A monotone Boolean (AND,OR) circuit computing a monotone Boolean function f is a read-k circuit if the polynomial produced (purely syntactically) by the arithmetic (+,x) version of the circuit has the property that for every prime implicant of f, the polynomial contains a monomial with the same set of variables, each appearing with degree at most k. Every monotone circuit is a read-k circuit for some k. We show that monotone read-1 circuits have the same power as: tropical (min,+) circuits solving 0/1 minimization problems, monotone arithmetic (+,x) circuits computing multilinear homogeneous polynomials, as well as multilinear non-monotone (AND,OR,NOT) circuits computing monotone homogeneous Boolean functions. Finally, we show that already monotone read-2 circuits can be exponentially smaller than monotone read-1 circuits.

翻译：单调 Boolean (AND,OR) 电路计算单调波函数 f 是一个读- k 电路, 如果由计算(+,x) 版本电路生成的多向线(纯合成) 具有以下属性:对于F的每一种主要不切入者,多向线(+,x) 含有一个单向的特性,每个变量都有相同的变量,每个变量都以不同程度显示, k。 每个单向线电路都是某些 k 的读- k 电路。 我们显示, 单调读-1 电路具有与热带( 分钟, +) 电路解决 0/1 最小化问题、 单向(+,x) 电路计算多线性同质多线性多线性多线性多线性多线性多线性多线(AND,OR,NOT) 电路,以及计算单调同质波伦功能的多线(AND,OR,NOT) 电路。 最后, 我们显示, 已经的单调读- 2 电路可以比单调读- 1 读- 读- 电路的指数小于单线小于单调读- 1 。