Analysing Complexity in Classes of Unary Automatic Structures

Liu, Jiamou; Minnes, Mia

doi:10.1007/978-3-642-00982-2_44

Cited by 2 publications

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“…In essence, a structure is automatic if the elements of the universe can be encoded by strings from a regular language and every relation of the structure can be recognized by a finite state automaton with several heads that proceed synchronously. Automatic structures received increasing interest over the last few years [6,7,14,25,27,29,30,31,35,36,38,44,48]. One of the main motivations for investigating automatic structures is that their first-order theories can be decided uniformly (i.e., the input is an automatic presentation and a first-order sentence).…”

Section: Introductionmentioning

confidence: 99%

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The isomorphism problem on classes of automatic structures with transitive relations

Kuske

Liu

Lohrey

2013

Trans. Amer. Math. Soc.

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Automatic structures are finitely presented structures where the universe and all relations can be recognized by finite automata. It is known that the isomorphism problem for automatic structures is complete for Σ 1 1 , the first existential level of the analytical hierarchy. Positive results on ordinals and on Boolean algebras raised hope that the isomorphism problem is simpler for transitive relations. We prove that this is not the case. More precisely, this paper shows: (i) The isomorphism problem for automatic equivalence relations is complete for Π 0 1 (the first universal level of the arithmetical hierarchy). (ii) The isomorphism problem for automatic trees of height n ≥ 2 is Π 0 2n−3complete. (iii) The isomorphism problem for well-founded automatic order trees is recursively equivalent to true first-order arithmetic. (iv) The isomorphism problem for automatic order trees is Σ 1 1-complete. (v) The isomorphism problem for automatic linear orders is Σ 1 1-complete. We also obtain Π 0 1-completeness of the elementary equivalence problem for several classes of automatic structures and Σ 1 1-completeness of the isomorphism problem for trees (resp., linear orders) consisting of a deterministic context-free language together with the prefix order (resp., lexicographic order). This solves several open questions ofÉsik, Khoussainov, Nerode, Rubin, and Stephan.

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Section: Introductionmentioning

confidence: 99%

“…Furthermore, the isomorphism problem of tree-automatic trees is studied in [26]. Lastly, we mention that for equivalence structures, linear orders, and order trees which have automatic presentations over a unary alphabet, the isomorphism problem is decidable in polynomial time [38].…”

Section: Introductionmentioning

confidence: 99%