Following research initiated by Tarski, Craig and Németi, and further pursued by Sain, we show that for certain subsets G of mappings from omega to omega, G-polyadic-algebras have the strong amalgamation property. Here omega denotes the set of natural numbers, and G-polyadic-algebras are obtained by restricting (the similarity type) and axiomatization of omega-dimensional polyadic algebras to finite quantifiers and substitutions in G. Using algebraic logic, we infer that for certain proper extensions of first order logic without equality, the theorems of Beth, Craig and Robinson hold.