Single letter languages accepted by alternating and probabilistic pushdown automata

We consider way k tape alternating pushdown automata k apa and way k tape alternating nite automata k afa We say that an alternating automaton accepts a language L with f n bounded maximal respectively minimal leaf size if arbitrary re spectively at least one accepting tree for any w L has no more than f jwj leaves The main results of the article are the following For way k tape alternating counter au tomata tape alternating pushdown automata and way synchronized alternating nite automata if theses automata accepts language L over one letter alphabet with o logn bounded maximal leaf size or o log log n bounded minimal leaf size then the language L is semilinear Moreover if a language L is accepted with o log log n bounded minimal re spectively o log n bounded maximal leafsize then it is accepted by constant bounded minimal respectively maximal leafsize by the same automaton To show that this bound is optimal we prove that apa and tape afa can accept a non semilinear languages over one letter alphabet with O log logn bounded minimal leaf size For maximal leaf size our bound is optimal due to King s results Languages over one letter alphabet accepted by probabilistic one way tape pushdown automata with isolated cutpoint are proved to be regular