A Homological Theory of Functions

Greg Yang

Arithmetic Geometry and Commutative Algebra mathscidoc:1912.43160

arXiv preprint arXiv:1701.02302
In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes Aot= B especially when Aot= B is known to be a subset of Aot= B . In this paper we introduce a homological theory of functions that can be used to establish complexity separations, while also providing other interesting consequences. We propose to associate a topological space Aot= B to each class of functions Aot= B , such that, to separate complexity classes Aot= B , it suffices to observe a change in" the number of holes", ie homology, in Aot= B as a subclass Aot= B of Aot= B is added to Aot= B . In other words, if the homologies of Aot= B and Aot= B are different, then Aot= B . We develop the underlying theory of functions based on combinatorial and homological commutative algebra and Stanley-Reisner theory, and recover Minsky and Papert's 1969 result that parity cannot be computed by nonmaximal degree polynomial threshold functions. In the process, we derive a" maximal principle" for polynomial threshold functions that is used to extend this result further to arbitrary symmetric functions. A surprising coincidence is demonstrated, where the maximal dimension of" holes" in Aot= B upper bounds the VC dimension of Aot= B , with equality for common computational cases such as the class of polynomial threshold functions or the class of linear functionals in Aot= B , or common algebraic cases such as when the Stanley-Reisner ring of Aot= B is Cohen-Macaulay. As another interesting application of our theory, we prove a result that a priori has nothing to do with complexity separation: it characterizes when a vector subspace intersects the positive
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  title={A Homological Theory of Functions},
  author={Greg Yang},
  booktitle={arXiv preprint arXiv:1701.02302},
Greg Yang. A Homological Theory of Functions. In arXiv preprint arXiv:1701.02302. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112642738707720.
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