Abstract
We consider the first-order theory of the monoid PA* of languages over a finite or infinite alphabet A with at least two letters endowed solely with concatenation lifted to sets: no set theoretical predicate or function, no constant. Coding a word u by the submonoid u* it generates, we prove that the operation u*, v* → uv* and the predicate {u*,X | e ∈ X, u ∈ X} are definable in . This allows to interpret the second-order theory of in the first-order theory of and prove the undecidability of the Π8 fragment of this last theory. These results involve technical difficulties witnessed by the logical complexity of the obtained definitions: the above mentioned predicates are respectively Δ5 and Δ7.
Published Version
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