This paper investigates two subjects in push-down automata (PDAs) and linear indexed grammars (LIGs), which are extended PDAs, focusing on eliminating the stack symbols. One of the subjects is concerned with
PDA and PI-LIG without
-transition rule, in which only input symbols are pushed down to the stack. It is shown that the class of languages of PI-LIGs is incomparable with that of PDAs, which is the class of context-free languages (CFLs). The other subject is a simple bottom-up parsing method for LIGs, in which the stack symbols are eliminated at the first step of the parsing. The paper shows several PI-LIGs, including PI-PDAs for fundamental context-free and context-sensitive languages, which are synthesized by a grammatical inference system LIG Learner.