Simplifications of Uniform Expressions Specified by Systems

Abstract

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 [Formula: see text] 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.