Higher-order model checking meets implicit automata: Finitary colored semantics of infinitary λ-terms

The higher-order model-checking problem (HOMC) concerns model-checking trees generated by recursion schemes. The second author and Melliès’s approached HOMC by evaluation in a őnitary semantics of the λY-calculus (simply typed λ-calculus with őxpoints) derived from the Scott model of linear logic augmented with a colouring modality [Gre16]. Since their semantics do not interpret arbitrary inőnitary λ-terms (or even Böhm trees), the decidability proof is more indirect than expected. This leads us to state some natural conjectures that would directly imply the decidability of HOMC (some results in this direction are sketched in Melliès’s follow-up work [Mel17]).

Implicit automata theory is an up-and-coming research programme led by the third and fourth authors among others that seeks to understand classes of formal languages (such as regular or starfree languages) and functions (e.g. regular or polyregular string-to-string functions) as corresponding to the expressive power of typed λ-calculi [HK96, NP20, Ngu21, PP24]. Our aforementioned results help us to extend this programme to ω-automata over inőnite strings and trees.