k+1 heads are better than k for PDA´s

We resolve the following long-standing conjecture of Harrison and Ibarra in 1968 [HI, p.462]: There are languages accepted by (k+1)-head 1-way deterministic pushdown automata ((k+1)-DPDA) but not by k-head 1-way pushdown automata (k-PDA), for every k. (Partial solutions for this conjecture can be found in [M1,M2,C].) On the assumption that their conjecture holds, [HI] also derived many important consequences. Now all those consequences become theorems. For example, the class of languages accepted by k-PDA´s is not closed under ∩ and complementation. Several other interesting consequences also follow: CFL ⊆∪kDPDA(k) and FA(2)⊆∪kDPDA(k), where DPDA (k)={L|L is accepted by a k-DPDA} and FA(2)={L|L is accepted by a 2-head FA). Our new proof itself is also interesting in the sense that the k+l versus k heads problems was solved by diagonalization methods [I2,M2,M3,M4,S] for stronger machines (2-way, etc). and by traditional counting arguments [S2,IK,YR,M1] for weaker machines (k-FA, k-head counter machine, etc).