Complexity of input-driven pushdown automata

In an input-driven pushdown automaton (IDPDA), the current input symbol determines whether the automaton performs a push operation, a pop operation, or does not touch the stack. Inputdriven pushdown automata, also known under alternative names of visibly pushdown automata and of nested word automata, have been intensively studied because of their desirable features: for instance, the model allows determinization, and the associated family of languages retains many of the strong closure and decidability properties of regular languages. This paper reports on various aspects of the complexity of input-driven pushdown automata, such as their descriptional complexity, the computational complexity of their membership problem and of other decision problems for input-driven languages. The research on IDPDAs has been associated to their complexity from the very beginning. When Mehlhorn [25] originally introduced the model, it was studied as a subclass of deterministic context-free languages with better space complexity. Further work on the model carried out in the 1980s [6, 12, 36] concentrated on improving the bounds on the complexity of the languages accepted by such automata, culminating in the proof of their containment in NC1. In 2004, the model was reintroduced by Alur and Madhusudan [2] under the name of visibly pushdown automata, and among their most important contributions were the first results on the descriptional complexity of the model, such as upper and lower bounds on the number of states in automata representing some operations on languages. Also, Alur and Madhusudan [2] established the computational complexity of several decision problems for the model. The paper by Alur and Madhusudan [2] sparked a renewed interest in IDPDAs, and inspired the research on various aspects of the model [1, 8, 10, 19, 39]. Alur and Madhusudan [3] also introduced an equivalent outlook on IDPDAs as automata operating on nested words, which provide a natural model for applications such as XML document processing, where data has a dual linear-hierarchical structure. Nested word automata have been studied in a number of recent papers [9, 11, 17, 35, 37]. Another equivalent outlook on IDPDAs is represented by pushdown forest automata [14], that are, roughly speaking, tree walking automata that traverse the tree in depth-first left-to-right order and are equipped with a synchronized pushdown.