Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity

Abstract

We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub-or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of 2 O(n) on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size n. (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of 2 2 O(n) following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing sub-or superword-closed languages is only NP-complete as opposed to PSPACE-complete for general NFAs. Finally, we extend our results into an approximation method to attack inequivalence problems for CFGs.This work was partially funded by the DFG project "Polynomial Systems on Semirings: Foundations, Algorithms, Applications".

The complexity of regular abstractions of one-counter languages

“…These results are in contrast with the exponential lower bounds known for both closures in the case of CFL [36]: Several constructions for L↑ and L↓ have been proposed in the literature (see, e.g., [8,11,18,36]), and the best in terms of the size of NFA are exponential, due to van Leeuwen [36] and Bachmeier, Luttenberger, and Schlund [8], respectively.…”

“…It can easily be turned into a walk of B as follows. Instead of a non-loop transition, we take its counterpart of type (8). Instead of a loop-transition, we take its counterpart of type (9).…”

The Ideal Approach to Computing Closed Subsets in Well-Quasi-orderings

“…In particular this implies that I ⊆ ↓ X Y (we say that I is "below Y ") and I ∩ Y = ∅ (we say that I is "crossing Y "). The converse implication does not hold, as witnessed by X = N, Y = [1, 3] ∪ [5,7] and I = ↓ 4.…”

“…In this chapter we are concerned with the issue of reasoning about, and computing with, downwards-closed and upwards-closed subsets of a WQO. These sets appear in program verification (prominently in model-checking of well structured systems [6], in verification of Petri nets [29], in separability problems [27,61], but also as an effective abstraction tool [5,60]). The question of how to handle downwards-closed subsets of WQOs in a generic way was first raised by Geeraerts et al: in [22] the authors postulated the existence of an adequate domain of limits satisfying some representation conditions.…”

On the state complexity of closures and interiors of regular languages with subwords and superwords

“…2 Supported by Grant MA 4938/21 of the DFG. 3 Supported by Grant ANR-11-BS02-001. upward closure of L, respectively.…”