Finite Presentations of Infinite Structures: Automata and Interpretations

Blumensath, Achim; Grädel, Erich

doi:10.1007/s00224-004-1133-y

Cited by 94 publications

(133 citation statements)

References 38 publications

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“…Indeed, the only subtree to which a ground tree rewriting rule (s, t) can be applied is the tree s. On stack trees, 2 This unusual definition is necessary because · is not associative. For example,…”

Section: Operation Automatamentioning

confidence: 99%

“…-(x, q) has a unique son (y, q ′ ), which has no other father such that (x, q) 2 − → (y, q ′ ). By definition of Trans, we have that ψ copy 2 n ,2 (x, y), and thus we have a (q, (q ′′ , q ′ )) ∈ ∆ and a z such that ψ copy 2 n ,1 (x, z) which is in Z q ′′ .…”

Section: C4 the Formula φ Associated With An Automatonmentioning

confidence: 99%

“…Also well-known are the automatic and tree-automatic structures (see for instance [2]), whose vertices are represented by words or trees and whose edges are characterised using finite automata running over tuples of vertices. The decidability of first-order logic (FO) over these graphs stems from the well-known closure properties of regular word and tree languages, but it can also be related to Rabin's result since tree-automatic graphs are precisely the class of graphs obtained from ∆ 2 using finite-set interpretations [10], a generalisation of WMSO interpretations mapping structures with a decidable MSO theory to structures with a decidable FO theory.…”

Section: Introductionmentioning

confidence: 99%

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Rewriting Higher-Order Stack Trees

Penelle

2015

Lecture Notes in Computer Science

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Abstract. Higher-order pushdown systems and ground tree rewriting systems can be seen as extensions of suffix word rewriting systems. Both classes generate infinite graphs with interesting logical properties. Indeed, the model-checking problem for monadic second order logic (respectively first order logic with a reachability predicate) is decidable on such graphs. We unify both models by introducing the notion of stack trees, trees whose nodes are labelled by higher-order stacks, and define the corresponding class of higher-order ground tree rewriting systems. We show that these graphs retain the decidability properties of ground tree rewriting graphs while generalising the pushdown hierarchy of graphs.

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Section: Operation Automatamentioning

confidence: 99%

Section: C4 the Formula φ Associated With An Automatonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rewriting Higher-Order Stack Trees

Penelle

2015

Lecture Notes in Computer Science

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“…For this, one has to pad shorter strings with the padding symbol ⋄ when defining the convolution of finite strings. More details on ω-automatic structures can be found in [BG04,HKMN08,KRB08]. In particular, a countable structure is ω-automatic if and only if it is automatic [KRB08].…”

Section: ω-Automatic Structuresmentioning

confidence: 99%

“…In essence, a graph is automatic if the elements of the universe can be represented as strings from a regular language and the edge relation can be recognized by a finite state automaton with several heads that proceed synchronously. Automatic graphs (and more general, automatic structures) received increasing interest over the last years [BG04, ⋆ The second and third author are supported by the DFG research project GELO. IKR02,KNRS07,KRS05,Rub08].…”

Section: Introductionmentioning

confidence: 99%

The Isomorphism Problem for ω-Automatic Trees

Kuske

Liu

Lohrey

2010

Lecture Notes in Computer Science

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Abstract. The main result of this paper states that the isomorphism for ω-automatic trees of finite height is at least has hard as second-order arithmetic and therefore not analytical. This strengthens a recent result by Hjorth, Khoussainov, Montalbán, and Nies [HKMN08] showing that the isomorphism problem for ω-automatic structures is not in Σ 1 2 . Moreover, assuming the continuum hypothesis CH, we can show that the isomorphism problem for ω-automatic trees of finite height is recursively equivalent with second-order arithmetic. On the way to our main results, we show lower and upper bounds for the isomorphism problem for ω-automatic trees of every finite height: (i) It is decidable (Π 0 1 -complete, resp,) for height 1 (2, resp.), (ii) Π 1 1 -hard and in Π 1 2 for height 3, and (iii) Π 1 n−3 -and Σ 1 n−3 -hard and in Π 1 2n−4 (assuming CH) for all n ≥ 4. All proofs are elementary and do not rely on theorems from set theory.

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Clique‐convergence is undecidable for automatic graphs

Cedillo

Pizaña

2020

Journal of Graph Theory

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The clique operator transforms a graph G into its clique graph K ( G ), which is the intersection graph of all the (maximal) cliques of G. Iterated clique graphs are then defined by K n ( G ) = K ( K n − 1 ( G ) ), K 0 ( G ) = G. If there are some n ≠ m such that K n ( G ) ≅ K m ( G ), then we say that G is clique‐convergent. The clique graph operator and iterated clique graphs have been studied extensively, but no characterization for clique‐convergence has been found so far. Automatic graphs are (not necessarily finite) graphs whose vertices and edges can be recognized by finite automata. Automatic graphs (and automatic structures) have strong decidability properties inherited from the finite automata defining them. Here we prove that clique‐convergence is algorithmically undecidable for the class of automatic graphs. Moreover, the problem remains undecidable, if we reduce to the class that contain only quasi‐clique‐Helly and bounded degree graphs. As a consequence, it follows that clique‐convergence for automatic graphs is not first‐order expressible.

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Finite Presentations of Infinite Structures: Automata and Interpretations

Cited by 94 publications

References 38 publications

Rewriting Higher-Order Stack Trees

Rewriting Higher-Order Stack Trees

The Isomorphism Problem for ω-Automatic Trees

Clique‐convergence is undecidable for automatic graphs

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