1.
Turing machine
–
Despite the models simplicity, given any computer algorithm, a Turing machine can be constructed that is capable of simulating that algorithms logic. The machine operates on an infinite memory tape divided into discrete cells, the machine positions its head over a cell and reads the symbol there. The Turing machine was invented in 1936 by Alan Turing, who called it an a-machine, thus, Turing machines prove fundamental limitations on the power of mechanical computation. Turing completeness is the ability for a system of instructions to simulate a Turing machine, a Turing machine is a general example of a CPU that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. More specifically, it is a capable of enumerating some arbitrary subset of valid strings of an alphabet. Assuming a black box, the Turing machine cannot know whether it will eventually enumerate any one specific string of the subset with a given program and this is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing. The Turing machine is capable of processing an unrestricted grammar, which implies that it is capable of robustly evaluating first-order logic in an infinite number of ways. This is famously demonstrated through lambda calculus, a Turing machine that is able to simulate any other Turing machine is called a universal Turing machine. The thesis states that Turing machines indeed capture the notion of effective methods in logic and mathematics. Studying their abstract properties yields many insights into computer science and complexity theory, at any moment there is one symbol in the machine, it is called the scanned symbol. The machine can alter the scanned symbol, and its behavior is in part determined by that symbol, however, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings, the Turing machine mathematically models a machine that mechanically operates on a tape. On this tape are symbols, which the machine can read and write, one at a time, in the original article, Turing imagines not a mechanism, but a person whom he calls the computer, who executes these deterministic mechanical rules slavishly. If δ is not defined on the current state and the current tape symbol, Q0 ∈ Q is the initial state F ⊆ Q is the set of final or accepting states. The initial tape contents is said to be accepted by M if it eventually halts in a state from F, Anything that operates according to these specifications is a Turing machine. The 7-tuple for the 3-state busy beaver looks like this, Q = Γ = b =0 Σ = q 0 = A F = δ = see state-table below Initially all tape cells are marked with 0. In the words of van Emde Boas, p.6, The set-theoretical object provides only partial information on how the machine will behave and what its computations will look like. For instance, There will need to be many decisions on what the symbols actually look like, and a failproof way of reading and writing symbols indefinitely

2.
Computational complexity theory
–
A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity measures are used, such as the amount of communication, the number of gates in a circuit. One of the roles of computational complexity theory is to determine the limits on what computers can. Closely related fields in computer science are analysis of algorithms. More precisely, computational complexity theory tries to classify problems that can or cannot be solved with appropriately restricted resources, a computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a problem is referred to as a problem instance. In computational complexity theory, a problem refers to the question to be solved. In contrast, an instance of this problem is a rather concrete utterance, for example, consider the problem of primality testing. The instance is a number and the solution is yes if the number is prime, stated another way, the instance is a particular input to the problem, and the solution is the output corresponding to the given input. For this reason, complexity theory addresses computational problems and not particular problem instances, when considering computational problems, a problem instance is a string over an alphabet. Usually, the alphabet is taken to be the binary alphabet, as in a real-world computer, mathematical objects other than bitstrings must be suitably encoded. For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices and this can be achieved by ensuring that different representations can be transformed into each other efficiently. Decision problems are one of the objects of study in computational complexity theory. A decision problem is a type of computational problem whose answer is either yes or no. A decision problem can be viewed as a language, where the members of the language are instances whose output is yes. The objective is to decide, with the aid of an algorithm, if the algorithm deciding this problem returns the answer yes, the algorithm is said to accept the input string, otherwise it is said to reject the input. An example of a problem is the following

3.
CC (complexity)
–
In computational complexity theory, CC is the complexity class containing decision problems which can be solved by comparator circuits of polynomial size. The most important problem which is complete for CC is a variant of the stable marriage problem. A comparator circuit is a network of wires and gates, each comparator gate, which is a directed edge connecting two wires, takes its two inputs and outputs them in sorted order. The input to any wire can be either a variable, its negation, one of the wires is designated as the output wire. The comparator circuit value problem is the problem of evaluating a comparator circuit given an encoding of the circuit, the complexity class CC is defined as the class of problems logspace reducible to CCVP. An equivalent definition is the class of problems AC0 reducible to CCVP, since there are sorting networks which can be constructed in AC0, this shows that the majority function is in CC. A problem in CC is CC-complete if every problem in CC can be reduced to it using a logspace reduction, the comparator circuit value problem is CC-complete. In the stable marriage problem, there is a number of men and women. Each person ranks all members of the opposite sex, a matching between men and women is stable if there are no unpaired man and woman who prefer each other over their current partners. Among the stable matchings, there is one in each woman gets the best man that she ever gets in any stable matching. The decision version of the matching problem is, given the rankings of all men and women, whether a given man. Although the classical Gale–Shapley algorithm cannot be implemented as a comparator circuit, another problem which is CC-complete is lexicographically-first maximal matching. In this problem, we are given a graph with an order on the vertices. The lexicographically-first maximal matching is obtained by successively matching vertices from the first bipartition to the minimal available vertices from the second bipartition, the problem asks whether the given edge belongs to this matching. Scott Aaronson showed that the model is CC-complete. The problem is to decide whether any pebbles are present in a particular pile after executing the program and he used this to show that the problem of deciding whether any balls reach a designated sink vertex in a Digi-Comp II-like device is also CC-complete. The comparator circuit evaluation problem can be solved in polynomial time, on the other hand, comparator circuits can solve directed reachability, and so CC contains NL. There is a world in which CC and NC are incomparable

4.
Big O notation
–
Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, in computer science, big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows. Big O notation characterizes functions according to their rates, different functions with the same growth rate may be represented using the same O notation. The letter O is used because the rate of a function is also referred to as order of the function. A description of a function in terms of big O notation usually only provides a bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, Big O notation is also used in many other fields to provide similar estimates. Let f and g be two functions defined on some subset of the real numbers. That is, f = O if and only if there exists a real number M. In many contexts, the assumption that we are interested in the rate as the variable x goes to infinity is left unstated. If f is a product of several factors, any constants can be omitted, for example, let f = 6x4 − 2x3 +5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms, 6x4, −2x3, and 5, of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the rule, 6x4 is a product of 6. Omitting this factor results in the simplified form x4, thus, we say that f is a big-oh of. Mathematically, we can write f = O, one may confirm this calculation using the formal definition, let f = 6x4 − 2x3 +5 and g = x4. Applying the formal definition from above, the statement that f = O is equivalent to its expansion, | f | ≤ M | x 4 | for some choice of x0 and M. To prove this, let x0 =1 and M =13, Big O notation has two main areas of application. In mathematics, it is used to describe how closely a finite series approximates a given function. In computer science, it is useful in the analysis of algorithms, in both applications, the function g appearing within the O is typically chosen to be as simple as possible, omitting constant factors and lower order terms

5.
NP-completeness
–
In computational complexity theory, a decision problem is NP-complete when it is both in NP and NP-hard. The set of NP-complete problems is often denoted by NP-C or NPC, the abbreviation NP refers to nondeterministic polynomial time. That is, the required to solve the problem using any currently known algorithm increases very quickly as the size of the problem grows. As a consequence, determining whether or not it is possible to solve problems quickly. NP-complete problems are addressed by using heuristic methods and approximation algorithms. A problem p in NP is NP-complete if every problem in NP can be transformed into p in polynomial time. NP-complete problems are studied because the ability to quickly verify solutions to a problem seems to correlate with the ability to solve that problem. It is not known whether every problem in NP can be quickly solved—this is called the P versus NP problem, because of this, it is often said that NP-complete problems are harder or more difficult than NP problems in general. A decision problem C is NP-complete if, C is in NP, C can be shown to be in NP by demonstrating that a candidate solution to C can be verified in polynomial time. Note that a problem satisfying condition 2 is said to be NP-hard, a consequence of this definition is that if we had a polynomial time algorithm for C, we could solve all problems in NP in polynomial time. The concept of NP-completeness was introduced in 1971, though the term NP-complete was introduced later, at 1971 STOC conference, there was a fierce debate among the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine. This is known as the question of whether P=NP, nobody has yet been able to determine conclusively whether NP-complete problems are in fact solvable in polynomial time, making this one of the great unsolved problems of mathematics. The Clay Mathematics Institute is offering a US $1 million reward to anyone who has a proof that P=NP or that P≠NP. Cook–Levin theorem states that the Boolean satisfiability problem is NP-complete, in 1972, Richard Karp proved that several other problems were also NP-complete, thus there is a class of NP-complete problems. For more details refer to Introduction to the Design and Analysis of Algorithms by Anany Levitin, an interesting example is the graph isomorphism problem, the graph theory problem of determining whether a graph isomorphism exists between two graphs. Two graphs are isomorphic if one can be transformed into the other simply by renaming vertices, consider these two problems, Graph Isomorphism, Is graph G1 isomorphic to graph G2. Subgraph Isomorphism, Is graph G1 isomorphic to a subgraph of graph G2, the Subgraph Isomorphism problem is NP-complete. The graph isomorphism problem is suspected to be neither in P nor NP-complete and this is an example of a problem that is thought to be hard, but is not thought to be NP-complete

6.
Non-deterministic Turing machine
–
In theoretical computer science, a Turing machine is a theoretical machine that is used in thought experiments to examine the abilities and limitations of computers. In essence, a Turing machine is imagined to be a computer that reads. It determines what action it should perform next according to its internal state, an example of one of a Turing Machines rules might thus be, If you are in state 2 and you see an A, change it to B and move left. In a deterministic Turing machine, the set of rules prescribes at most one action to be performed for any given situation, by contrast, a non-deterministic Turing machine may have a set of rules that prescribes more than one action for a given situation. For example, an X on the tape in state 3 might make the DTM write a Y on the tape, move the one position to the right. For example, an X on the tape in state 3 might allow the NTM to write a Y, move right, and switch to state 5, or to write an X, move left, and stay in state 3. L is the movement to the left, and R is to the right, the difference with a standard Turing machine is that for those, the transition relation is a function. How does the NTM know which of these actions it should take, there are two ways of looking at it. One is to say that the machine is the luckiest possible guesser, it always picks a transition that eventually leads to an accepting state, the other is to imagine that the machine branches into many copies, each of which follows one of the possible transitions. Whereas a DTM has a single computation path that it follows, If at least one branch of the tree halts with an accept condition, we say that the NTM accepts the input. NTMs can compute the results as DTMs, that is, they are capable of computing the same values. The time complexity of these varies, however, as is discusssed below. NTMs effectively include DTMs as special cases, so it is clear that DTMs are not more powerful. The 3-tape DTMs are easily simulated with a normal single-tape DTM, therefore, the length of an accepting computation of the DTM is, in general, exponential in the length of the shortest accepting computation of the NTM. This is considered to be a property of simulations of NTMs by DTMs, the most famous unresolved question in computer science. The time complexity of NTMs is not the same as for DTMs and it is a common misconception that quantum computers are NTMs. It is believed but has not been proven that the power of computers is incomparable to that of NTMs. That is, problems likely exist that an NTM could efficiently solve that a computer cannot

7.
AC0
–
AC0 is a complexity class used in circuit complexity. It is the smallest class in the AC hierarchy, and consists of all families of circuits of depth O and polynomial size, with unlimited-fanin AND gates and it thus contains NC0, which has only bounded-fanin AND and OR gates. Integer addition and subtraction are computable in AC0, but multiplication is not, in 1984 Furst, Saxe, and Sipser showed that calculating the parity of an input cannot be decided by any AC0 circuits, even with non-uniformity. It follows that AC0 is not equal to NC1, because a family of circuits in the class can compute parity. More precise bounds follow from switching lemma, using them, it has been shown that there is an oracle separation between the polynomial hierarchy and PSPACE

8.
ACC0
–
ACC0, sometimes called ACC, is a class of computational models and problems defined in circuit complexity, a field of theoretical computer science. The class is defined by augmenting the class AC0 of constant-depth alternating circuits with the ability to count, specifically, a problem belongs to ACC0 if it can be solved by polynomial-size, constant-depth circuits of unbounded fan-in gates, including gates that count modulo a fixed integer. ACC0 corresponds to computation in any solvable monoid, more formally, a language belongs to AC0 if it can be computed by a family of circuits C1, C2. A language belongs to ACC0 if it belongs to AC0 for some m, in some texts, ACCi refers to a hierarchy of circuit classes with ACC0 at its lowest level, where the circuits in ACCi have depth O and polynomial size. The class ACC0 can also be defined in terms of computations of nonuniform deterministic finite automata over monoids. In this framework, the input is interpreted as elements from a fixed monoid, the class ACC0 is the family of languages accepted by a NUDFA over some monoid that does not contain an unsolvable group as a subsemigroup. This inclusion is strict, because a single MOD-2 gate computes the parity function, more generally, the function MODm can not be computed in AC0 for prime p unless m is a power of p. The class ACC0 is included in TC0 and it is conjectured that ACC0 is unable to compute the majority function of its inputs, but this remains unresolved as of July 2014. Every problem in ACC0 can be solved by circuits of depth 2, with AND gates of polylogarithmic fan-in at the inputs, the proof follows ideas of the proof of Todas theorem. Williams proves that ACC0 does not contain NEXPTIME, the proof uses many results in complexity theory, including the time hierarchy theorem, IP = PSPACE, derandomization, and the representation of ACC0 via SYM+ circuits. It is known that computing the permanent is impossible for logtime-uniform ACC0 circuits, which implies that the complexity class PP is not contained in logtime-uniform ACC0

9.
BPP (complexity)
–
BPP is one of the largest practical classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run quickly on real modern machines. BPP also contains P, the class of problems solvable in time with a deterministic machine. Alternatively, BPP can be defined using only deterministic Turing machines, for some applications this definition is preferable since it does not mention probabilistic Turing machines. In practice, a probability of 1⁄3 might not be acceptable, however. It can be any constant between 0 and 1⁄2 and the set BPP will be unchanged and this makes it possible to create a highly accurate algorithm by merely running the algorithm several times and taking a majority vote of the answers. For example, if one defined the class with the restriction that the algorithm can be wrong with probability at most 1⁄2100, besides the problems in P, which are obviously in BPP, many problems were known to be in BPP but not known to be in P. The number of problems is decreasing, and it is conjectured that P = BPP. For a long time, one of the most famous problems that was known to be in BPP, in other words, is there an assignment of values to the variables such that when a nonzero polynomial is evaluated on these values, the result is nonzero. It suffices to choose each variables value uniformly at random from a subset of at least d values to achieve bounded error probability. If the access to randomness is removed from the definition of BPP, in the definition of the class, if we replace the ordinary Turing machine with a quantum computer, we get the class BQP. Adding postselection to BPP, or allowing computation paths to have different lengths, BPPpath is known to contain NP, and it is contained in its quantum counterpart PostBQP. A Monte Carlo algorithm is an algorithm which is likely to be correct. Problems in the class BPP have Monte Carlo algorithms with polynomial bounded running time and this is compared to a Las Vegas algorithm which is a randomized algorithm which either outputs the correct answer, or outputs fail with low probability. Las Vegas algorithms with polynomial bound running times are used to define the class ZPP, alternatively, ZPP contains probabilistic algorithms that are always correct and have expected polynomial running time. This is weaker than saying it is a polynomial time algorithm, since it may run for super-polynomial time and it is known that BPP is closed under complement, that is, BPP = co-BPP. BPP is low for itself, meaning that a BPP machine with the power to solve BPP problems instantly is not any more powerful than the machine without this extra power. The relationship between BPP and NP is unknown, it is not known whether BPP is a subset of NP, NP is a subset of BPP or neither. If NP is contained in BPP, which is considered unlikely since it would imply practical solutions for NP-complete problems, then NP = RP and it is known that RP is a subset of BPP, and BPP is a subset of PP

10.
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

11.
Polynomial hierarchy
–
In computational complexity theory, the polynomial hierarchy is a hierarchy of complexity classes that generalize the classes P, NP and co-NP to oracle machines. It is a counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. There are multiple equivalent definitions of the classes of the polynomial hierarchy. If any Σ k P = Σ k +1 P, or if any Σ k P = Π k P, then the hierarchy collapses to level k, in particular, if P = NP, then the hierarchy collapses completely. The union of all classes in the hierarchy is the complexity class PH. The polynomial hierarchy is an analogue of the hierarchy and arithmetical hierarchy. It is known that PH is contained within PSPACE, but it is not known whether the two classes are equal. One useful reformulation of this problem is that PH = PSPACE if, if the polynomial hierarchy has any complete problems, then it has only finitely many distinct levels. Since there are PSPACE-complete problems, we know that if PSPACE = PH, then the hierarchy must collapse. Each class in the hierarchy contains ≤ m P -complete problems. Furthermore, each class in the hierarchy is closed under ≤ m P -reductions, meaning that for a class C in the hierarchy. These two facts together imply that if K i is a problem for Σ i P, then Σ i +1 P = N P K i. For instance, Σ2 P = N P S A T, in other words, if a language is defined based on some oracle in C, then we can assume that it is defined based on a complete problem for C. Complete problems therefore act as representatives of the class for which they are complete, the Sipser–Lautemann theorem states that the class BPP is contained in the second level of the polynomial hierarchy. Kannans theorem states that for any k, Σ2 is not contained in SIZE, todas theorem states that the polynomial hierarchy is contained in P#P. EXPTIME Exponential hierarchy Arithmetic hierarchy A. R. Meyer and L. J. Stockmeyer. The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space, in Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, pp. 125–129,1972. The paper that introduced the polynomial hierarchy, theoretical Computer Science, vol.3, pp. 1–22,1976. Michael R. Garey and David S. Johnson, computers and Intractability, A Guide to the Theory of NP-Completeness