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Logical Theory of the Monoid of Languages over a Non Tally Alphabet

Choffrut C., Grigorieff S.

Fundamenta Informaticae

2015

Vol. 138, nr 1/2

159--177

We consider the first-order theory of the monoid P(A*) 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) | ε ∈ X, u ∈ X} are definable in P(A*); •,=i. This allows to interpret the second-order theory of A*; •,=i in the first-order theory of P(A*); •,=i 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.

Grigorieff S., Margenstern M.

Fundamenta Informaticae

2004

Vol. 61, nr 1

19--27

In this paper, we consider cellular automata on special grids of the hyperbolic plane: the grids are constructed on infinigons, i.e. polygons with infinitely many sides. We show that the truth of arithmetical formulas can be decided in finite time with infinite initial recursive configurations. Next, we define a new kind of cellular automata, endowed with data and more powerful operations which we call register cellular automata. This time, starting from finite configurations, it is possible to decide the truth of arithmetic formulas in linear time with respect to the size of the formula.