In this paper, we study the finite satisfiability problem for the logic BE under the homogeneity assumption. BE is the cornerstone of Halpern and Shoham's interval temporal logic, and features modal operators corresponding to the prefix (a.k.a. "Begins") and suffix (a.k.a. "Ends") relations on intervals. In terms of complexity, BE lies in between the "Chop logic C", whose satisfiability problem is known to be non-elementary, and the PSPACE-complete interval logic D of the sub-interval (a.k.a. "During") relation. BE was shown to be EXPSPACE-hard, and the only known satisfiability procedure is primitive recursive, but not elementary. Our contribution consists of tightening the complexity bounds of the satisfiability problem for BE, by proving it to be EXPSPACE-complete. We do so by devising an equi-satisfiable normal form with boundedly many nested modalities. The normalization technique resembles Scott's quantifier elimination, but it turns out to be much more involved due to the limitations enforced by the homogeneity assumption.

翻译：本文研究了同质性假设下BE逻辑的有限可满足性问题。BE逻辑是Halpern与Shoham区间时序逻辑的基石，包含对应区间上前缀（又称“开始”）与后缀（又称“结束”）关系的模态算子。就复杂性而言，BE逻辑介于可满足性问题已知为非初等的“切分逻辑C”与PSPACE完备的区间逻辑D（对应子区间关系，又称“期间”）之间。已有研究表明BE逻辑具有EXPSPACE难度，而已知的可满足性算法属于原始递归但非初等。我们的贡献在于通过证明BE逻辑的可满足性问题为EXPSPACE完备，收紧其复杂性界限。为此，我们设计了一种具有有界嵌套模态的等价可满足范式，该范式规范化技术类似于斯科伦量词消去，但由于同质性假设所施加的限制，其实现过程更为复杂。