Incompleteness Theorems, Large Cardinals, and Automata over Finite Words

Olivier Finkel (1)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

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 Tn =:ZFC + "There exist (at least) n inaccessible cardinals", for integers n ≥ 0. 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 Z ^{3×3} of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA.

Mots clés: Automata and formal languages; logic in computer science; finite words; context-free grammars; 2-tape automaton; Post Correspondence Problem; weighted automaton; finitely generated matrix subsemigroups of Z^{3×3}

Domaines: Logique [math.LO]; Logique en informatique [cs.LO]

Olivier Finkel. Incompleteness Theorems, Large Cardinals, and Automata over Finite Words. International Journal of Foundations of Computer Science, 2019. ⟨hal-02318263⟩