automata engineers

Abstract: A semilinear set is stratifiable if it is a finite union of linear sets whose periods have at most two nonzero coordinates, with no two periods in one component connecting alternating pairs of coordinates. We give a computable many-one reduction from the existence of a rational zero of an integer polynomial to deciding stratifiability. After the classical reduction to quadratic systems, we reduce rational solvability of these systems to a problem we call finite separation by Cartesian products of rational linear forms. Its input consists of rational subspaces $D_1,\ldots,D_k\le\mathbb Q^r\times\mathbb Q^r$. A separator is a map $(\boldsymbol x,\boldsymbol y)\mapsto(u(\boldsymbol x),v(\boldsymbol y))$, where $u,v\colon\mathbb Q^r\to\mathbb Q$ are nonzero rational linear forms. The problem asks whether finitely many separators suffice so that every pair outside $\bigcup_iD_i$ receives, from some separator, a value attained at no pair in that union. We then reduce this problem to stratifiability. Ginsburg and Spanier's characterization of letter-bounded context-free languages} also induces a reduction from rational polynomial solvability to context-freeness of letter-bounded languages specified by semilinear exponent sets. An OpenAI preprint claims undecidability of rational polynomial solvability; if correct, our result settles the decidability of general stratifiability, a problem open since 1966.

Abstract: It is well known that the pumping lemma for regular languages over finite alphabets does not extend to infinite alphabets. Instead, the generalized version of the pumping lemma for infinite alphabets states that the pumped patterns need not be identical, but only equivalent up to a finite-order permutation of the alphabet. This generalized variant of the pumping lemma involves two parameters: the length of the pumped pattern and an upper bound on the order of the permutation. We improve the known permutation-order bounds from factorial to tight bounds governed by Landau's function. We also show that, if the pumping length is sufficiently large, then permutations of order at most two always suffice. Our main structural result concerns the one-register case: every one-register automaton satisfies the classical pumping lemma, with no alphabet permutation. We prove a quadratic upper bound on the pumping length and give a matching quadratic lower bound up to constant factors. The same classical pumping phenomenon is shown for hierarchical register automata. Finally, we prove that, in general, deciding whether a quasi-regular language is pumpable for given constants is undecidable, while the minimal classical pumping length is computable for non-guessing one-register automata.