Incompleteness Theorems, Large Cardinals, and Automata over Finite Words

Abstract : 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.
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Olivier Finkel. Incompleteness Theorems, Large Cardinals, and Automata over Finite Words. 14th Annual Conference on Theory and Applications of Models of Computation, TAMC 2017., Apr 2017, Bern, Switzerland. pp.231 - 246, ⟨10.1007/978-3-319-55911-7⟩. ⟨hal-01588572⟩

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