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Definability of Cai-Fürer-Immerman Problems in Choiceless Polynomial Time

Wied Pakusa, Svenja Schalthöfer, Erkal Selman
2018 ACM Transactions on Computational Logic  
CPT can define many interesting Ptime queries, including (the original version of) the Cai-Fürer-Immerman (CFI) query.  ...  Choiceless Polynomial Time (CPT) is one of the most promising candidates in the search for a logic capturing Ptime.  ...  Most importantly, it is open whether the Cai-Fürer-Immerman problem over general graphs can be expressed in Choiceless Polynomial Time.  ... 
doi:10.1145/3154456 fatcat:b4nso4uqa5eqlfofh7lgczf2km

Is Polynomial Time Choiceless? [chapter]

Erich Grädel, Martin Grohe
2015 Lecture Notes in Computer Science  
A long time ago, Yuri Gurevich made precise the problem of whether there is a logic capturing polynomial-time on arbitrary finite structures, and conjectured that no such logic exists.  ...  Nevertheless, together with Andreas Blass and Saharon Shelah, he has also proposed what still seems to be the most promising candidate for a logic for polynomial time, namely Choiceless Polynomial Time  ...  Cyclic equation systems arise for instance in Cai-Fürer-Immerman (CFI) constructions.  ... 
doi:10.1007/978-3-319-23534-9_11 fatcat:n6rczztse5g4nezyzahgrmis3q

Choiceless Polynomial Time, Counting and the Cai–Fürer–Immerman Graphs

Anuj Dawar, David Richerby, Benjamin Rossman
2006 Electronical Notes in Theoretical Computer Science  
We consider Choiceless Polynomial Time ( e CPT), a language introduced by Blass, Gurevich and Shelah, and show that it can express a query originally constructed by Cai, Fürer and Immerman to separate  ...  used, even in e CPT(Card), an extension of e CPT with counting.  ...  Cai, Fürer and Immerman showed that the extension IFP + C of IFP with counting terms is still too weak to capture P [3] .  ... 
doi:10.1016/j.entcs.2005.05.024 fatcat:au55aqwhandrlkdvekciabidke

Choiceless Logarithmic Space

Erich Grädel, Svenja Schalthöfer
2019 Leibniz International Proceedings in Informatics  
One of the strongest candidates for a logic capturing Ptime is Choiceless Polynomial Time (CPT).  ...  Choiceless Polynomial Time is the restriction of this model to polynomial-time resources. It is known that CPT is strictly more powerful than fixed-point logic with counting.  ...  By the same arguments as used in the proof of Theorem 40 in [8] , this implies that the Cai-Fürer-Immerman query cannot be defined in CLogspace. As a consequence we get Theorem 27.  ... 
doi:10.18154/rwth-2019-07940 fatcat:367nu3is4bbl5n5sadl7uwyxqi

Choiceless polynomial time, counting and the Cai–Fürer–Immerman graphs

Anuj Dawar, David Richerby, Benjamin Rossman
2008 Annals of Pure and Applied Logic  
We consider Choiceless Polynomial Time (CPT), a language introduced by Blass, Gurevich and Shelah, and show that it can express a query originally constructed by Cai, Fürer and Immerman to separate fixed-point  ...  The program we present uses sets of unbounded finite rank: we demonstrate that this is necessary by showing that the query cannot be computed by any program that has a constant bound on the rank of sets  ...  We are grateful to the referee for pointing out a serious error in Section 5 of that paper. This section has been completely rewritten as Section 7 of the present paper. The  ... 
doi:10.1016/j.apal.2007.11.011 fatcat:hfmyhfbrevfbfo6uewhwzy5jyu

On Symmetric and Choiceless Computation [chapter]

Anuj Dawar
2016 Lecture Notes in Computer Science  
[4] define the class CPTC of problems decidable in choiceless polynomial time with counting and ask the question whether this includes all of P.  ...  Cai-Fürer Immerman Graphs The final ingredient in proving a lower bound is the construction of pairs of graphs that are not isomorphic, but are equivalent in the relation ≡ k .  ... 
doi:10.1007/978-3-319-28678-5_2 fatcat:p6ku7i4gcjfefbbjbgiygmhzmq

Logics with Rank Operators

Anuj Dawar, Martin Grohe, Bjarki Holm, Bastian Laubner
2009 2009 24th Annual IEEE Symposium on Logic In Computer Science  
The logics we define have data complexity contained in polynomial time and all known examples of polynomial time queries that are not definable in FP+C are definable in FP+rk, the extension of FP with  ...  cardinality of a definable set.  ...  However, Cai, Fürer, and Immerman [5] proved that FP+C does not capture polynomial time (on the class of all finite structures).  ... 
doi:10.1109/lics.2009.24 dblp:conf/lics/DawarGHL09 fatcat:emhilp4hvnexzp5r7d2yee4dji

On polynomial time computation over unordered structures

Andreas Blass, Yuri Gurevich, Saharon Shelah
2002 Journal of Symbolic Logic (JSL)  
We show that the examples in a paper of Cai, Fürer, and Immerman, when suitably padded, are in choiceless polynomial time yet not in fixpoint plus counting.  ...  We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time.  ...  ) Cai, Fürer, Immerman graphs G 0 m and G 1 m (as defined in Section 4) for all m?  ... 
doi:10.2178/jsl/1190150152 fatcat:xojjlyg65rbplcz4gc7gbhmxr4

On Polynomial Time Computation Over Unordered Structures [article]

Andreas Blass and Yuri Gurevich and Saharon Shelah
2001 arXiv   pre-print
We show that the examples in a paper of Cai, Furer, and Immerman, when suitably padded, are in choiceless polynomial time yet not in fixpoint plus counting.  ...  We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time.  ...  We point out that, although we have formulated these questions for CPT+Card, the logic in which we are primarily interested, the first two of them are open also forCPT, and the last two are open also for  ... 
arXiv:math/0102059v1 fatcat:io6j7jst3zfbvjlmrqj4hqpuzi

Characterising Choiceless Polynomial Time with First-Order Interpretations

Erich Gradel, Wied Pakusa, Svenja Schalthofer, Lukasz Kaiser
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
Choiceless Polynomial Time (CPT) is one of the candidates in the quest for a logic for polynomial time.  ...  Our main result is that PIL has precisely the same expressive power as Choiceless Polynomial Time.  ...  In [13] Dawar, Richerby and Rossman show that CPT can define the well-known isomorphism problem for the Cai, Fürer, Immerman graphs (which are 2-bounded structures).  ... 
doi:10.1109/lics.2015.68 dblp:conf/lics/GradelPSK15 fatcat:o6zudj655jdxzk54icw22ulauu

Fixed-Point Definability and Polynomial Time on Chordal Graphs and Line Graphs [article]

Martin Grohe
2010 arXiv   pre-print
The question of whether there is a logic that captures polynomial time was formulated by Yuri Gurevich in 1988.  ...  The introductory part of this paper is a short survey of the state-of-the-art in the quest for a logic capturing polynomial time.  ...  Acknowledgements I would like to thank Yijia Chen and Bastian Laubner for valuable comments on an earlier version of this paper.  ... 
arXiv:1001.2572v2 fatcat:wklf3k4p65eobmoxatvy2sfvpe

Choiceless Polynomial Time, Symmetric Circuits and Cai-Fürer-Immerman Graphs [article]

Benedikt Pago
2021 arXiv   pre-print
A prominent example of a decision problem in PTIME that is not known to be CPT-definable is the isomorphism problem on unordered Cai-Fürer-Immerman graphs (the CFI-query).  ...  Choiceless Polynomial Time (CPT) is currently the only candidate logic for capturing PTIME (that is, it is contained in PTIME and has not been separated from it).  ...  Cai-Fürer-Immerman graphs and their symmetries The CFI-construction Let G = (V, E) be an undirected (and unordered) connected graph.  ... 
arXiv:2107.03778v1 fatcat:5wuqyd46yrfv5d64ko3jvadsze

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting [article]

Matthew Anderson, Anuj Dawar, Bjarki Holm
2013 arXiv   pre-print
We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems.  ...  polynomial time with counting.  ...  Acknowledgments The authors would like to thank Siddharth Barman for his helpful comments on an early draft of this paper and the anonymous reviewers for their constructive suggestions.  ... 
arXiv:1304.6870v1 fatcat:jp4v3sxrvfb6rolqnpkpumrsei

Solving Linear Programs without Breaking Abstractions

Matthew Anderson, Anuj Dawar, Bjarki Holm
2015 Journal of the ACM  
We use this to show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [Blass et al. 1999 ].  ...  In particular, we demonstrate that the solvability of linear programs can be expressed in fixed-point logic with counting (FPC) as long as the program is given by a separation oracle that is itself definable  ...  ACKNOWLEDGMENT The authors would like to thank Siddharth Barman for his helpful comments on an early draft of this paper and the anonymous reviewers for their constructive suggestions.  ... 
doi:10.1145/2822890 fatcat:qhnbmivddvg5xc7phjga3i4znq

Towards Capturing PTIME with no Counting Construct (but with a Choice Operator) [article]

Eugenia Ternovska
2021 arXiv   pre-print
The central open question in Descriptive Complexity is whether there is a logic that characterizes deterministic polynomial time (PTIME) on relational structures.  ...  Towards this goal, we define a logic that is obtained from first-order logic with fixed points, FO(FP), by a series of transformations that include restricting logical connectives and adding a dynamic  ...  However, even extended with counting terms, FO(FP) + C, fixed-point logic is too weak to express all polynomial-time properties on structures without such order, as shown by Cai, Fürer and Immerman in  ... 
arXiv:2111.07978v1 fatcat:s2jiasyeafd7hb2kp4uka5ugii
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