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A Dichotomy for First-Order Reducts of Unary Structures
[article]

2017
*
arXiv
*
pre-print

Moreover, we study the subclass C

arXiv:1601.04520v5
fatcat:psjkbouz6fhzdgprwfthjinje4
*of*CSPs*for**structures**A*that are*reducts**of**a**structure*with*a**unary*language. Also this class C properly extends the class*of*all finite-domain CSPs. ... Many natural decision problems can be formulated as constraint satisfaction problems*for**reducts**A**of*finitely bounded homogeneous*structures*. ... We study the computational complexity*of*CSP(*A*)*for*the class*of*all finite-signature*first*-*order**reducts**A**of**unary**structures*. ...##
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A dichotomy theorem for mono-unary algebras

2000
*
Fundamenta Mathematicae
*

We study the isomorphism relation

doi:10.4064/fm-163-1-25-37
fatcat:v5tk5vie6zbl3brtk2nm66lnaq
*of*invariant Borel classes*of*countable mono-*unary*algebras and prove*a*strong*dichotomy*theorem. ... The proof*of*the main theorem will actually establish*a*stronger result*for**first*-*order*theories*of*mono-*unary*algebras. Theorem 1.2. Let T be*a**first*-*order*theory*of*mono-*unary*algebras. ... Moreover, we obtain*a*stronger result about*first*-*order*theories*of*monounary algebras from the proofs*of*preceding sections. Theorem 5.3. Let T be*a**first*-*order*theory*of*mono-*unary*algebras. ...##
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Full Constraint Satisfaction Problems

2006
*
SIAM journal on computing (Print)
*

We prove

doi:10.1137/s0097539703427197
fatcat:d5v55mqylzarhn4n7dfvpe4che
*a*more general version, extending the*dichotomy*to so-called three-inclusive*structures*, i.e.,*structures*which contain, with any*unary*relation R, all*unary*relations R ′*for*subsets R ′ ⊆ R ... to W -full input*structures*are NP-complete or 'quasi-polynomial' (*of**order*n O(log n) ). ... . , n}, and φ is*a*quantifier-free*first**order*expression. ...##
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Generic Expression Hardness Results for Primitive Positive Formula Comparison
[chapter]

2011
*
Lecture Notes in Computer Science
*

We present two generic hardness results

doi:10.1007/978-3-642-22012-8_27
fatcat:sgyanmkrubbo3m3diy5r23sepq
*for*the studied problems, and discuss evidence that they are optimal and yield,*for*each*of*the problems,*a*complexity trichotomy. ... In particular, we study the complexity*of*these problems relative to finite relational*structures*. ... Matt Valeriote acknowledges the support*of*the Natural Sciences and Engineering Research Council*of*Canada. ...##
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Generic expression hardness results for primitive positive formula comparison

2013
*
Information and Computation
*

We present two generic hardness results

doi:10.1016/j.ic.2012.10.008
fatcat:g454vqddzjdofmgronukrsgcee
*for*the studied problems, and discuss evidence that they are optimal and yield,*for*each*of*the problems,*a*complexity trichotomy. ... In particular, we study the complexity*of*these problems relative to finite relational*structures*. ... Matt Valeriote acknowledges the support*of*the Natural Sciences and Engineering Research Council*of*Canada. ...##
###
Complexity of Homogeneous Co-Boolean Constraint Satisfaction Problems
[article]

2010
*
arXiv
*
pre-print

In this short note we show

arXiv:1011.4744v1
fatcat:5ptx3rduizfplfin4keg34iypm
*a**dichotomy*theorem*for*every finite domain D*of*CSP built upon graphs*of*homogeneous co-Boolean functions, i.e.,*unary*functions sharing the Boolean range 0, 1. ... It has been proved that*for*each CSP problem over*a*given set*of*relations there exists*a*corresponding CSP problem over graphs*of**unary*functions belonging to the same complexity class. ... Thus, CSPs over*unary*functions are as powerful as general CSP problems. Graphs*of**unary*functions give us*a*very*structural*template which is really convenient to work with. ...##
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Smooth approximations and CSPs over finitely bounded homogeneous structures

2022
*
Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science
*

We introduce the novel machinery

doi:10.1145/3531130.3533353
fatcat:5o3qocdjxjdizp6v7vf2xfoy7e
*of*smooth approximations, and apply it to confirm the CSP*dichotomy*conjecture*for**first*-*order**reducts**of*the random tournament, and to give new short proofs*of*the conjecture ...*For*all but the last*structure*, we moreover characterize*for*the*first*time those CSPs which are solvable by local consistency methods, again using the same machinery. ... In Section 5, we reprove the P/NP-complete*dichotomy**for**first*-*order**reducts**of*the*order**of*the rationals. ...##
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Constraint Satisfaction Problems for Reducts of Homogeneous Graphs

2019
*
SIAM journal on computing (Print)
*

We moreover show

doi:10.1137/16m1082974
fatcat:piol3ztb7ffjde4tkpewjxxfze
*a*similar complexity*dichotomy**for*all*structures*whose relations are*first*-*order*definable in*a*homogeneous graph whose reflexive closure is an equivalence relation. ... But*for*natural classes*of*such CSPs there is often the potential*for**structured*classifications, and this has proved to be the case*for**structures**first*-*order*definable over the*order*(Q, <)*of*the rationals ... That is,*for*us*a**reduct**of*∆ is as the classical definition*of**a**reduct*with the difference that we*first*allow*a**first*-*order*expansion*of*∆. ...##
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Universal algebra and hardness results for constraint satisfaction problems

2009
*
Theoretical Computer Science
*

These criteria also give non-expressibility results

doi:10.1016/j.tcs.2008.12.048
fatcat:qm6drmlez5cy3iqz6xv5cvjazi
*for*various restrictions*of*Datalog. Furthermore, we show that if CSP(Γ ) is not*first*-*order*definable then it is L-hard. ... We also infer*a*partial classification theorem*for*the complexity*of*CSP(Γ ) when the associated algebra*of*Γ is the full idempotent*reduct**of**a*preprimal algebra. ... We are also grateful to the organizers*of*the Dagstuhl Seminar on the Complexity*of*Constraints where much*of*this work was completed. ...##
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Dichotomy for Holant Problems of Boolean Domain
[chapter]

2011
*
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
*

We prove

doi:10.1137/1.9781611973082.132
dblp:conf/soda/CaiLX11
fatcat:ffvzsbxwifchtmakjek22qxm6u
*a*complexity*dichotomy*theorem*for*Holant * (F), where F is*a*set*of*constraint functions on Boolean variables and taking complex values. ... The main proof technique and indeed the formulation*of*the theorem use holographic algorithms and*reductions*. ... The symmetric*dichotomy*can be viewed as*a*special case*of*the*dichotomy*in this paper and on the other hand also servers as the starting point*for*our*reduction*. ...##
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Characterizations of monadic NIP

2021
*
Transactions of the American Mathematical Society. Series B
*

We give several characterizations

doi:10.1090/btran/94
fatcat:euv3sy6xkrcrpedn2sgvhk3zcu
*of*when*a*complete*first*-*order*theory T T is monadically NIP, i.e. when expansions*of*T T by arbitrary*unary*predicates do not have the independence property. ... As an application, we prove non-*structure*results*for*hereditary classes*of*finite substructures*of*non-monadically NIP models that eliminate quantifiers. ... Introduction It is well known that many*first*-*order*theories whose models are tame can become unwieldy after naming*a**unary*predicate. ...##
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Front Matter, Table of Contents, Preface, Conference Organization

2020
*
Annual Conference for Computer Science Logic
*

It further proves

doi:10.4230/lipics.csl.2020.0
dblp:conf/csl/X20
fatcat:xrswfikjybh4pjsl7h6fd2vt24
*dichotomies**for*two significant classes*of**structures*that are*first*-*order**reducts**of*(Z, <, +). ...*A**first*contribution in this line deals with "numeric CSPs": CSPs over*structures*whose relations are*first*-*order*definable in*a**reduct**of*the integers with the linear*ordering*relation. ...##
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Characterizations of monadic NIP
[article]

2021
*
arXiv
*
pre-print

We give several characterizations

arXiv:2104.12989v2
fatcat:vvtbbz7k4baqbj5z7i4e3qgw4y
*of*when*a*complete*first*-*order*theory T is monadically NIP, i.e. when expansions*of*T by arbitrary*unary*predicates do not have the independence property. ... As an application, we prove non-*structure*results*for*hereditary classes*of*finite substructures*of*non-monadically NIP models that eliminate quantifiers. ... Introduction It is well known that many*first**order*theories whose models are tame can become unwieldy after naming*a**unary*predicate. ...##
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Dichotomies for classes of homomorphism problems involving unary functions

2004
*
Theoretical Computer Science
*

We prove

doi:10.1016/j.tcs.2003.12.015
fatcat:tm3uyu76jfesda735fsrv4x4um
*a**dichotomy*result*for*the class*of*non-uniform constraint satisfaction problems where the template is*a*λ 2 -*structure*with the property that * . ... We establish*a**dichotomy*result*for*the class*of*non-uniform constraint satisfaction problems over the signature consisting*of*one*unary*function symbol by showing that every such problem is either complete ... Acknowledgement The second two authors would like to thank Eric Allender*for**a*useful discussion and guidance. ...##
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Tractable Triangles
[chapter]

2011
*
Lecture Notes in Computer Science
*

We study the computational complexity

doi:10.1007/978-3-642-23786-7_17
fatcat:c35lgphdprg7ddlgavzt5cxgiu
*of*binary valued constraint satisfaction problems (VCSP) given by allowing only certain types*of*costs in every triangle*of*variable-value assignments to three distinct ... We show that*for*several computational problems, including CSP, Max-CSP, finite-valued VCSP, and general-valued VCSP, the only non-trivial tractable classes are the well known maximum matching problem ... In the case*of*finite-valued and general-valued VCSP, we have obtained*a*complete*dichotomy*with respect to the equivalence classes which naturally follow from the total*order*on the valuation*structure*...
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