Transcendence of numbers related to Episturmian words

This thesis relates combinatorial properties of sequences to arithmetic properties of the numbers that they represent. A guiding principle is that numbers whose b- expansion has low subword complexity are either rational or transcendental. This heuristic was confirmed by Ferenczi and Maduit, who proved that for b > 1 an inte- ger, numbers whose b-expansion is a Sturmian sequence or an Arnoux-Rauzy sequence are transcendental. (These can be considered as the simplest non-ultimately-periodic sequences in terms of their subword complexity.) Subsequent work of Adamczewski and Bugeaud extended this result by proving that all numbers whose b-expansion has linear subword complexity are rational or transcendental, again for b an integer. The latter authors obtained related results to the case of b an algebraic base under certain combinatorial properties of the sequence, which depend on b. The main contribution of this thesis is providing a transcendence result which applies to arbitrary algebraic bases. We introduce a new combinatorial condition on sequences and prove a transcendence result for numbers of the form α := Σ∞n=1 unβ−n where β is any algebraic number such that |β| > 1 and u = u1u2 ··· is a sequence of algebraic numbers satisfying the above-mentioned criterion. In particular we prove that all Episturmian (a generalisation of Arnoux-Rauzy words) words satisfy this criterion

Regular Methods for Operator Precedence Languages

The operator precedence languages (OPLs) represent the largest known subclass of the context-free languages which enjoys all desirable closure and decidability properties. This includes the decidability of language inclusion, which is the ultimate verification problem. Operator precedence grammars, automata, and logics have been investigated and used, for example, to verify programs with arithmetic expressions and exceptions (both of which are deterministic pushdown but lie outside the scope of the visibly pushdown languages). In this paper, we complete the picture and give, for the first time, an algebraic characterization of the class of OPLs in the form of a syntactic congruence that has finitely many equivalence classes exactly for the operator precedence languages. This is a generalization of the celebrated Myhill-Nerode theorem for the regular languages to OPLs. As one of the consequences, we show that universality and language inclusion for nondeterministic operator precedence automata can be solved by an antichain algorithm. Antichain algorithms avoid determinization and complementation through an explicit subset construction, by leveraging a quasi-order on words, which allows the pruning of the search space for counterexample words without sacrificing completeness. Antichain algorithms can be implemented symbolically, and these implementations are today the best-performing algorithms in practice for the inclusion of finite automata. We give a generic construction of the quasi-order needed for antichain algorithms from a finite syntactic congruence. This yields the first antichain algorithm for OPLs, an algorithm that solves the ExpTime-hard language inclusion problem for OPLs in exponential time

On transcendence of numbers related to Sturmian and Arnoux-Rauzy words

We consider numbers of the form Sβ(u) := P∞ n=0 un βn , where u = ⟨un⟩ ∞n=0 is an infinite word over a finite alphabet and β ∈ C satisfies |β| > 1. Our main contribution is to present a combinatorial criterion on u, called echoing, that implies that Sβ(u) is transcendental whenever β is algebraic. We show that every Sturmian word is echoing, as is the Tribonacci word, a leading example of an Arnoux-Rauzy word. We furthermore characterise Q-linear independence of sets of the form {1, Sβ(u1), . . . , Sβ(uk)}, where u1, . . . , uk are Sturmian words having the same slope. Finally, we give an application of the above linear independence criterion to the theory of dynamical systems, showing that for a contracted rotation on the unit circle with algebraic slope, its limit set is either finite or consists exclusively of transcendental elements other than its endpoints 0 and 1. This confirms a conjecture of Bugeaud, Kim, Laurent, and Nogueira

LIPIcs

The operator precedence languages (OPLs) represent the largest known subclass of the context-free languages which enjoys all desirable closure and decidability properties. This includes the decidability of language inclusion, which is the ultimate verification problem. Operator precedence grammars, automata, and logics have been investigated and used, for example, to verify programs with arithmetic expressions and exceptions (both of which are deterministic pushdown but lie outside the scope of the visibly pushdown languages). In this paper, we complete the picture and give, for the first time, an algebraic characterization of the class of OPLs in the form of a syntactic congruence that has finitely many equivalence classes exactly for the operator precedence languages. This is a generalization of the celebrated Myhill-Nerode theorem for the regular languages to OPLs. As one of the consequences, we show that universality and language inclusion for nondeterministic operator precedence automata can be solved by an antichain algorithm. Antichain algorithms avoid determinization and complementation through an explicit subset construction, by leveraging a quasi-order on words, which allows the pruning of the search space for counterexample words without sacrificing completeness. Antichain algorithms can be implemented symbolically, and these implementations are today the best-performing algorithms in practice for the inclusion of finite automata. We give a generic construction of the quasi-order needed for antichain algorithms from a finite syntactic congruence. This yields the first antichain algorithm for OPLs, an algorithm that solves the ExpTime-hard language inclusion problem for OPLs in exponential time