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A Completeness Property of Wilke's Tree Algebras [chapter]

Saeed Salehi
2003 Lecture Notes in Computer Science  
In this paper a completeness property of these operations is studied.  ...  Wilke's tree algebra formalism for characterizing families of tree languages is based on six operations involving letters, binary trees and binary contexts.  ...  Acknowledgement I am much grateful to professor Magnus Steinby for reading drafts of this paper and for his fruitful ideas, comments and support.  ... 
doi:10.1007/978-3-540-45138-9_60 fatcat:7t7yvjwo3rgnlox5skqqo5ooie

Tree algebras and varieties of tree languages

Saeed Salehi, Magnus Steinby
2007 Theoretical Computer Science  
We consider several aspects of Wilke's [T. Wilke, An algebraic characterization of frontier testable tree languages, Theoret. Comput.  ...  A convergent term rewriting system yields normal form representations of binary trees and contexts, as well as a new completeness proof and a computational decision method for the axiomatization of tree  ...  Syntactic Γ -algebras The basic properties of Wilke's [34] syntactic tree algebra congruences and syntactic tree algebras of binary tree languages can be obtained conveniently by considering more generally  ... 
doi:10.1016/j.tcs.2007.02.006 fatcat:qfptqgpyxvhprcdz5eez7vtjfq

Tree algebra of sofic tree languages

Nathalie Aubrun, Marie–Pierre Béal
2014 RAIRO - Theoretical Informatics and Applications  
For this goal, we define a tree algebra which is finer than the classical syntactic tree algebra based on contexts.  ...  Each class corresponds to a class of subshifts of trees which is invariant by conjugacy.  ...  This congruence is a three-sorted algebra (labels, trees, contexts) called Wilke's tree algebra (see [21] ).  ... 
doi:10.1051/ita/2014018 fatcat:d7qijubctnectpwxov367wfiyi

AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

OLIVIER CARTON, THOMAS COLCOMBET, GABRIELE PUPPIS
2018 Journal of Symbolic Logic (JSL)  
Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.  ...  AbstractWe develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings.  ...  Acknowledgements We are grateful to Achim Blumensath for his numerous comments on this work and to Alexander Rabinovich for the discussions on the subject and for introducing us to the question of definability  ... 
doi:10.1017/jsl.2018.7 fatcat:iwwjewuuxre6ba7bt666e7ukfe

An algebraic approach to MSO-definability on countable linear orderings [article]

Olivier Carton, Gabriele Puppis
2018 arXiv   pre-print
Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.  ...  We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings.  ...  Acknowledgements We are grateful to Achim Blumensath for his numerous comments on this work and to Alexander Rabinovich for the discussions on the subject and for introducing us to the question of definability  ... 
arXiv:1702.05342v2 fatcat:gas7y463ujfxte64ovfrky44lm

Eilenberg Theorems for Free [article]

Henning Urbat, Jiří Adámek, Liang-Ting Chen, Stefan Milius
2017 arXiv   pre-print
Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory.  ...  We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality.  ...  The algebraic theory corresponding to binary trees is given by Wilke's tree algebras [47] .  ... 
arXiv:1602.05831v3 fatcat:rkuqhflgobfnhihwkauoolx5rm

Synchronous Parallel Database Transformations [chapter]

Klaus-Dieter Schewe, Qing Wang
2012 Lecture Notes in Computer Science  
For the characterisation of XML database transformations the natural first step is therefore to define the appropriate tree-based backgrounds, which draw on hereditarily finite trees, tree algebra operations  ...  This defines a computational model for XML database transformation using a variant of Abstract State Machines.  ...  In terms of manipulating tree structures, our tree algebra is powerful enough to express operations provided by the forest algebra [Bojanczyk and Walukiewicz, 2007] and Wilke's tree algebra [Wilke  ... 
doi:10.1007/978-3-642-28472-4_21 fatcat:2hjtynvwj5cdtpwqgdhpmrhowi

Factorization forests for infinite words and applications to countable scattered linear orderings

Thomas Colcombet
2010 Theoretical Computer Science  
The theorem of factorization forests of Imre Simon shows the existence of nested factorizations -à la Ramsey -for finite words.  ...  We use this last variant in a simplified proof of the translation of recognizable languages over countable scattered linear orderings to languages accepted by automata.  ...  I am also grateful to the two anonymous referees for their numerous comments which greatly improved the quality of this work.  ... 
doi:10.1016/j.tcs.2009.10.013 fatcat:52ivbabt6jbehev23capc2uapq

A SURVEY ON SMALL FRAGMENTS OF FIRST-ORDER LOGIC OVER FINITE WORDS

VOLKER DIEKERT, PAUL GASTIN, MANFRED KUFLEITNER
2008 International Journal of Foundations of Computer Science  
We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments.  ...  We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere.  ...  We thank the referee for his very detailed report which gave us many helpful suggestions and for pointing out some inaccuracies in a preliminary version of this paper.  ... 
doi:10.1142/s0129054108005802 fatcat:bqeokxgje5fq3lwfgjgx2p6gmm

How big must complete XML query languages be?

Clemens Ley, Michael Benedikt
2009 Proceedings of the 12th International Conference on Database Theory - ICDT '09  
trees.  ...  How can one extend XPath to get a first-order complete language? Marx has shown that Conditional XPath -an extension of XPath with an "Until" operator -is first order complete.  ...  CTL * is not complete over trees, since it has no means of determining the number of children of a node with a certain property.  ... 
doi:10.1145/1514894.1514917 dblp:conf/icdt/LeyB09 fatcat:kkzcfwghxjg7folp5uvxku5gsi

Some model theory for the modal μ-calculus: syntactic characterisations of semantic properties [article]

Gaëlle Fontaine, Yde Venema
2018 arXiv   pre-print
Each of the properties that we consider is, in a similar way, associated with one of the following special kinds of subset of a tree model: singletons, finite sets, finitely branching subtrees, noetherian  ...  More in particular, we discuss a number of semantic properties pertaining to formulas of the modal μ-calculus.  ...  Acknowledgment The authors are deeply indebted to the two referees for many helpful comments on an earlier version of this paper.  ... 
arXiv:1801.05994v1 fatcat:5jcelk3fe5aodm2ulx4ecthlju

First-order separation over countable ordinals [article]

Thomas Colcombet, Sam van Gool, Rémi Morvan
2022 arXiv   pre-print
a new operation capturing the loss of precision induced by first-order indistinguishability.  ...  For this, we develop the algebraic concept of monoid (resp. ω-semigroup, resp. ordinal monoid) with aperiodic merge, an extension of monoids (resp. ω-semigroup, resp. ordinal monoid) that explicitly includes  ...  Our completeness proof for the algorithm follows a scheme similar to the one followed by Place and Zeitoun in the context of finite and ω-words [PZ16] , which was inspired by Wilke's characterisation  ... 
arXiv:2201.03089v1 fatcat:k4nolvsumjc4rmjdw2o2wrev24

Parity Games and Automata for Game Logic (Extended Version) [article]

Helle Hvid Hansen, Clemens Kupke, Johannes Marti, Yde Venema
2017 arXiv   pre-print
They are essentially a graph-based view of the alternating tree automata that were introduced by Wilke in the study of modal μ-calculus.  ...  Our proof makes extensive use of structures that we call syntax graphs that combine the ease-of-use of syntax trees of formulas with the flexibility and succinctness of automata.  ...  Furthermore, game logic constitutes a very general dynamic logic that makes very few assumptions on the algebraic properties of the modal operators.  ... 
arXiv:1709.00777v1 fatcat:hxpnvu3uqfgnzf7ayq6pirkuya

Parity Games and Automata for Game Logic [chapter]

Helle Hvid Hansen, Clemens Kupke, Johannes Marti, Yde Venema
2018 Lecture Notes in Computer Science  
They are essentially a graph-based view of the alternating tree automata that were introduced by Wilke in the study of modal µ-calculus. Position b P(b) Moves E[b]  ...  Our proof makes extensive use of structures that we call syntax graphs that combine the ease-of-use of syntax trees of formulas with the flexibility and succinctness of automata.  ...  Furthermore, game logic constitutes a very general dynamic logic that makes very few assumptions on the algebraic properties of the modal operators.  ... 
doi:10.1007/978-3-319-73579-5_8 fatcat:cw3dll3vn5arrodfehzzcdt36m

Simulations and Antichains for Efficient Handling of Finite Automata [article]

Lukáš Holík
2017 arXiv   pre-print
We give efficient algorithms for computing tree automata simulations and we also introduce a new type of relation that arises from a combination of tree automata downward and upward simulation and that  ...  For size reduction of tree automata, we adapt the simulation quotient technique that is well established for finite word automata.  ...  This makes tree automata a convenient tool for modelling and reasoning about various kinds of structured objects such as syntactical trees, structured documents, configurations of complex systems, algebraic  ... 
arXiv:1706.03208v1 fatcat:qmzaknz5xnfy5lmjp426jjlf5a
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