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There is a fundamental difficulty in generalizing weighted automata to thecase of infinite words: in general the infinite sum-of-products from which theweight of a given word is derived will diverge. Many solutions to this problemhave been proposed, including restricting the type of weights used andemploying a different valuation function that forces convergence. In this paperwe describe an alternative approach that, rather than seeking to avoid theinevitable divergences, instead embraces them as a source of usefulinformation. Specifically, rather than taking coefficients from an arbitrarysemiring S we instead take them from S^N. Doing this is useful because it givesus information about how the weight of an infinite word does or does notdiverge, and if it does diverge what form the divergence takes --- e.g.,polynomial, exponential, etc. This approach has proved to be incredibly usefulin the field of quantum simulation because when studying infinite systems,information about how quantities of interest, such as energy or magnetization,diverge is exactly what we want.  In this paper we introduce a new kind of automaton which we call a divergingautomaton that maps infinite words to sequences of weights from a semiring andwhich employs a Buchi-like boundary condition. We then develop a theory fordiverging power series and prove a Kleene Theorem connecting rational divergingpower series to diverging automata. Afterward we repeat this process byintroducing bidiverging automata which map biinfinite words to elements in S^(Zx N), developing a theory for bidiverging power series, and proving anotherKleene Theorem. We conclude by describing how bidiverging automata are appliedto simulate biinfinite quantum systems.

Embracing divergence: a formalism for when your semiring is simply not complete, with applications in quantum simulation