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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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 A!= B especially when A is known to be a subset of 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 S_A to each class of functions A, such that, to separate complexity classes A from a superclass B', it suffices to observe a change in" the number of holes", ie homology, in S_A as a subclass B of B'is added to A. In other words, if the homologies of S_A and S_ {A union B} are different, then A!= B'. We develop the underlying theory of functions based on homological commutative algebra and Stanley-Reisner theory, and prove a" maximal principle" for polynomial threshold functions that is used to recover Aspnes, Beigel, Furst, and Rudich's characterization of the polynomial threshold degree of symmetric functions. A surprising coincidence is demonstrated, where, roughly speaking, the maximal dimension of" holes" in S_A upper bounds the VC dimension of A, with equality for common computational cases such as the class of polynomial threshold functions or the class of linear functionals over the finite field of 2 elements, or common algebraic cases such as when the Stanley-Reisner ring of S_A 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 cone, in terms of homological

This paper explores the Nash equilibria of a variant of the Colonel Blotto game, which we call the Asymmetric Colonel Blotto game. In the Colonel Blotto game, two players simultaneously distribute forces across n battlefields. Within each battlefield, the player that allocates the higher level of force wins. The payo↵ of the game is the proportion of wins on the individual battlefields. In the asymmetric version, the levels of force distributed to the battlefields must be nondecreasing. In this paper, we find a family of Nash equilibria for the case with three battlefields and equal levels of force and prove the uniqueness of the marginal distributions. We also find the unique equilibrium payo↵ for all possible levels of force in the case with two battlefields, and obtain partial results for the unique equilibrium payo↵ for asymmetric levels of force in the case with three battlefields.