State Complexity of the Quotient Operation on Input-Driven Pushdown Automata

The quotient of a formal language [Formula: see text] by another language [Formula: see text] is the set of all strings obtained by taking a string from [Formula: see text] that ends with a suffix of a string from [Formula: see text], and removing that suffix. The quotient of a regular language by any language is always regular, whereas the context-free languages and many of their subfamilies, such as the linear and the deterministic languages, are not closed under the quotient operation. This paper establishes the closure of the family of languages recognized by input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the quotient operation. A construction of automata representing the result of the operation is given, and its state complexity with respect to nondeterministic IDPDA is shown to be exactly [Formula: see text], where [Formula: see text] and [Formula: see text] are the numbers of states in the automata recognizing [Formula: see text] and [Formula: see text], respectively.

Visibly Pushdown Transducers with Well-Nested Outputs

Visibly pushdown transducers (VPTs) are visibly pushdown automata extended with outputs. They have been introduced oto model transformations of nested words, i.e. words with a call/return structure. When outputs are also structured and well nested words, VPTs are a natural formalism to express tree transformations evaluated in streaming. We prove the class of VPTs with well-nested outputs to be decidable in PTIME. Moreover, we show that this class is closed under composition and that its type-checking against visibly pushdown languages is decidable.