When can $n$ given numbers be combined using arithmetic operators from a given subset of $\{+, -, \times, \div\}$ to obtain a given target number? We study three variations of this problem of Arithmetic Expression Construction: when the expression (1) is unconstrained; (2) has a specified pattern of parentheses and operators (and only the numbers need to be assigned to blanks); or (3) must match a specified ordering of the numbers (but the operators and parenthesization are free). For each of these variants, and many of the subsets of $\{+,-,\times,\div\}$, we prove the problem NP-complete, sometimes in the weak sense and sometimes in the strong sense. Most of these proofs make use of a "rational function framework" which proves equivalence of these problems for values in rational functions with values in positive integers.

中文翻译：

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算术表达式构造

何时可以使用算术运算符从$ \ {+，-，\ times，\ div \} $的给定子集中将给定的$ n $个数字组合起来以获得给定的目标数？我们研究了此算术表达式构造问题的三个变体：当表达式（1）不受约束时；（2）具有指定的括号和运算符模式（并且只需要将数字分配给空格）；或（3）必须匹配数字的指定顺序（但运算符和括号是免费的）。对于这些变体中的每一个，以及$ \ {+，-，\ times，\ div \} $的许多子集，我们证明问题NP完全，有时是弱的，有时是强的。这些证明大多数都利用“有理函数框架”来证明这些问题对于有理函数中的值与正整数中的值是等效的。