Abstract
We examine the common and seemingly simple specification that the output stream equals the input stream. We show that this is not in full generality expressible in first-order or temporal logic by an infinite set of sentences or a recursive specification, but requires certain extra assumptions, such as the existence of a clock or discrete input values.
The main negative results are stated for first-order expressibility and have direct corollaries for inexpressibility in first-order temporal logic: output equals input with arbitrary delay is not expressible by a (perhaps infinite) theory (Theorems 2 and 3), even with a timestamp (Theorem 8), and is not expressible for an ω timeline by a sentence, even with a timestamp (Theorem 10). Output equals input with constant delay cannot be expressed for ω timeline by a sentence with extra unary predicates over the timeline.
As an example of the positive results, we show output equals input can be expressed by a sentence in the language with a (weak) clock if the base model contains either an extra function (Theorem 14), or arithmetic (Theorem 15).
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Marcus, L., Menas, T. Expressibility of output equals input. Acta Informatica 29, 645–662 (1992). https://doi.org/10.1007/BF01185565
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DOI: https://doi.org/10.1007/BF01185565