The Inclusion Problem of Context-Free Languages: Some Tractable Cases

Abstract

We study the problem of testing whether a context-free language is included in a fixed set L0, where L0 is the language of words reducing to the empty word in the monoid defined by a complete string rewrite system. We prove that, if the monoid is cancellative, then our inclusion problem is polynomially reducible to the problem of testing equivalence of straight-line programs in the same monoid. As an application, we obtain a polynomial time algorithm for testing if a context-free language is included in a Dyck language (the best previous algorithm for this problem was doubly exponential).

Visibly Pushdown Transducers with Well-Nested Outputs

“…Then, as shown in [8], one can build in polynomial time a context-free grammar G A ′ generating the range of A ′ . Finally, we appeal to [2] to conclude.…”

“…Proof. This proof heavily relies on results from [2] showing that deciding whether a context-free language is included into a Dyck language can be solved in PTIME.…”

“…-Criterion (C1): consider only words u 2 such that height(u 2 ) ≤ |Q| 2 The non-deterministic algorithm follows from the claim: in order to exhibit a witness of the fact that A ∈ awnVPT, the algorithm guesses whether (C1) or (C2) is violated, and a pair of states (p, q) and on the fly, one or two runs, according to the criterion, of exponential size, which can be done in polynomial space (using counters keeping track of the current length of the guessed run(s)). Using Proposition 6 it also verifies that there exists a stack σ such that (p, σ) is reachable and (q, σ) is co-reachable.…”

“…Then based on these criteria, we prove the class awnVPT to be decidable in PSPACE. Regarding the class gwnVPT, using a recent result of [2], we prove it is decidable in PTIME. Finally, we prove that the two classes gwnVPT and awnVPT enjoy good properties: they are closed under composition and type-checking is decidable against visibly pushdown languages.…”

“…wn , and there exists a run 2 , then one can find in this run a decomposition according to criterion (C2), and by induction hypothesis on u, the corresponding matched loops have a null effect in the computation of B. When removing these matched loops, we thus obtain new input words u ′ 1 and u ′ 3 with output words…”