Combinatorial Problems, Combinator Equations and Normal Forms

Some cases of solvability of the convertibility equation MX=N, where M,N and X are normal forms of the λ-K-β-calculus are considered and their relationship to pattern-matching problems is discussed. In particular, the solvability of such an equation is decided in the cases: i) N ≡ I and M = BF ii) N ≡ I and M = CBF where F is a normal form iii) N is an arbitrary normal form.