  We investigate the satisfiability problem for Horn fragments of the
Halpern-Shoham interval temporal logic depending on the type (box or diamond)
of the interval modal operators, the type of the underlying linear order
(discrete or dense), and the type of semantics for the interval relations
(reflexive or irreflexive). For example, we show that satisfiability of Horn
formulas with diamonds is undecidable for any type of linear orders and
semantics. On the contrary, satisfiability of Horn formulas with boxes is
tractable over both discrete and dense orders under the reflexive semantics and
over dense orders under the irreflexive semantics, but becomes undecidable over
discrete orders under the irreflexive semantics. Satisfiability of binary Horn
formulas with both boxes and diamonds is always undecidable under the
irreflexive semantics.
