Arithmetic circuit complexity

In computational complexity theory, arithmetic circuits are the standard model for computing polynomials. Informally, an arithmetic circuit takes as inputs either variables or numbers, and is allowed to either add or multiply two expressions it has already computed. Arithmetic circuits provide a formal way to understand the complexity of computing polynomials. The basic type of question in this line of research is "what is the most efficient way to compute a given polynomial ?"

Definitions

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A simple arithmetic circuit.

An arithmetic circuit   over the field   and the set of variables   is a directed acyclic graph as follows. Every node in it with indegree zero is called an input gate and is labeled by either a variable   or a field element in   Every other gate is labeled by either   or   in the first case it is a sum gate and in the second a product gate. An arithmetic formula is a circuit in which every gate has outdegree one (and so the underlying graph is a directed tree).

A circuit has two complexity measures associated with it: size and depth. The size of a circuit is the number of gates in it, and the depth of a circuit is the length of the longest directed path in it. For example, the circuit in the figure has size six and depth two.

An arithmetic circuit computes a polynomial in the following natural way. An input gate computes the polynomial it is labeled by. A sum gate   computes the sum of the polynomials computed by its children (a gate   is a child of   if the directed edge   is in the graph). A product gate computes the product of the polynomials computed by its children. Consider the circuit in the figure, for example: the input gates compute (from left to right)   and   the sum gates compute   and   and the product gate computes  

Overview

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Given a polynomial   we may ask ourselves what is the best way to compute it — for example, what is the smallest size of a circuit computing   The answer to this question consists of two parts. The first part is finding some circuit that computes   this part is usually called upper bounding the complexity of   The second part is showing that no other circuit can do better; this part is called lower bounding the complexity of   Although these two tasks are strongly related, proving lower bounds is usually harder, since in order to prove a lower bound one needs to argue about all circuits at the same time.

Note that we are interested in the formal computation of polynomials, rather than the functions that the polynomials define. For example, consider the polynomial   over the field of two elements this polynomial represents the zero function, but it is not the zero polynomial. This is one of the differences between the study of arithmetic circuits and the study of Boolean circuits. In Boolean complexity, one is mostly interested in computing a function, rather than some representation of it (in our case, a representation by a polynomial). This is one of the reasons that make Boolean complexity harder than arithmetic complexity. The study of arithmetic circuits may also be considered as one of the intermediate steps towards the study of the Boolean case,[1] which we hardly understand.

Upper bounds

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As part of the study of the complexity of computing polynomials, some clever circuits (alternatively algorithms) were found. A well-known example is Strassen's algorithm for matrix product. The straightforward way for computing the product of two   matrices requires a circuit of size order   Strassen showed that we can, in fact, multiply two matrices using a circuit of size roughly   Strassen's basic idea is a clever way for multiplying   matrices. This idea is the starting point of the best theoretical way for multiplying two matrices that takes time roughly  

Another interesting story lies behind the computation of the determinant of an   matrix. The naive way for computing the determinant requires circuits of size roughly   Nevertheless, we know that there are circuits of size polynomial in   for computing the determinant. These circuits, however, have depth that is linear in   Berkowitz came up with an improvement: a circuit of size polynomial in   but of depth  [2]

We would also like to mention the best circuit known for the permanent of an   matrix. As for the determinant, the naive circuit for the permanent has size roughly   However, for the permanent the best circuit known has size roughly   which is given by Ryser's formula: for an   matrix  

 

(this is a depth three circuit).

Lower bounds

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In terms of proving lower bounds, our knowledge is very limited. Since we study the computation of formal polynomials, we know that polynomials of very large degree require large circuits, for example, a polynomial of degree   require a circuit of size roughly   So, the main goal is to prove lower bound for polynomials of small degree, say, polynomial in   In fact, as in many areas of mathematics, counting arguments tell us that there are polynomials of polynomial degree that require circuits of superpolynomial size. However, these counting arguments usually do not improve our understanding of computation. The following problem is the main open problem in this area of research: find an explicit polynomial of polynomial degree that requires circuits of superpolynomial size.

The state of the art is a   lower bound for the size of a circuit computing, e.g., the polynomial   given by Strassen and by Baur and Strassen. More precisely, Strassen used Bézout's theorem to show that any circuit that simultaneously computes the   polynomials   is of size   and later Baur and Strassen showed the following: given an arithmetic circuit of size   computing a polynomial   one can construct a new circuit of size at most   that computes   and all the   partial derivatives of   Since the partial derivatives of   are   the lower bound of Strassen applies to   as well.[3] This is one example where some upper bound helps in proving lower bounds; the construction of a circuit given by Baur and Strassen implies a lower bound for more general polynomials.

The lack of ability to prove lower bounds brings us to consider simpler models of computation. Some examples are: monotone circuits (in which all the field elements are nonnegative real numbers), constant depth circuits, and multilinear circuits (in which every gate computes a multilinear polynomial). These restricted models have been studied extensively and some understanding and results were obtained.

Algebraic P and NP

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The most interesting open problem in computational complexity theory is the P vs. NP problem. Roughly, this problem is to determine whether a given problem can be solved as easily as it can be shown that a solution exists to the given problem. In his seminal work Valiant[4] suggested an algebraic analog of this problem, the VP vs. VNP problem.

The class VP is the algebraic analog of P; it is the class of polynomials   of polynomial degree that have polynomial size circuits over a fixed field   The class VNP is the analog of NP. VNP can be thought of as the class of polynomials   of polynomial degree such that given a monomial we can determine its coefficient in   efficiently, with a polynomial size circuit.

One of the basic notions in complexity theory is the notion of completeness. Given a class of polynomials (such as VP or VNP), a complete polynomial   for this class is a polynomial with two properties: (1) it is part of the class, and (2) any other polynomial   in the class is easier than   in the sense that if   has a small circuit then so does   Valiant showed that the permanent is complete for the class VNP. So in order to show that VP is not equal to VNP, one needs to show that the permanent does not have polynomial size circuits. This remains an outstanding open problem.

Depth reduction

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One benchmark in our understanding of the computation of polynomials is the work of Valiant, Skyum, Berkowitz and Rackoff.[5] They showed that if a polynomial   of degree   has a circuit of size   then   also has a circuit of size polynomial in   and   of depth   For example, any polynomial of degree   that has a polynomial size circuit, also has a polynomial size circuit of depth roughly   This result generalizes the circuit of Berkowitz to any polynomial of polynomial degree that has a polynomial size circuit (such as the determinant). The analog of this result in the Boolean setting is believed to be false.

One corollary of this result is a simulation of circuits by relatively small formulas, formulas of quasipolynomial size: if a polynomial   of degree   has a circuit of size   then it has a formula of size   This simulation is easier than the depth reduction of Valiant el al. and was shown earlier by Hyafil.[6]

See also

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  • Polynomial evaluation for a more general and less formal discussion of the complexity of polynomial evaluation.

Further reading

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  • Bürgisser, Peter (2000). Completeness and reduction in algebraic complexity theory. Algorithms and Computation in Mathematics. Vol. 7. Berlin: Springer-Verlag. ISBN 978-3-540-66752-0. Zbl 0948.68082.
  • Bürgisser, Peter; Clausen, Michael; Shokrollahi, M. Amin (1997). Algebraic complexity theory. Grundlehren der Mathematischen Wissenschaften. Vol. 315. With the collaboration of Thomas Lickteig. Berlin: Springer-Verlag. ISBN 978-3-540-60582-9. Zbl 1087.68568.
  • von zur Gathen, Joachim (1988). "Algebraic complexity theory". Annual Review of Computer Science. 3: 317–347. doi:10.1146/annurev.cs.03.060188.001533.

Footnotes

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  1. ^ L. G. Valiant. Why is Boolean complexity theory difficult? Proceedings of the London Mathematical Society symposium on Boolean function complexity, pp. 84–94, 1992.
  2. ^ S. J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Inf. Prod. Letters 18, pp. 147–150, 1984.
  3. ^ Shpilka, Amir; Yehudayoff, Amir (2010). "Arithmetic Circuits: a survey of recent results and open questions" (PDF). Foundations and Trends in Theoretical Computer Science. 5 (3–4): 207-388. doi:10.1561/0400000039.
  4. ^ Valiant, L. G. (1979). Completeness classes in algebra. ACM Press. doi:10.1145/800135.804419.
  5. ^ Valiant, L. G.; Skyum, S.; Berkowitz, S.; Rackoff, C. (1983). "Fast Parallel Computation of Polynomials Using Few Processors". SIAM Journal on Computing. 12 (4): 641–644. doi:10.1137/0212043. ISSN 0097-5397.
  6. ^ Hyafil, Laurent (1979). "On the Parallel Evaluation of Multivariate Polynomials". SIAM Journal on Computing. 8 (2): 120–123. doi:10.1137/0208010. ISSN 0097-5397.