allowed between the two players, this game-theoretic view of the computational problems can be used to characterize some familiar complexity classes.

As a simple example, consider the famous N P-complete problem SAT. To prove that a Boolean formula F is satisfiable, the prover simply sends a truth assignment t on the variables that occurred in $F$ to the verifier, then the verifier verifies, in polynomial time, whether $t$ satisfies $F$ or not and accepts $F$ if $t$ indeed satisfies $F$. Thus, we say that SAT has a proof system with the following properties:
(1) The verifier has the power of a polynomial-time DTM, and the prover has the unlimited computational power.
(2) The game lasts for one round only, with the prover making the first move.
(3) A formula F is in SAT if and only if the prover is able to win the game on F.

数学代考|计算复杂性理论代写computatiknal complexity theory代考|Arthur–Merlin Proof Systems

In order to analyze the computational power of interactive proof systems, we first study a weaker type of probabilistic proof system called the Arthur-Merlin proof system. An Arthur-Merlin proof system is similar to an interactive proof system where Arthur is a verifier with the power of a polynomial-time PTM and Merlin is a prover, except that Merlin is even more powerful than an ordinary prover so that he is able to read the whole history of the computation of Arthur on the given input, including the random numbers generated by Arthur. If we examine the interactive proof systems of Examples $10.1$ and $10.2$, we can see that the secrecy of the random bits used by the verifier is critical to the correctness of the system. Indeed, if in Example $10.1$ the prover knows game by sending back the bit $b$, regardless of whether $G_{1} \cong G_{2}$ or not. Thus, the extra power of Merlin appears to be a strong restriction to the Thus, the extra power of Merlin appears to be a strong restriction to the restriction is not really so strong and that the two notions of proof systems are essentially equivalent. Yet, the simplicity of the Arthur-Merlin proof systems allows us to perform detailed analysis of its computational power.

As Merlin is so powerful that Arthur is not able to hide his random numbers from Merlin, we may as well require that in an Arthur-Merlin proof system, Arthur always sends the random numbers he used to Merlin. Furthermore, Merlin can always simulate Arthur’s computation to compute the next query to be asked by Arthur from the history of the computation and the new random numbers received from Arthur. Thus, Arthur really does not have to send Merlin anything except the new random numbers generated by the random number generator. In other words, Arthur really plays a passive role whose only task is to verify, at the end, whether the computation satisfies a predetermined condition and whether to accept the input. Formally, we define an Arthur machine to be an interactive TM such that
(1) each new query of Arthur is simply a new sequence of random bits; and
(2) every random number generated by Arthur must be sent to Merlin as a query, even if Arthur does not expect an answer from Merlin so that the last random number generated by Arthur counts as one round of communication between the two players.

数学代考|计算复杂性理论代写computatiknal complexity theory代考|AM Hierarchy Versus Polynomial-Time Hierarchy

In this section, we consider the complexity classes $A M_{k}$ of sets having Arthur-Merlin proof systems of a bounded number of rounds. It is clear that these classes form a hierarchy:
$$A M_{0} \subseteq A M_{1} \subseteq \cdots \subseteq A M_{k} \subseteq A M_{k+1} \subseteq \cdots .$$
We are going to see that this hierarchy collapses to the second level. In addition, they are contained in the second level $\Pi_{2}^{P}$ of the polynomial-time hierarchy.

To begin with, we first give an alternating quantifier characterization for the $A M$ hierarchy like that for the polynomial-time hierarchy in Theorem 3.8. Intuitively, a move by Merlin demonstrates a new piece of proof to Arthur, and so it corresponds to an existential quantifier $\exists$, while a move by Arthur is simply a sequence of random bits, and to a probabilistic quantifier $\exists^{+}$. (Recall that $\left(\exists_{r}^{+} y,|y|=m\right)$ means “for at least $r \cdot 2^{m}$ strings $y$ of length $m$,” and $\exists^{+}$is the abbreviation for $\exists_{3 / 4}^{+}$.)

Recall that two predicates $R_{1}$ and $R_{0}$ are complementary if $R_{1}$ implies $\neg R_{0}$.

数学代考|计算复杂性理论代写COMPUTATIKNAL COMPLEXITY THEORY代考|EXAMPLES AND DEFINITIONS

1验证者拥有多项式时间 DTM 的能力，而证明者拥有无限的计算能力。
2游戏只持续一轮，证明者先走一步。
3当且仅当证明者能够在 F 上赢得比赛时，公式 F 在 SAT 中。

数学代考|计算复杂性理论代写COMPUTATIKNAL COMPLEXITY THEORY代考|ARTHUR–MERLIN PROOF SYSTEMS

1Arthur 的每个新查询都只是一个新的随机位序列；和
2Arthur 生成的每一个随机数都必须作为查询发送给 Merlin，即使 Arthur 并不期待 Merlin 的回答，因此 Arthur 生成的最后一个随机数算作两个玩家之间的一轮通信。

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