Observers, Symmetries, and the Hierarchy of Language Classes: A Theory of Computation Parameterized by the Observer

Abstract

We introduce the \emph{observational hierarchy}, a new axis of classification for formal languages, orthogonal to the Chomsky hierarchy. An observer is a function $O : \Sigma^* \to S$ that determines which information about the input is accessible to a computational system. The order-blind automaton, which perceives the input as a multiset of symbols rather than a sequence, constitutes the paradigmatic case. We prove that the class of languages recognisable by any machine equipped with such an observer coincides exactly with the permutation-closed languages. We then define a partial order on observers that induces a hierarchy of language classes parametrised not by the computational power of the machine, but by the structure of the observer. We prove that this hierarchy has the structure of a partial order with a diamond-shaped profile sub-lattice, comprising the length branch $O_\bot \prec O_{\mathrm{len}} \prec O_{\mathrm{prof}} \prec O_\top$ and the parity branch $O_\bot \prec O_{\mathrm{par}} \prec O_{\mathrm{prof}} \prec O_\top$, with $O_{\mathrm{len}}$ and $O_{\mathrm{par}}$ incomparable, and an infinite subsequence branch $O_\bot \prec O_1 \prec O_2 \prec \cdots \prec O_\top$, both converging to the complete observer. We prove that the observational hierarchy is strictly incomparable with the Chomsky hierarchy, and introduce the notion of \emph{observational complexity} of a language. We further define observer-parametrised complexity classes $\mathbf{P}_O$ and $\mathbf{NP}_O$, and show that computational hardness and structural blindness are two independent phenomena. In particular, $\mathbf{P}_{O_{\mathrm{prof}}} = \mathbf{NP}_{O_{\mathrm{prof}}}$ holds as a structural collapse strictly inside $\mathbf{P}$.