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We show how some problems in additive number theory can be attacked in a novel way, using techniques from the theory of finite automata. We start by recalling the relationship between first-order logic and finite automata, and use this relationship to solve several problems involving sums of numbers defined by their base-2 and Fibonacci representations. Next, we turn to harder results. Recently, Cilleruelo, Luca, & Baxter proved, for all bases b ≥ 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome (Cilleruelo et al., Math. Comput. 87, 3023–3055, 2018). However, the cases b = 2, 3, 4 were left unresolved. We prove that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem of palindromes as an additive basis. We consider some other variations on this problem, and prove similar results. We argue that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof. [ABSTRACT FROM AUTHOR]

The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragment Σ2 over infinite words. Here, Σ2 consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is 픹Σ2. Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an ω-regular language L, it is decidable whether L is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun’s recent decidability result for 픹Σ2 from finite to infinite words. [ABSTRACT FROM AUTHOR]

Semiautomatic structures generalise automatic structures in the sense that for some of the relations and functions in the structure one only requires the derived relations and functions are automatic when all but one input are filled with constants. One can also permit that this applies to equality in the structure so that only the sets of representatives equal to a given element of the structure are regular while equality itself is not an automatic relation on the domain of representatives. It is shown that one can find semiautomatic representations for the field of rationals and also for finite algebraic field extensions of it. Furthermore, one can show that infinite algebraic extensions of finite fields have semiautomatic representations in which the addition and equality are both automatic. Further prominent examples of semiautomatic structures are term algebras, any relational structure over a countable domain with a countable signature and any permutation algebra with a countable domain. Furthermore, examples of structures which fail to be semiautomatic are provided. [ABSTRACT FROM AUTHOR]

Higher-order pushdown systems and ground tree rewriting systems can be seen as extensions of suffix word rewriting systems. Both classes generate infinite graphs with interesting logical properties. Indeed, the model-checking problem for monadic second order logic (respectively first order logic with a reachability predicate) is decidable on such graphs. We unify both models by introducing the notion of stack trees, trees whose nodes are labelled by higher-order stacks, and define the corresponding class of higher-order ground tree rewriting systems. We show that these graphs retain the decidability properties of ground tree rewriting graphs while generalising the pushdown hierarchy of graphs. [ABSTRACT FROM AUTHOR]