An Approach to Computing Downward Closures

Abstract

Abstract. The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful abstraction, algorithms for computing a finite automaton for the downward closure of a given language have been established only for few language classes.This work presents a simple general method for computing downward closures. For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property.This result is used to prove that downward closures are computable for (i) every language class with effectively semilinear Parikh images that are closed under rational transductions, (ii) matrix languages, and (iii) indexed languages (equivalently, languages accepted by higher-order pushdown automata of order 2).

Intersection Types for Unboundedness Problems

“…ϕ, says that ϕ holds for arbitrarily large finite sets X . Let us also remark that decidability of SUP implies that given a language defined by a nondeterministic recursion scheme, it is possible to compute its downward closure [30], and given two such languages, it is possible to decide whether they can be separated by a piecewise testable language [10].…”

The Diagonal Problem for Higher-Order Recursion Schemes is Decidable

“…By Corollary 2.2, word languages generated by schemes are closed under rational transductions. In this case, Theorem 3.1 together with a result of Zetzsche [27] can be used to compute the downward closure of a language generated by a HORS.…”

“…The diagonal problem is a decision problem with a number of interesting algorithmic consequences. It is a central subproblem for computing the downward closure of languages of words [27], as well as for the problem of separability by piecewise-testable languages [11]. It is used in deciding reachability of a certain type of parameterized concurrent systems [25].…”

“…It is well-known that schemes have a decidable emptiness problem [23], and it can be shown that they are closed under rational linear transductions, and in particular they form a full trio when restricted to finite word languages. In this context, a result by Zetzsche [27] entails computability of the downward closure of languages of words recognized by higher-order schemes. Moreover, a recent result by Czerwiński, Martens, van Rooijen, and Zeitoun [10] entails that the separability by piecewise testable languages is decidable for languages recognized by higher-order schemes.…”

“…also [4]), for Petri-net languages [14], for stacked counter automata [28], and context-free FIFO rewriting systems and 0L-systems [1]. More recently, Zetzsche [27] has given an algorithm for indexed grammars, or equivalently for second-order pushdown automata. Hague, Kochems, and Ong [16] have made an important further advance by showing how to compute the downward closure of the language of pushdown automata of arbitrary order.…”