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Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy
[article]
2013
arXiv
pre-print
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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]
2012
arXiv
pre-print
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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
2017
ACM Transactions on Computational Logic
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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
2009
Logical Methods in Computer Science
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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
2006
Information and Computation
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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]
1996
Lecture Notes in Computer Science
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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
1998
Theoretical Computer Science
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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]
2017
arXiv
pre-print
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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]
2015
arXiv
pre-print
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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]
2016
Lecture Notes in Computer Science
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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
2012
Electronic Proceedings in Theoretical Computer Science
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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
2002
Information and Computation
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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
2014
Theoretical Computer Science
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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]
2014
arXiv
pre-print
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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
1996
Annals of Pure and Applied Logic
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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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