A monotone Boolean (OR,AND) 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 at least one 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 already read-1 (OR,AND) circuits are not weaker than monotone arithmetic constant-free (+,x) circuits computing multilinear polynomials, are not weaker than non-monotone multilinear (OR,AND,NOT) circuits computing monotone Boolean functions, and have the same power as tropical (min,+) circuits solving combinatorial minimization problems. Finally, we show that read-2 (OR,AND) circuits can be exponentially smaller than read-1 (OR,AND) circuits.

翻译：单调布尔（OR,AND）电路在计算单调布尔函数f时，若其算术版本（+,×）电路（纯语法层面）生成的多项式满足：对于f的每个素蕴含项，该多项式至少包含一个具有相同变量集且每个变量出现次数不超过k次的单项式，则称该电路为read-k电路。所有单调电路均属于某k值下的read-k电路。本文证明：read-1（OR,AND）电路既不弱于计算多线性多项式的单调算术无常数（+,×）电路，也不弱于计算单调布尔函数的非单调多线性（OR,AND,NOT）电路，且与求解组合最小化问题的热带（min,+）电路具有相同能力。最后证明：read-2（OR,AND）电路的规模可能指数级小于read-1（OR,AND）电路。