Invariants of Automatic Presentations and Semi-synchronous Transductions

2006 ◽  
pp. 289-300 ◽  
Author(s):  
Vince Bárány
Keyword(s):  
Automatic Presentations

scholarly journals UNARY FA-PRESENTABLE SEMIGROUPS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250038 ◽  
Author(s):  
ALAN J. CAIN ◽  
NIK RUŠKUC ◽  
RICHARD M. THOMAS
Keyword(s):  
Difficult Problem ◽  
Decision Problems ◽  
Finite Model ◽  
Locally Finite ◽  
Letter Alphabet ◽  
Significant Interest ◽  
Automatic Presentations ◽  
The Relationship ◽  
Completely Simple

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classification of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a difficult problem in general. A restricted problem, also of significant interest, is to ask this question for unary automatic presentations: automatic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally finite, but non-finitely generated unary FA-presentable semigroups may be infinite. Every unary FA-presentable semigroup satisfies some Burnside identity. We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classification is given of the unary FA-presentable completely simple semigroups.

scholarly journals Automatic Presentations and Semigroup Constructions

2009 ◽  
Vol 47 (2) ◽  
pp. 568-592 ◽  
Author(s):  
Alan J. Cain ◽  
Graham Oliver ◽  
Nik Ruškuc ◽  
Richard M. Thomas
Keyword(s):  
Automatic Presentations

Automatic Presentations for Cancellative Semigroups

2008 ◽  
pp. 149-159 ◽  
Author(s):  
Alan J. Cain ◽  
Graham Oliver ◽  
Nik Ruškuc ◽  
Richard M. Thomas
Keyword(s):  
Automatic Presentations

scholarly journals Automatic presentations for semigroups

2009 ◽  
Vol 207 (11) ◽  
pp. 1156-1168 ◽  
Author(s):  
Alan J. Cain ◽  
Graham Oliver ◽  
Nik Ruškuc ◽  
Richard M. Thomas
Keyword(s):  
Automatic Presentations

Automata Presenting Structures: A Survey of the Finite String Case

2008 ◽  
Vol 14 (2) ◽  
pp. 169-209 ◽  
Author(s):  
Sasha Rubin
Keyword(s):  
Order Theory ◽  
Isomorphism Problem ◽  
First Order ◽  
First Order Theory ◽  
Automatic Presentations ◽  
The Relationship

AbstractA structure has a (finite-string)automatic presentationif the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

scholarly journals Automatic structures of bounded degree revisited

2011 ◽  
Vol 76 (4) ◽  
pp. 1352-1380 ◽  
Author(s):  
Dietrich Kuske ◽  
Markus Lohrey
Keyword(s):  
Order Theory ◽  
Upper Bounds ◽  
Bounded Degree ◽  
First Order ◽  
Automatic Structures ◽  
First Order Theory ◽  
Automatic Structure ◽  
Automatic Presentations ◽  
Exponential Space

AbstractThe first-order theory of a string automatic structure is known to be decidable, but there are examples of string automatic structures with nonelementary first-order theories. We prove that the first-order theory of a string automatic structure of bounded degree is decidable in doubly exponential space (for injective automatic presentations, this holds even uniformly). This result is shown to be optimal since we also present a string automatic structure of bounded degree whose first-order theory is hard for 2EXPSPACE. We prove similar results also for tree automatic structures. These findings close the gaps left open in [28] by improving both the lower and the upper bounds.

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