On Polynomial Recursive Sequences

Cadilhac, Michaël; Mazowiecki, Filip; Paperman, Charles; Pilipczuk, Michał; Sénizergues, Géraud

doi:10.1007/s00224-021-10046-9

Cited by 4 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…holonomic) and polynomial recursive sequences [5]. The zeroness problem for P-recursive sequences is decidable [36] and the same holds for polynomial recursive sequences (as a corollary of the existence of cancelling polynomials [5,Theorem 11]). However, no complexity upper bounds are known for those more general classes.…”

Section: Discussionmentioning

confidence: 99%

“…In order to apply (5) we require that L is a deterministic class efficiently closed under complement (i.e., a representation for the complement is constructible in PTIME) and that the class M is closed under disjoint unions and intersections with languages from L . Most deterministic languages classes, such as those recognised by deterministic finte automata, deterministic context-free grammars, deterministic Parikh automata, deterministic register automata, etc., satisfy the first requirement 6 .…”

Section: From Inclusion To Universalitymentioning

confidence: 99%

“…The reduction applies also to undecidable instances of the language inclusion problem such as CFG ⊆ DCFG, however in this case it is of no use since DCFG ⊆ DCFG is known to be undecidable [19, Theorem 10.7, Point 2] 4. The construction below can easily be adapted to infinite alphabets of the form Σ × A, where Σ is finite and A is an infinite set of data values[3] 5. Languages of infinite words can be handled similarly.…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Clemente

2020

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: From Inclusion To Universalitymentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

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

Clemente

2020

Preprint

View full text Add to dashboard Cite

show abstract

On Rational Recursion for Holonomic Sequences

Teguia Tabuguia,

Worrell

2024

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Simple C2-finite Sequences

Nuspl

Pillwein

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

A sequence is called C-finite, if it satisfies a linear recurrence with constant coefficients and holonomic, if it satisfies a linear recurrence with polynomial coefficients. The class of C 2 -finite sequences is a natural generalization of holonomic sequences and consists of sequences satisfying a linear recurrence with C-finite coefficients whose leading coefficient has no zero terms. Recently, we investigated computational properties of C 2 -finite sequences: we showed that these sequences form a difference ring and provided methods to compute in this ring.From an algorithmic point of view, some of these results were not as far reaching as we hoped for. In this paper, we define the class of simple C 2 -finite sequences and show that it satisfies the same computational properties, but does not share the same technical issues. In particular, we are able to derive bounds for the asymptotic behavior, can compute closure properties more efficiently, and have a characterization via the generating function. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic algorithms; • Mathematics of computing → Combinatorial algorithms.

show abstract

On Polynomial Recursive Sequences

Cited by 4 publications

References 20 publications

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

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

On Rational Recursion for Holonomic Sequences

Simple C2-finite Sequences

Contact Info

Product

Resources

About