We study automata-theoretic classes of string-to-string functions whose output may grow faster than linearly in their input. Our central contribution is to introduce a new such class, with polynomial growth and three equivalent definitions: the smallest class containing regular functions and closed under a "composition by substitution" operation; a restricted variant of pebble transducers; a $\lambda$-calculus with a linear type system. As their name suggests, these comparison-free polyregular functions form a subclass of polyregular functions; we prove that the inclusion is strict. Other properties of our new function class that we show are incomparability with HDT0L transductions and closure under composition. Finally, we look at the recently introduced layered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that a function can be obtained by composing such transducers together if and only if it is polyregular, and that k-layered SSTs (or k-marble transducers) are equivalent to a corresponding notion of (k+1)-layered HDT0L systems.

中文翻译：

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免比较的多正则函数

我们研究了字符串间函数的自动机理论类，这些类的输出输出可能比其输入的线性增长快。我们的主要贡献是引入一个具有多项式增长和三个等效定义的新此类，即最小类包含规则函数并且在“替换组成”运算下关闭；卵石换能器的受限制的变体；具有线性类型系统的$ \ lambda $微积分。顾名思义，这些无比较的多正则函数构成了多正则函数的子类。我们证明包含是严格的。我们展示的新函数类的其他属性是与HDT0L转导和组成下的闭包无法比拟的。最后，我们来看一下最近推出的分层流式串换能器（SST）或等效的k大理石换能器。