The Power of Programs over Monoids in J
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The model of programs over (finite) monoids, introduced by Barrington and Thérien, gives an interesting way to characterise the circuit complexity class NC^1 and its subclasses and showcases deep connections with algebraic automata theory. In this article, we investigate the computational power of programs over monoids in J, a small variety of finite aperiodic monoids. First, we give a fine hierarchy within the class of languages recognised by programs over monoids from J, based on the length of programs but also some parametrisation of J. Second, and most importantly, we make progress in understanding what regular languages can be recognised by programs over monoids in J. We show that those programs actually can recognise all languages from a class of restricted dot-depth one languages, using a non-trivial trick, and conjecture that this class suffices to characterise the regular languages recognised by programs over monoids in J.
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