  Hyperproperties lift conventional trace-based languages from a set of
execution traces to a set of sets of executions. From a formal-language
perspective, these are sets of sets of words, namely hyperlanguages.
Hyperautomata are based on classical automata models that are lifted to handle
hyperlanguages. Finite hyperautomata (NFH) have been suggested to express
regular hyperproperties. We study the realizability problem for regular
hyperlanguages: given a set of languages, can it be precisely described by an
NFH? We show that the problem is complex already for singleton hyperlanguages.
We then go beyond regular hyperlanguages, and study context-free
hyperlanguages. We show that the natural extension to context-free
hypergrammars is highly undecidable. We then suggest a refined model, namely
synchronous hypergrammars, which enables describing interesting non-regular
hyperproperties, while retaining many decidable properties of context-free
grammars.
