The equational theory of the Weihrauch lattice with (iterated) composition

Cécilia Pradic
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Abstract

We study the equational theory of the Weihrauch lattice with composition and iterations, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the composition operator $\star$ and its iteration $(-)^\diamond$ , which are true however we substitute (slightly extended) Weihrauch degrees for the variables. We characterize them using B\"uchi games on finite graphs and give a complete axiomatization that derives them. The term signature and the axiomatization are reminiscent of Kleene algebras, except that we additionally have meets and the lattice operations do not fully distributes over composition. The game characterization also implies that it is decidable whether an equation is universally valid. We give some complexity bounds; in particular, the problem is Pspace-hard in general and we conjecture that it is solvable in Pspace.
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具有(迭代)构成的魏赫劳赫网格的等式理论
我们研究的是具有组成和迭代的魏赫劳赫格的等式理论,即由变量、格运算 $\sqcup$, $\sqcap$, 组成算子$\star$及其迭代$(-)^\diamond$构成的项之间的等式集合。我们用有限图上的渊博博弈来描述它们,并给出了一个完整的公理化来解释它们。术语签名和公理化都让人想起克莱因代数,只是我们另外有了相遇,而且网格操作并不完全分布于组成。博弈表征也意味着方程是否普遍有效是可以判定的。我们给出了一些复杂性边界;特别是,这个问题一般来说是 Pspace-hard(Pspace-hard)的,我们猜想它在 Pspace 中是可解的。
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