In this article, we study the impact of applying simple reduction rules to random syntactic formulas encoded as trees. We assume that there is an operator that has an absorbing pattern and prove that if we use this property to simplify a uniform random expression with n nodes, then the expected size of the result is bounded by a constant. The same holds for higher moments, establishing the lack of expressivity of uniform random expressions. Our framework is quite general as we consider expressions defined by systems of combinatorial equations. For our proofs, we rely on Drmota’s multidimensional theorem for systems of generating functions.

中文翻译：

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系统指定的统一表达式的简化

在本文中，我们研究了将简单归约规则应用于编码为树的随机句法公式的影响。我们假设存在一个具有吸收模式的算子，并证明如果我们使用这个属性来简化一个均匀随机表达式n节点，则结果的预期大小由一个常数限定。这同样适用于更高的时刻，确定了统一随机表达式缺乏表现力。当我们考虑由组合方程系统定义的表达式时，我们的框架非常通用。对于我们的证明，我们依赖 Drmota 的多维定理来生成函数系统。