On the complexity of the universality and inclusion problems for unambiguous context-free grammars (technical report)

Abstract

We study the computational complexity of universality and inclusion problems for unambiguous finite automata and context-free grammars. We observe that several such problems can be reduced to the universality problem for unambiguous context-free grammars. The latter problem has long been known to be decidable and we propose a PSPACE algorithm that works by reduction to the zeroness problem of recurrence equations with convolution. We are not aware of any non-trivial complexity lower bounds. However, we show that computing the coin-flip measure of an unambiguous contextfree language, a quantitative generalisation of universality, is hard for the long-standing open problem SQRTSUM. * This work has been partially supported by the Polish NCN grant 2017/26/D/ST6/00201. This is a technical report of an invited paper of the same title to appear in the proceedings of VPT 2020.1 In a later book, Kuich and Salomaa reprove decidability [23, Corollary 16.25] by using variable elimination, which is arguably closer to algebraic geometry than formal languages.1. We observe that in many cases the inclusion problem L ⊆ M reduces in polynomial time to the subcase where L is deterministic (Section 3.1.1). One application is lower bounds: Once we know that CFG ⊆ NFA is EXPTIME-hard [20, Theorem 2.1], we can immediately deduce that the same lower bound carries over to DCFG ⊆ NFA [20, Theorem 3.1].2. We observe that in many cases the inclusion problem L ⊆ M with L deterministic reduces in polynomial time to the universality problem (Section 3.1.2). One application is upper bounds (combined with the previous point): For instance, from the fact that UFA = Σ * is in PTIME we can deduce that the more general problem NFA ⊆ UFA is also in PTIME (Theorem 7), which seems to be a new observation.3. We apply the last two points to show that the following inclusion problems A ⊆ B reduce to UUCFG: A ∈ {DCFG, UCFG, CFG} and B = UFA (Theorem 9); A ∈ {DFA, UFA, NFA} and B = UCFG (Theorem 8). Since UUCFG is a special instance of the latter set of problems, they are PTIME interreducible with UUCFG.

On the Complexity of the Universality and Inclusion Problems for Unambiguous Context-Free Grammars

“…There are other problems which are known to be undecidable for CFG but decidable for DCFG, such as the regularity problem [32,34,31]. An interesting open problem 9 The sum of all coefficients is at most 1.…”

“…We aim at obtaining µ(X i ) = x i . By comparing (10) with (9) we deduce that the nonterminals C i and A must generate languages of measure µ(C i ) = c i , resp., µ(A) = a n+1 = 1 2(n+1) . Since the measures a, c i are rational, it suffices to find regular languages L(A), L(C i ).…”

“…The second constraint is handled by the following lemma. A full proof is available in the technical report [9].…”

“…* This work has been partially supported by the Polish NCN grant 2017/26/D/ST6/00201. A full technical report is available [9].…”

“…* This work has been partially supported by the Polish NCN grant 2017/26/D/ST6/00201. A full technical report is available [9].1 In a later book, Kuich and Salomaa reprove decidability [24, Corollary 16.25] by using variable elimination, which is arguably closer to algebraic geometry than formal languages.…”