在本文中，我们研究了代数分支程序（ABP）的顺序和变量的读取次数的限制。给定$n$变量的置换$π$，对于$π$排序的ABP（$π$-OABP），对于从源到宿的任何有向路径$p$，变量可以在$p$上最多出现一次，并且变量在$p$上出现的顺序必须遵守$π$。如果任何变量在$A$中最多出现$r$次，则ABP $A$被称为读$r$。我们的主要结果涉及到身份测试问题。在任何字段$F$和黑盒模型中，即只给出对多项式的查询访问，我们有以下结果：读取$r$ $π$ -OABP可计算多项式可以在$\DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]$中测试。我们的下一组结果研究了OABP的计算限制。它表明，任何OABP计算行列式或永久需要大小$Ω(2^n/n)$和读取$Ω(2^n/n^2)$。我们给出了一个多线性多项式$p$在$2n+1$变量在一些特定的选择领域$G$，使任何OABP计算$p$必须读取一些变量至少$2^n$倍。我们表明，在$n$变量的程度$r$的基本对称多项式可以计算的大小$O(rn)$读取$r$ OABP，但不是由读取$(r-1)$ OABP，任何$0 < 2r-1 \leq n$。最后，我们给出了一个多项式$p$和两个变量阶数$π\neq π'$的例子，使得$p$可以由一次读取的$π$-OABP计算，但是其中任何计算$p$的$π'$-OABP必须至少读取$2^n$的某个变量
英文摘要：
In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. Given a permutation $π$ of $n$ variables, for a $π$-ordered ABP ($π$-OABP), for any directed path $p$ from source to sink, a variable can appear at most once on $p$, and the order in which variables appear on $p$ must respect $π$. An ABP $A$ is said to be of read $r$, if any variable appears at most $r$ times in $A$. Our main result pertains to the identity testing problem. Over any field $F$ and in the black-box model, i.e. given only query access to the polynomial, we have the following result: read $r$ $π$-OABP computable polynomials can be tested in $\DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]$.
Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size $Ω(2^n/n)$ and read $Ω(2^n/n^2)$. We give a multilinear polynomial $p$ in $2n+1$ variables over some specifically selected field $G$, such that any OABP computing $p$ must read some variable at least $2^n$ times. We show that the elementary symmetric polynomial of degree $r$ in $n$ variables can be computed by a size $O(rn)$ read $r$ OABP, but not by a read $(r-1)$ OABP, for any $0 < 2r-1 \leq n$. Finally, we give an example of a polynomial $p$ and two variables orders $π\neq π'$, such that $p$ can be computed by a read-once $π$-OABP, but where any $π'$-OABP computing $p$ must read some variable at least $2^n$
