The HOM problem is decidable

Godoy, Guillem; Giménez, Omer; Ramos, Lander; Àlvarez, Carme

doi:10.1145/1806689.1806757

Cited by 11 publications

(4 citation statements)

References 27 publications

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“…So, one could try to solve that problem, first. Recall that the problem is decidable in the setting of finite trees and homomorphisms by [CGGR16,GG13].…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

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Regular matching problems for infinite trees

Camino¹,

Diekert²,

Dundua³

et al. 2022

Logical Methods in Computer Science

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We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $\sigma$ is a substitution such that $\sigma(x)$ is a set of trees in $T(\Sigma\cup H)\setminus H$ for all $x\in \mathcal{X}$. Here, $H$ denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if $L$ and $R$ are regular, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma(L)\subseteq R$?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when $L,R$ are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if $L\subseteq T(\Sigma\cup\mathcal{X})$ and $R\subseteq T(\Sigma)$ are regular tree languages over finite or infinite trees, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma_{\mathrm{io}}(L)\subseteq R$?" is decidable. Here, the subscript "$\mathrm{io}$" in $\sigma_{\mathrm{io}}(L)$ refers to "inside-out". Moreover, there are only finitely many maximal solutions $\sigma$, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension $\sigma_{\mathrm{oi}}$ remains open, even in the restricted setting of finite trees.

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“…So, one could try to solve that problem, first. Recall that the problem is decidable in the setting of finite trees and homomorphisms by [CGGR16,GG13].…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…The inputs are a homomorphism ϕ and a regular tree language L. The question is whether ϕ(L) is regular. The problem is decidable in the setting of finite trees by [GG13]. It is Dexptime-complete by [CGGR16].…”

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confidence: 99%

Regular matching problems for infinite trees

Camino¹,

Diekert²,

Dundua³

et al. 2022

Logical Methods in Computer Science

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“…For instance, they cannot ensure that certain subtrees of input trees are equal [10], much like the classical (string) automata cannot ensure that the number of a's and b's in a word is equal. This defect was tackled with extensions proposed in [25] and [3,12,13] where tree automata with constraints can explicitly require or forbid certain subtrees to be equal. Such devices have played a crucial part in deciding the HOMproblem: This long-standing open question [2] asks, given a regular tree language and a tree homomorphism, whether the image is again regular.…”

Section: Introductionmentioning

confidence: 99%

Solving the Weighted HOM-Problem With the Help of Unambiguity

Nász

2023

Electron. Proc. Theor. Comput. Sci.

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The HOM-problem, which asks whether the image of a regular tree language under a tree homomorphism is again regular, is known to be decidable by [Godoy, Giménez, Ramos, Àlvarez: The HOM problem is decidable. STOC (2010)]. Research on the weighted version of this problem, however, is still in its infancy since it requires customized investigations. In this paper we address the weighted HOM-problem and strive to keep the underlying semiring as general as possible. In return, we restrict the input: We require the tree homomorphism h to be tetris-free, a condition weaker than injectivity, and for the given weighted tree automaton, we propose an ambiguity notion with respect to h. These assumptions suffice to ensure decidability of the thus restricted HOM-problem for all zero-sum free semirings by allowing us to reduce it to the (decidable) unweighted case.

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“…Over the last two decades, the decidability results have been pushed to stronger and stronger models. As one remarkable result, the theory of these automata has provided tools for solving a long standing open question, namely the decidability of the "HOM problem" [14,15], which asks for a given regular language T of finite trees and a tree homomorphism h, whether the image h(T ) of T under h is a regular tree language. Another motivation is the development of automaton models for capturing constraints in XML specification languages, like monadic key constraints for XML documents [1].…”

Section: Introductionmentioning

confidence: 99%

Automata on Infinite Trees with Equality and Disequality Constraints Between Siblings

Carayol¹,

Löding

Serre

2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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This article is inspired by two works from the early 90s. The first one is by Bogaert and Tison who considered a model of automata on finite ranked trees where one can check equality and disequality constraints between direct subtrees: they proved that this class of automata is closed under Boolean operations and that both the emptiness and the finiteness problem of the accepted language are decidable. The second one is by Niwinski who showed that one can compute the cardinality of any ω-regular language of infinite trees.Here, we generalise the model of automata of Tison and Bogaert to the setting of infinite binary trees. Roughly speaking we consider parity tree automata where some transitions are guarded and can be used only when the two direct sub-trees of the current node are equal/disequal. We show that the resulting class of languages encompasses the one of ω-regular languages of infinite trees while sharing most of its closure properties, in particular it is a Boolean algebra. Our main technical contribution is then to prove that it also enjoys a decidable cardinality problem. In particular, this implies the decidability of the emptiness problem.Categories and Subject Descriptors Theory of Computation [Formal languages and automata theory]: Tree languages Keywords Automata on infinite trees, Automata with equality and disequality constraints, Emptiness problem, Finiteness problem.

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The HOM problem is decidable

Cited by 11 publications

References 27 publications

Regular matching problems for infinite trees

Regular matching problems for infinite trees

Solving the Weighted HOM-Problem With the Help of Unambiguity

Automata on Infinite Trees with Equality and Disequality Constraints Between Siblings

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