On the complexity of stream equality

Abstract

We study the complexity of deciding the equality of streams specified by systems of equations. There are several notions of stream models in the literature, each generating a different semantics of stream equality. We pinpoint the complexity of each of these notions in the arithmetical or analytical hierarchy. Their complexity ranges from low levels of the arithmetical hierarchy such as Π 0 2 for the most relaxed stream models, to levels of the analytical hierarchy such as Π 1 1 and up to subsuming the entire analytical hierarchy for more restrictive but natural stream models. Since all these classes properly include both the semi-decidable and co-semi-decidable classes, it follows that regardless of the stream semantics employed, there is no complete proof system or algorithm for determining equality or inequality of streams. We also discuss several related problems, such as the existence and uniqueness of stream solutions for systems of equations, as well as the equality of such solutions.

Transducer degrees: atoms, infima and suprema

“…Since z + h is even, we get τ | h > 0 from (18). As ϕ( − → α , h) = k, we conclude β| k > 0 by (16). However, by (16) and (23), τ | i > 0 for every i ∈ {k 1 , .…”

“…This flexibility can be convenient for reasoning about transformations of infinite words (or infinite terms). On the other hand, the increased power results in undecidability of many properties such as equivalence [9,15,16]. As a rewrite system, the properties termination (normalisation) and confluence are of interest; see for instance [7,21,22,28,38].…”

Proving Looping and Non-Looping Non-Termination by Finite Automata

“…Finally, we note that equality of streams [17,18,38,19] (infinite sequences of symbols) can be rendered as a non-termination problem (a comparison program running indefinitely if the streams are equal, and terminating as soon as a difference is found). It remains to be investigated whether non-termination techniques can be employed fruitfully for proving stream equality.…”