Interval Extensions of Orders and Temporal Approximation Spaces

Studying the algorithmic properties ofinterval extensions of dense linear orders,in particular,the complexity degrees(namely,the $ s\Sigma $-degree)of the extensions,we show thatcontinuity is a necessary and sufficient conditionfor the equality between the complexity degreesof an order and its interval extension.We treat temporal approximation spaces over interval extensionsas mathematical models of verb semantics in natural languages.We show thatthe continuity of order impliesthe effectiveness of checking the validity of$ \Delta_{0}^{DL} $-formulasin spaces over$ sc $-simple enrichments.As a corollary,we obtain an effective description of the intervalscorresponding to various verb tenses in English.