Exact Descriptional Complexity of Determinization of Input-Driven Pushdown Automata

The number of states and stack symbols needed to determinize nondeterministic input-driven pushdown automata (NIDPDA) working over a fixed alphabet is determined precisely. It is proved that in the worst case exactly \(2^{n^2}\) states are needed to determinize an n-state NIDPDA, and the proof uses witness automata with a stack alphabet \(\varGamma = \{0,1\}\) working on strings over a 4-symbol input alphabet (Only an asymptotic lower bound was known before in the case of a fixed alphabet). Also, the impact of NIDPDA determinization on the size of stack alphabet is determined precisely for the first time: it is proved that \(s(2^{n^2}-1)\) stack symbols are necessary in the worst case to determinize an n-state NIDPDA working over an input alphabet of size \(s+5\) with s left brackets (The previous lower bound was only asymptotic in the number of states and did not depend on the number of left brackets).