We investigate the closure properties of read-once oblivious Algebraic Branching Programs (roABPs) under various natural algebraic operations and prove the following. - Non-closure under factoring: There is a sequence of explicit polynomials $(f_n(x_1,\ldots, x_n))_n$ that have $\mathsf{poly}(n)$-sized roABPs such that some irreducible factor of $f_n$ does not have roABPs of superpolynomial size in any order. - Non-closure under powering: There is a sequence of polynomials $(f_n(x_1,\ldots, x_n))_n$ with $\mathsf{poly}(n)$-sized roABPs such that any super-constant power of $f_n$ does not have roABPs of polynomial size in any order (and $f_n^n$ requires exponential size in any order). - Non-closure under symmetric compositions: There are symmetric polynomials $(f_n(e_1,\ldots, e_n))_n$ that have roABPs of polynomial size such that $f_n(x_1,\ldots, x_n)$ do not have roABPs of subexponential size. (Here, $e_1,\ldots, e_n$ denote the elementary symmetric polynomials in $n$ variables.) These results should be viewed in light of known results on models such as algebraic circuits, (general) algebraic branching programs, formulas and constant-depth circuits, all of which are known to be closed under these operations. To prove non-closure under factoring, we construct hard polynomials based on expander graphs using gadgets that lift their hardness from sparse polynomials to roABPs. For symmetric compositions, we show that the circulant polynomial requires roABPs of exponential size in every variable order.

翻译：本文研究了只读一次无记忆代数分支程序（roABPs）在多种自然代数运算下的闭包性质，并证明了以下结论。 - 在因式分解下的非闭包性：存在一个显式的多项式序列 $(f_n(x_1,\ldots, x_n))_n$，其具有 $\mathsf{poly}(n)$ 规模的 roABPs，但 $f_n$ 的某个不可约因子在任何变量顺序下都不具有超多项式规模的 roABPs。 - 在幂运算下的非闭包性：存在一个多项式序列 $(f_n(x_1,\ldots, x_n))_n$，其具有 $\mathsf{poly}(n)$ 规模的 roABPs，但 $f_n$ 的任何超常数次幂在任何变量顺序下都不具有多项式规模的 roABPs（且 $f_n^n$ 在任何顺序下都需要指数规模）。 - 在对称复合下的非闭包性：存在对称多项式序列 $(f_n(e_1,\ldots, e_n))_n$，其具有多项式规模的 roABPs，但 $f_n(x_1,\ldots, x_n)$ 不具有亚指数规模的 roABPs。（此处 $e_1,\ldots, e_n$ 表示 $n$ 个变量的初等对称多项式。）这些结果应与已知的关于代数电路、（一般）代数分支程序、公式以及常数深度电路等模型的结果对照来看，已知这些模型在上述运算下都是封闭的。为了证明在因式分解下的非闭包性，我们基于扩展图构造了困难多项式，并利用特定构件将其从稀疏多项式的困难性提升至 roABPs 的困难性。对于对称复合，我们证明了循环多项式在任何变量顺序下都需要指数规模的 roABPs。