computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by ...

, an integer circuit is a circuit

model of computation In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how ...

in which inputs to the circuit are

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

s of

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s and each gate of the circuit computes either a set operation or an arithmetic operation on its input sets. As an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

ic problem, the possible questions are to find if a given integer is an element of the output node or if two circuits compute the same set. The decidability is still an open question, but there are results on restriction of those circuits. Finding answers to some questions about this model could serve as a proof to many important mathematical conjectures, like

Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold ...

. It is a natural extension of the

circuits over sets of natural numbers Circuits over natural numbers are a mathematical model used in studying computational complexity theory. They are a special case of circuits. The object is a labeled directed acyclic graph the nodes of which evaluate to sets of natural numbers, th ...

when the considered set contains also negative integers, the definitions, which does not change, will not be repeated on this page. Only the differences will be mentioned.

Complexity of the membership problem

The membership problem is the problem of deciding, given an integer circuit ''C'', an input to the circuit ''X'', and a specific integer ''n'', whether the integer ''n'' is in the output of the circuit ''C'' when provided with input ''X''. The computational complexity of this problem depends on the type of gates allowed in the circuit ''C''. The table below summarizes the computational complexity of the membership problem for various classes of integer circuits. Here, MF

_

(O) denotes the classes defined by O-formulae, which are O-circuits with maximal fan-out 1.

References

{{Reflist Computational complexity theory Arithmetic