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In computational complexity theory, the class IP (which stands for Interactive Polynomial time) is the class of …

General representation of an interactive proof protocol.

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1. IP (complexity) – In computational complexity theory, the class IP is the class of problems solvable by an interactive proof system. The concept of a proof system was first introduced by Shafi Goldwasser, Silvio Micali. These two machines exchange a number, p, of messages and once the interaction is completed. At most two additional rounds of interaction are required to replicate the effect of a private-coin protocol, the opposite inclusion is straightforward, because the verifier can always send to the prover the results of their private coin tosses, which proves that the two types of protocols are equivalent. The proof can be divided in two parts, we show that IP ⊆ PSPACE and PSPACE ⊆ IP, in order to demonstrate that IP ⊆ PSPACE, we present a simulation of an interactive proof system by a polynomial space machine. This expression is the average of NMj+1, weighted by the probability that the verifier sent message mj+1, take M0 to be the empty message sequence, here we will show that NM0 can be computed in polynomial space, and that NM0 = Pr. First, to compute NM0, an algorithm can calculate the values NMj for every j. Since the depth of the recursion is p, only polynomial space is necessary, the second requirement is that we need NM0 = Pr, the value needed to determine whether w is in A. We use induction to prove this as follows and we must show that for every 0 ≤ j ≤ p and every Mj, NMj = Pr, and we will do this using induction on j. The base case is to prove for j = p, then we will use induction to go from p down to 0. The base case of j = p is fairly simple, since mp is either accept or reject, if mp is accept, NMp is defined to be 1 and Pr =1 since the message stream indicates acceptance, thus the claim is true. If mp is reject, the argument is very similar, for the inductive hypothesis, we assume that for some j+1 ≤ p and any message sequence Mj+1, NMj = Pr and then prove the hypothesis for j and any message sequence Mj. If j is even, mj+1 is a message from V to P, by the definition of NMj, N M j = ∑ m j +1 Pr r N M j +1. Then, by the hypothesis, we can say this is equal to ∑ m j +1 Pr r ∗ Pr. Finally, by definition, we can see that this is equal to Pr, If j is odd, mj+1 is a message from P to V. By definition, N M j = max m j +1 N M j +1, then, by the inductive hypothesis, this equals max m j +1 ∗ Pr. This is equal to Pr since, max m j +1 Pr ≤ Pr because the prover on the side could send the message mj+1 to maximize the expression on the left-hand side. And, max m j +1 Pr ≥ Pr Since the same Prover cannot do any better than send that same message, thus, this holds whether i is even or odd and the proof that IP ⊆ PSPACE is complete

2. Quantum computing – Quantum computing studies theoretical computation systems that make direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from binary digital electronic computers based on transistors, a quantum Turing machine is a theoretical model of such a computer, and is also known as the universal quantum computer. The field of computing was initiated by the work of Paul Benioff and Yuri Manin in 1980, Richard Feynman in 1982. A quantum computer with spins as quantum bits was also formulated for use as a quantum space–time in 1968, there exist quantum algorithms, such as Simons algorithm, that run faster than any possible probabilistic classical algorithm. A classical computer could in principle simulate a quantum algorithm, as quantum computation does not violate the Church–Turing thesis, on the other hand, quantum computers may be able to efficiently solve problems which are not practically feasible on classical computers. A classical computer has a made up of bits, where each bit is represented by either a one or a zero. A quantum computer maintains a sequence of qubits, in general, a quantum computer with n qubits can be in an arbitrary superposition of up to 2 n different states simultaneously. A quantum computer operates by setting the qubits in a drift that represents the problem at hand. The sequence of gates to be applied is called a quantum algorithm, the calculation ends with a measurement, collapsing the system of qubits into one of the 2 n pure states, where each qubit is zero or one, decomposing into a classical state. The outcome can therefore be at most n classical bits of information, Quantum algorithms are often probabilistic, in that they provide the correct solution only with a certain known probability. Note that the term non-deterministic computing must not be used in case to mean probabilistic. An example of an implementation of qubits of a computer could start with the use of particles with two spin states, down and up. This is true because any such system can be mapped onto an effective spin-1/2 system, a quantum computer with a given number of qubits is fundamentally different from a classical computer composed of the same number of classical bits. This means that when the state of the qubits is measured. To better understand this point, consider a classical computer that operates on a three-bit register, if there is no uncertainty over its state, then it is in exactly one of these states with probability 1. However, if it is a computer, then there is a possibility of it being in any one of a number of different states. The state of a quantum computer is similarly described by an eight-dimensional vector. Here, however, the coefficients a k are complex numbers, and it is the sum of the squares of the absolute values, ∑ i | a i |2

3. String (computer science) – In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, a string is generally understood as a data type and is often implemented as an array of bytes that stores a sequence of elements, typically characters, using some character encoding. A string may also more general arrays or other sequence data types and structures. When a string appears literally in source code, it is known as a literal or an anonymous string. In formal languages, which are used in logic and theoretical computer science. Let Σ be a non-empty finite set of symbols, called the alphabet, no assumption is made about the nature of the symbols. A string over Σ is any sequence of symbols from Σ. For example, if Σ =, then 01011 is a string over Σ, the length of a string s is the number of symbols in s and can be any non-negative integer, it is often denoted as |s|. The empty string is the string over Σ of length 0. The set of all strings over Σ of length n is denoted Σn, for example, if Σ =, then Σ2 =. Note that Σ0 = for any alphabet Σ, the set of all strings over Σ of any length is the Kleene closure of Σ and is denoted Σ*. In terms of Σn, Σ ∗ = ⋃ n ∈ N ∪ Σ n For example, if Σ =, although the set Σ* itself is countably infinite, each element of Σ* is a string of finite length. A set of strings over Σ is called a language over Σ. For example, if Σ =, the set of strings with an number of zeros, is a formal language over Σ. Concatenation is an important binary operation on Σ*, for any two strings s and t in Σ*, their concatenation is defined as the sequence of symbols in s followed by the sequence of characters in t, and is denoted st. For example, if Σ =, s = bear, and t = hug, then st = bearhug, String concatenation is an associative, but non-commutative operation. The empty string ε serves as the identity element, for any string s, therefore, the set Σ* and the concatenation operation form a monoid, the free monoid generated by Σ. In addition, the length function defines a monoid homomorphism from Σ* to the non-negative integers, a string s is said to be a substring or factor of t if there exist strings u and v such that t = usv

4. Shafi Goldwasser – Shafrira Goldwasser is an American-born Israeli computer scientist. She is a professor of engineering and computer science at MIT. She joined MIT in 1983, and in 1997 became the first holder of the RSA Professorship and she became a professor at the Weizmann Institute of Science, concurrent to her professorship at MIT, in 1993. She is a member of the Theory of Computation group at MIT Computer Science, Goldwasser was a co-recipient of the 2012 Turing Award. Goldwassers research areas include computational complexity theory, cryptography and computational number theory and her work in complexity theory includes the classification of approximation problems, showing that some problems in NP remain hard even when only an approximate solution is needed. Goldwasser has twice won the Gödel Prize in theoretical science, first in 1993. In 2001 she was elected to the American Academy of Arts and Sciences, in 2004 she was elected to the National Academy of Science and she was selected as an IACR Fellow in 2007. Goldwasser received the 2008-2009 Athena Lecturer Award of the Association for Computing Machinerys Committee on Women in Computing and she is the recipient of The Franklin Institutes 2010 Benjamin Franklin Medal in Computer and Cognitive Science. She received the IEEE Emanuel R. Piore Award in 2011, and was awarded the 2012 Turing Award along with Silvio Micali for their work in the field of cryptography