🤖 AI Summary
This work challenges the traditional Chomsky hierarchy by highlighting its neglect of how an observer’s perception of input influences language recognizability. Introducing the notion of an “observer,” the paper proposes an orthogonal observational hierarchy that classifies formal languages based on the structure of observer functions rather than computational power. Integrating formal language theory, automata models, and poset analysis, the authors define observational complexity and parameterized complexity classes, proving that multiset observers correspond precisely to permutation-closed language classes. They construct an observational hierarchy exhibiting a diamond-shaped sublattice structure and demonstrate that under full-profile observation, P_O = NP_O, thereby revealing that computational hardness and structural blindness are independent phenomena.
📝 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 $O : Σ^* \to S$ 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 $\mathbf{P}$.