# ALL (complexity)

In computability and complexity theory, **ALL** is the class of all decision problems.

## Relations to other classes[edit]

**ALL** contains all of the complex classes of decision problems, including **RE** and **co-RE**.

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## Relations to other classes[edit]

## External links[edit]

In computability and complexity theory, **ALL** is the class of all decision problems.

**ALL** contains all of the complex classes of decision problems, including **RE** and **co-RE**.

RELATED RESEARCH TOPICS

1. 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

2. Decision problem – In computability theory and computational complexity theory, a decision problem is a question in some formal system that can be posed as a yes-no question, dependent on the input values. For example, the given two numbers x and y, does x evenly divide y. is a decision problem. The answer can be yes or no, and depends upon the values of x and y. A method for solving a problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the problem given two numbers x and y, does x evenly divide y. would give the steps for determining whether x evenly divides y. One such algorithm is long division, taught to school children. If the remainder is zero the answer produced is yes, otherwise it is no, a decision problem which can be solved by an algorithm, such as this example, is called decidable. The field of computational complexity categorizes decidable decision problems by how difficult they are to solve, difficult, in this sense, is described in terms of the computational resources needed by the most efficient algorithm for a certain problem. The field of theory, meanwhile, categorizes undecidable decision problems by Turing degree. A decision problem is any arbitrary yes-or-no question on a set of inputs. Because of this, it is traditional to define the decision problem equivalently as and these inputs can be natural numbers, but may also be values of some other kind, such as strings over the binary alphabet or over some other finite set of symbols. The subset of strings for which the problem returns yes is a formal language, alternatively, using an encoding such as Gödel numberings, any string can be encoded as a natural number, via which a decision problem can be defined as a subset of the natural numbers. A classic example of a decision problem is the set of prime numbers. It is possible to decide whether a given natural number is prime by testing every possible nontrivial factor. Although much more efficient methods of primality testing are known, the existence of any method is enough to establish decidability. A decision problem A is called decidable or effectively solvable if A is a recursive set, a problem is called partially decidable, semidecidable, solvable, or provable if A is a recursively enumerable set. Problems that are not decidable are called undecidable, the halting problem is an important undecidable decision problem, for more examples, see list of undecidable problems. Decision problems can be ordered according to many-one reducibility and related to feasible reductions such as polynomial-time reductions

3. 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

4. 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