Incompleteness Theorems, Large Cardinals, and Automata Over Finite Words
Olivier Finkel · International Journal of Foundations of Computer Science · 2019
We prove that one can construct various kinds of automata over finite words for which some elementary properties are actually independent from strong set theories like [Formula: see text] [Formula: see text] “There exist (at least) [Formula: see text] inaccessible cardinals”, for integers [Formula: see text]. In particular, we prove independence results for languages of finite words generated by context-free grammars, or accepted by 2-tape or 1-counter automata. Moreover we get some independence results for weighted automata and for some related finitely generated subsemigroups of the set [Formula: see text] of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA.