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Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy [article]

Christoph Berkholz, Andreas Krebs, Oleg Verbitsky
2013 arXiv   pre-print
Given two structures G and H distinguishable in k (first-order logic with k variables), let A^k(G,H) denote the minimum alternation depth of a k formula distinguishing G from H.  ...  For k> 3 the last lower bound holds also over uncolored trees, while the alternation hierarchy of 2 collapses even over all uncolored graphs.  ...  Acknowledgement The authors are thankful to Oleg Pikhurko for his useful suggestions.  ... 
arXiv:1212.2747v4 fatcat:lzati3rb4nbqfe63k5zp53gdfq

An effective characterization of the alternation hierarchy in two-variable logic [article]

Andreas Krebs, Howard Straubing
2012 arXiv   pre-print
We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities.  ...  This implies decidability of the individual levels. More generally we show that the two-sided semidirect product of a decidable variety with the variety J is decidable.  ...  Acknowledgements We are grateful to Manfred Kufleitner, Benjamin Steinberg, and Pascal Weil for detailed discussions of this work.  ... 
arXiv:1205.4802v1 fatcat:aj7tzkfpmzg7vjio6r65ovk3mm

An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

Andreas Krebs, Howard Straubing
2017 ACM Transactions on Computational Logic  
We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities.  ...  This implies decidability of the individual levels. More generally we show that two-sided semidirect products with J as the right-hand factor preserve decidability.  ...  Acknowledgements We are grateful to Manfred Kufleitner, Benjamin Steinberg, and Pascal Weil for detailed discussions of this work, and to the anonymous referees for their helpful suggestions.  ... 
doi:10.1145/3149822 fatcat:uyjaqipt7fbb7gpwoipokqznya

Structure Theorem and Strict Alternation Hierarchy for FO2on Words

Philipp Weis, Neil Immerman, Thomas Schwentick
2009 Logical Methods in Computer Science  
For both languages, our structure theorems show exactly what is expressible using a given quantifier depth, n, and using m blocks of alternating quantifiers, for any m \leq n.  ...  Using these characterizations, we prove, among other results, that there is a strict hierarchy of alternating quantifiers for both languages.  ...  the quantifier depth and on the alternation depth.  ... 
doi:10.2168/lmcs-5(3:4)2009 fatcat:hmlrvjrwqrgolprc2z6o4cvqwm

The alternation hierarchy in fixpoint logic with chop is strict too

Martin Lange
2006 Information and Computation  
We develop a game-theoretic characterisation of its model checking problem and use these games to show that the alternation hierarchy in this logic is strict.  ...  The structure of this result follows the lines of Arnold's proof showing that the alternation hierarchy in the modal -calculus is strict over the class of binary trees.  ...  The most valuable one, however, was the idea to extend Arnold's proof of the strict hierarchy in the modal -calculus.  ... 
doi:10.1016/j.ic.2006.05.001 fatcat:i5mn5wtlcjhwhohtufv5he5uei

The modal mu-calculus alternation hierarchy is strict [chapter]

J. C. Bradfield
1996 Lecture Notes in Computer Science  
One of the open questions about the modal mu-calculus is whether the alternation hierarchy collapses; that is, whether all modal fixpoint properties can be expressed with only a few alternations of least  ...  In this paper, we resolve this question by showing that the hierarchy does not collapse. @ 1998 -Elsevier Science B.V. All rights reserved  ...  unbounded quantifers by quantifiers bounded in a.  ... 
doi:10.1007/3-540-61604-7_58 fatcat:rqvgzyzj3neo7f7rpt465s5iga

The modal mu-calculus alternation hierarchy is strict

J.C. Bradfield
1998 Theoretical Computer Science  
One of the open questions about the modal mu-calculus is whether the alternation hierarchy collapses; that is, whether all modal fixpoint properties can be expressed with only a few alternations of least  ...  In this paper, we resolve this question by showing that the hierarchy does not collapse. @ 1998 -Elsevier Science B.V. All rights reserved  ...  unbounded quantifers by quantifiers bounded in a.  ... 
doi:10.1016/s0304-3975(97)00217-x fatcat:5s5cbfkq3ncdngny4qtoyqcl3e

Going Higher in First-Order Quantifier Alternation Hierarchies on Words [article]

Thomas Place, Marc Zeitoun
2017 arXiv   pre-print
We investigate quantifier alternation hierarchies in first-order logic on finite words. Levels in these hierarchies are defined by counting the number of quantifier alternations in formulas.  ...  We prove that one can decide membership of a regular language in the levels BΣ_2 (finite boolean combinations of formulas having only one alternation) and Σ_3 (formulas having only two alternations and  ...  For each problem, we actually present a reduction to same problem for the corresponding fragment in the order hierarchy, and decidability then follows from the results of the previous sections.  ... 
arXiv:1707.05696v1 fatcat:mi6mid5llfawxcnekst73fm7ku

Level Two of the Quantifier Alternation Hierarchy over Infinite Words [article]

Manfred Kufleitner, Tobias Walter
2015 arXiv   pre-print
The study of various decision problems for logic fragments has a long history in computer science.  ...  This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment.  ...  On the level of logic, the difference between the Straubing-Thérien hierarchy and the dot-depth hierarchy is that formulas for the dot-depth hierarchy may also use the successor predicate.  ... 
arXiv:1509.06207v1 fatcat:agg5o2xkizg7hcs46ie2izpw5a

Level Two of the Quantifier Alternation Hierarchy over Infinite Words [chapter]

Manfred Kufleitner, Tobias Walter
2016 Lecture Notes in Computer Science  
The study of various decision problems for logic fragments has a long history in computer science.  ...  This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment.  ...  On the level of logic, the difference between the Straubing-Thérien hierarchy and the dot-depth hierarchy is that formulas for the dot-depth hierarchy may also use the successor predicate.  ... 
doi:10.1007/978-3-319-34171-2_16 fatcat:hztry7ymsjh75j6xgnjwukti24

The μ-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity

Julian Gutierrez, Felix Klaedtke, Martin Lange
2012 Electronic Proceedings in Theoretical Computer Science  
It is known that the alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict.  ...  infinite nested words and finite graphs with feedback vertex sets of a bounded size.  ...  Acknowledgments The authors thank Christian Dax for initial discussions on the topic of this paper and Julian Bradfield for advice on the alternation hierarchy.  ... 
doi:10.4204/eptcs.96.9 fatcat:ilfyzzs5fjc5xdaqufboz3pdk4

The Monadic Quantifier Alternation Hierarchy over Grids and Graphs

Oliver Matz, Nicole Schweikardt, Wolfgang Thomas
2002 Information and Computation  
The monadic second-order quantifier alternation hierarchy over the class of finite graphs is shown to be strict.  ...  It is notable that one can obtain sets of graphs which occur arbitrarily high in the monadic hierarchy but are already definable in the first-order closure of existential monadic second-order logic.  ...  It is open whether in this generalized context the alternation depth of the second-order quantifiers again induces a strict hierarchy. Further open questions are listed in the Conclusion.  ... 
doi:10.1006/inco.2002.2955 fatcat:iowkj2wiwzfm3p23gwygm63qgi

The μ-calculus alternation hierarchy collapses over structures with restricted connectivity

Julian Gutierrez, Felix Klaedtke, Martin Lange
2014 Theoretical Computer Science  
For instance, over the class of infinite words the alternation-free fragment of the µ-calculus is already as expressive as the full logic.  ...  The alternation hierarchy of least and greatest fixpoint operators in the µ-calculus is strict.  ...  The authors thank Christian Dax for initial discussions on the topic of this article, Julian Bradfield for advice on the alternation hierarchy and Florian Bruse for helping to correct the translation from  ... 
doi:10.1016/j.tcs.2014.03.027 fatcat:mknbv5jjb5gopfjyphovwepglq

Going higher in the First-order Quantifier Alternation Hierarchy on Words [article]

Thomas Place, Marc Zeitoun
2014 arXiv   pre-print
We investigate the quantifier alternation hierarchy in first-order logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas.  ...  We prove that one can decide membership of a regular language to the levels BΣ_2 (boolean combination of formulas having only 1 alternation) and Σ_3 (formulas having only 2 alternations beginning with  ...  Set i a level in the quantifier alternation hierarchy. We define the game for Σ i (<).  ... 
arXiv:1404.6832v1 fatcat:rcc2s66srrfznlhkb2rpw7pmdy

Arity and alternation in second-order logic

J.A. Makowsky, Y.B. Pnueli
1996 Annals of Pure and Applied Logic  
For first-order logic FOL, with alternation of quantifiers bounded by n, AUTOSAT (FOL,) is definable in AA(3, n + 4). AUTOSAT(AA(k, n)) is definable in AA(k + cl, n + d,) for some cl, d,.  ...  For first-order logic FOL with unbounded alternation of quantifiers AUTOSAT(FOL) is PSpacrcomplete.  ...  Fagin, who provided us with Proposition 2, and to the participants of the 1994 Oberwolfach Meeting on Finite Model Theory for various useful comments. We are also indebted to B.  ... 
doi:10.1016/0168-0072(95)00013-5 fatcat:b3iwj3l4sze6plwzp7jw2nmvsu
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