The machinery of the human brain -- analog, probabilistic, embodied -- can be characterized computationally, but what machinery confers what computational powers? Any such system can be abstractly cast in terms of two computational components: a finite state machine carrying out computational steps, whether via currents, chemistry, or mechanics; plus a set of allowable memory operations, typically formulated in terms of an information store that can be read from and written to, whether via synaptic change, state transition, or recurrent activity. Probing these mechanisms for their information content, we can capture the difference in computational power that various systems are capable of. Most human cognitive abilities, from perception to action to memory, are shared with other species; we seek to characterize those (few) capabilities that are ubiquitously present among humans and absent from other species. Three realms of formidable constraints -- a) measurable human cognitive abilities, b) measurable allometric anatomic brain characteristics, and c) measurable features of specific automata and formal grammars -- illustrate remarkably sharp restrictions on human abilities, unexpectedly confining human cognition to a specific class of automata (nested stack), which are markedly below Turing machines.
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