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Why do some polynomials require large circuits? A deep analysis of their computational complexity!
In computational complexity theory, arithmetic circuits have become the standard model for computing polynomials. Typically, arithmetic circuits take variables or numbers as input and can calculate expressions by addition or multiplication. These circuits not only provide a formal way to understand the complexity of computing polynomials, but also allow us to explore how to efficiently compute specific polynomials.
Every circuit has two complexity metrics: size and depth.
The size of a circuit refers to the number of gates in it, while the depth represents the length of the longest path in the graph. For example, if a circuit has a size of six and a depth of two, then its computational power can be reasonably expected. The structure of the circuit is a directed acyclic graph, and the outputs of the input gates are used to calculate the final value of the polynomial.
Given a polynomial f, we often ask what is the best way to evaluate it. For example, how to make the circuit that computes f as small as possible. The answer to this question usually has two parts: first, find a circuit that can compute f, which is called an upper bound on the complexity of f; second, prove that No other circuit can be more efficient than this one, and this is a lower bound on the complexity of f.
Lower bounds are usually more difficult to prove than upper bounds, because they involve proving all circuits simultaneously.
While the two tasks are closely related, the difficulty of proving lower bounds is often daunting, especially when we have to consider very large polynomials. Past research has shown that the computational resources required for certain polynomials rise dramatically as their degree increases. This point has been widely discussed in computational complexity theory.
When talking about algorithms, examples like Strassen's algorithm come to mind. This algorithm can perform the multiplication of two n × n matrices in a size of approximately n^2.807, while the traditional method requires a circuit size of n^3 . Behind all this is a profound mathematical wisdom that changes the way mathematical operations are calculated.
The study highlights the delicate balance between upper and lower bounds on polynomial complexity.
In addition, we also observed some interesting phenomena in the process of calculating the determinant. Traditional computational methods require circuits of size approximately n!, but in practice there are circuits that scale polynomially and require only linear depth. These advances show the power of mathematical research in the search for streamlined ways to compute.
However, our knowledge of the situation with retrospective lower bounds is quite limited. Some key problems remain unsolved, especially finding an example that points out an obvious polynomial to prove that the lower bound of the circuit is a superpolynomial, which will become a major challenge for the academic community. Compared with polynomial degree calculations, the academic community's exploration of some simplified models, such as monotonic circuits, constant depth circuits, and multilinear circuits, has shown considerable potential. These models provide rich perspectives in understanding.
In this whole process, the most striking issue is the relationship between P and NP. The central question of this theory is whether a given problem can be solved as easily as the solution can be tested. The VP and VNP problems proposed by Vaillant attempt to explore the same problem from an algebraic perspective. VP is an analog of algebraic P, containing polynomials with polynomial circuits, while VNP is considered algebraic NP. There is currently no conclusive evidence to show whether VP is equal to VNP.
Proving the connection between benchmarks and complexity theory continues to challenge the boundaries of our knowledge.
As we gain a deeper understanding of how to compute polynomials efficiently, some apparent gaps between theory and practice emerge. In the future, how circuit design can adapt to the changes in these theories will be a topic that the computer science community needs to continue to explore. One can’t help but wonder, as technology advances, what creative solutions can be born in this complex computing world to meet the ever-growing challenges?
