Monotone interpretations

The domain of computation of an program 〈X , C,F ,R〉 is the constructor algebra T (C). A substitution σ is a mapping from variables to terms and a ground substitution is one which ranges over constructor terms of T (C). Observe that constructor terms are normal forms for the program. For each function symbol f ∈ F , the program 〈X , C,F ,R〉 computes a partial function JfK : T (C) → T (C) defined by: For all vi ∈ T (C), JfK(v1, · · · , vn) = w iff f(v1, · · · , vn) ∗ →w and w is in T (C). Otherwise JfK(v1, · · · , vn) is undefined. A term t which contains n variables x1, · · · , xn defines a function φt such that for all vi ∈ T (C), φt(v1, · · · , vn) = JtσK where σ is a ground substitution such that xiσ = vi for any i.