可定义性的可判定性
Decidability of definability
摘要：
对于具有有限关系签名τ的固定可数无限结构Γ，我们研究了如下计算问题：输入是无量词τ-公式_0，_1，.，定义关系R_0，R_1，.，R_n over Γ.问题是关系R_0是否是可由R_1，. R_n，即，可由一阶公式定义，该公式仅使用R_1，...，R_n，等式，合取，存在量化（禁止析取，否定和全称量化）。我们证明了这个问题的可判定性，所有的结构，有一个一阶定义在一个有序的齐次结构Δ有限的关系签名，其年龄是一个拉姆齐类，并确定了许多禁止的子结构。具有这种性质的结构Γ的例子有有理数的序、随机图、齐次泛偏序集、随机竞赛图、所有齐次泛C-关系等等。当我们用存在的正可定义性或存在的可定义性来代替原始的正可定义性时，我们也得到了问题的可判定性。我们的证明利用了普遍的代数和模型理论的概念，拉姆齐理论，以及最近的Ramsey类拓扑动力学的表征。
英文摘要：
For a fixed countably infinite structure Γ with finite relational signature τ, we study the following computational problem: input are quantifier-free τ-formulas ϕ_0,ϕ_1,...,ϕ_n that define relations R_0,R_1,...,R_n over Γ. The question is whether the relation R_0 is primitive positive definable from R_1,...,R_n, i.e., definable by a first-order formula that uses only relation symbols for R_1,..., R_n, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden). We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
