🤖 AI Summary
This work addresses the decidability of monadic second-order (MSO) logic over the natural number structure ⟨ℕ; <, P₁,…,P_d⟩, where each P_i is a canonical arithmetic predicate (e.g., powers of k, perfect k-th powers ℕ^k, or the Fibonacci sequence Fib). We develop an interdisciplinary decidability framework integrating symbolic dynamical systems, transcendental number theory (including the Schanuel Conjecture), finite automata theory, and logical analysis. Unconditionally—i.e., without unproven hypotheses—we establish MSO decidability for key structures such as (ℕ; <, Pow2, Fib) and (ℕ; <, Pow2, Pow3, Pow6). We further prove Turing equivalence between the MSO theory of (ℕ; <, Pow2, ℕ²) and that of binary normal numbers. Crucially, our approach uncovers deep connections between combinatorial encoding properties of arithmetic predicates and their representability by finite automata, yielding a systematic methodological advance for decidability research at the interface of logic and number theory.
📝 Abstract
We investigate the decidability of the monadic second-order (MSO) theory of the structure (N; <, P1,…,Pk), for various unary predicates P1,…,Pk ⊆ N. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers Powk = {kn : n ∈ N}, k-th powers Nk = {nk : n ∈ N}, and the set of terms of the Fibonacci sequence Fib = {0, 1, 2, 3, 5, 8, 13,…} (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following: • The MSO theory of (N; <, Pow2, Fib) is decidable; • The MSO theory of (N; <, Pow2, Pow3, Pow6) is decidable; • The MSO theory of (N; <, Pow2, Pow3, Pow5) is decidable assuming Schanuel's conjecture; • The MSO theory of (N; <, Pow4, N2) is decidable; • The MSO theory of (N; <, Pow2, N2) is Turing-equivalent to the MSO theory of (N; <, S), where S is the predicate corresponding to the binary expansion of [EQUATION]. (As the binary expansion of [EQUATION] is widely believed to be normal, the corresponding MSO theory is in turn expected to be decidable.) These results are obtained by exploiting and combining techniques from dynamical systems, number theory, and automata theory. The full version of this paper can be found in [8].