Orbit Problems are a class of fundamental reachability questions that arise in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. Instances of the problem comprise a dimension \(d\in \mathbb {N}\), a square matrix \(A\in \mathbb {Q}^{d\times d}\), and a query regarding the behaviour of some sets under repeated applications of A. For instance, in the Semialgebraic Orbit Problem, we are given semialgebraic source and target sets \(S,T\subseteq \mathbb {R}^{d}\), and the query is whether there exists \(n\in {\mathbb {N}}\) and x ∈ S such that Aⁿx ∈ T. The main contribution of this paper is to introduce a unifying formalism for a vast class of orbit problems, and show that this formalism is decidable for dimension d ≤ 3. Intuitively, our formalism allows one to reason about any first-order query whose atomic propositions are a membership queries of orbit elements in semialgebraic sets. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory—Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of \(\mathbb {R}^{d}\) for which membership is decidable. On the other hand, previous work has shown that in dimension d = 4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

轨道问题是一类基本的可到达性问题，它在分析离散时间线性动力系统（例如自动机，马尔可夫链，递归序列和线性while循环）时出现。问题的实例包括维度\（d \ in \ mathbb {N} \），方阵\（A \ in \ mathbb {Q} ^ {d \ times d} \）以及关于以下行为的查询：重复应用A的一些集。例如，在“半代数轨道问题”中，我们得到了半代数源集和目标集\（S，T \ subseteq \ mathbb {R} ^ {d} \），并且查询的是在{\ mathbb {N}} \）和X ∈小号使得甲^Ñ X ∈ Ť。本文的主要贡献是针对大量轨道问题引入了统一的形式主义，并表明这种形式主义对于维数d是可决定的。≤3。直觉上，我们的形式主义允许人们对任何一阶查询进行推理，这些查询的原子命题是半代数集中的轨道元素的隶属关系查询。我们的决策程序依赖于代数数的分离边界以及先验数论的经典结果-贝克定理，即对数代数的线性形式。此外，我们认为，我们的主要结果代表了对单个矩阵轨道的可决定性（相对于可达性）的自然限制。一方面，半代数集可以说是\（\ mathbb {R} ^ {d} \）的子集的最大一般类，其成员资格是可确定的。另一方面，先前的工作表明在维d中= 4，给出具有单例源集S和多目标目标集T的轨道问题特殊情况的决策程序，将需要在Diophantine逼近方面取得重大突破。