Recent constraint logic programming (CLP) languages, such as HAL and Mercury, require type, mode and determinism declarations for predicates. This information allows the generation of efficient target code and the detection of many errors at compile-time. Unfortunately, mode checking in such languages is difficult. One of the main reasons is that, for each predicate mode declaration, the compiler is required to appropriately re-order literals in the predicate's definition. The task is further complicated by the need to handle complex instantiations (which interact with type declarations and higher-order predicates) and automatic initialization of solver variables. Here we define mode checking for strongly typed CLP languages which require reordering of clause body literals. In addition, we show how to handle a simple case of polymorphic modes by using the corresponding polymorphic types.

An optimal algorithm is presented for computing the smallest set of linear inequalities that define the integer hull of a possibly unbounded two-dimensional convex polygon R. Input to the algorithm is a set of linear inequalities defining R, and the integer hull computed is the convex hull of the integer points of R. It is proven that the integer hull has at most O(n log A max ) inequalities, where n is the number of input inequalities and A max is the magnitude of the largest input coefficient. It is shown that the algorithm presented has complexity O(n log A max ) and that this is optimal by proving that the integer hull may have $\Omega(n \log A_{max})$ inequalities in the worst case.