776 research outputs found
The HOM problem is decidable
We close affirmatively a question which has been open for 35 years: decidability of the HOM problem. The HOM problem consists in deciding, given a tree homomorphism and a regular tree languagle represented by a tree automaton, whether is regular. For deciding the HOM problem, we develop new constructions and techniques which are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular languages through tree homomorphisms. Our contributions are based on the following new results. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with disequality constraints recognizing the complementary language. We also define a new class of automaton with arbitrary disequality constraints and a particular kind of equality constraints. This new class essentially recognizes the intersection of a tree automaton with disequality constraints and the image of a regular language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism. The above constructions are combined adequately to provide an algorithm deciding the HOM problem.Postprint (published version
Solving the Weighted HOM-Problem With the Help of Unambiguity
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,
Gim\'enez, Ramos, \`Alvarez: 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.Comment: In Proceedings AFL 2023, arXiv:2309.0112
Algorithmic recognition of infinite cyclic extensions
We prove that one cannot algorithmically decide whether a finitely presented
-extension admits a finitely generated base group, and we use this
fact to prove the undecidability of the BNS invariant. Furthermore, we show the
equivalence between the isomorphism problem within the subclass of unique
-extensions, and the semi-conjugacy problem for deranged outer
automorphisms.Comment: 24 page
Effective lambda-models vs recursively enumerable lambda-theories
A longstanding open problem is whether there exists a non syntactical model
of the untyped lambda-calculus whose theory is exactly the least lambda-theory
(l-beta). In this paper we investigate the more general question of whether the
equational/order theory of a model of the (untyped) lambda-calculus can be
recursively enumerable (r.e. for brevity). We introduce a notion of effective
model of lambda-calculus calculus, which covers in particular all the models
individually introduced in the literature. We prove that the order theory of an
effective model is never r.e.; from this it follows that its equational theory
cannot be l-beta or l-beta-eta. We then show that no effective model living in
the stable or strongly stable semantics has an r.e. equational theory.
Concerning Scott's semantics, we investigate the class of graph models and
prove that no order theory of a graph model can be r.e., and that there exists
an effective graph model whose equational/order theory is minimum among all
theories of graph models. Finally, we show that the class of graph models
enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34
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