computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...

, an integer circuit is a circuit model of computation 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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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 Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

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 list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the ...

. It is a natural extension of the

circuits over sets of natural numbers Circuit may refer to: Science and technology Electrical engineering * Electrical circuit, a complete electrical network with a closed-loop giving a return path for current ** Analog circuit, uses continuous signal levels ** Balanced circ ...

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