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

Manuel Bodirsky, Antoine Mottet
2017 arXiv   pre-print
Moreover, we study the subclass C 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.  ... 
arXiv:1601.04520v5 fatcat:psjkbouz6fhzdgprwfthjinje4

A dichotomy theorem for mono-unary algebras

Su Gao
2000 Fundamenta Mathematicae  
We study the isomorphism relation 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.  ... 
doi:10.4064/fm-163-1-25-37 fatcat:v5tk5vie6zbl3brtk2nm66lnaq

Full Constraint Satisfaction Problems

Tomás Feder, Pavol Hell
2006 SIAM journal on computing (Print)  
We prove 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.  ... 
doi:10.1137/s0097539703427197 fatcat:d5v55mqylzarhn4n7dfvpe4che

Generic Expression Hardness Results for Primitive Positive Formula Comparison [chapter]

Simone Bova, Hubie Chen, Matthew Valeriote
2011 Lecture Notes in Computer Science  
We present two generic hardness results 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.  ... 
doi:10.1007/978-3-642-22012-8_27 fatcat:sgyanmkrubbo3m3diy5r23sepq

Generic expression hardness results for primitive positive formula comparison

Simone Bova, Hubie Chen, Matthew Valeriote
2013 Information and Computation  
We present two generic hardness results 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.  ... 
doi:10.1016/j.ic.2012.10.008 fatcat:g454vqddzjdofmgronukrsgcee

Complexity of Homogeneous Co-Boolean Constraint Satisfaction Problems [article]

Florian Richoux
2010 arXiv   pre-print
In this short note we show 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.  ... 
arXiv:1011.4744v1 fatcat:5ptx3rduizfplfin4keg34iypm

Smooth approximations and CSPs over finitely bounded homogeneous structures

Antoine Mottet, Michael Pinsker
2022 Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science  
We introduce the novel machinery 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.  ... 
doi:10.1145/3531130.3533353 fatcat:5o3qocdjxjdizp6v7vf2xfoy7e

Constraint Satisfaction Problems for Reducts of Homogeneous Graphs

Manuel Bodirsky, Barnaby Martin, Michael Pinsker, András Pongrácz
2019 SIAM journal on computing (Print)  
We moreover show 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 ∆.  ... 
doi:10.1137/16m1082974 fatcat:piol3ztb7ffjde4tkpewjxxfze

Universal algebra and hardness results for constraint satisfaction problems

Benoît Larose, Pascal Tesson
2009 Theoretical Computer Science  
These criteria also give non-expressibility results 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.  ... 
doi:10.1016/j.tcs.2008.12.048 fatcat:qm6drmlez5cy3iqz6xv5cvjazi

Dichotomy for Holant Problems of Boolean Domain [chapter]

Jin-Yi Cai, Pinyan Lu, Mingji Xia
2011 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms  
We prove 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.  ... 
doi:10.1137/1.9781611973082.132 dblp:conf/soda/CaiLX11 fatcat:ffvzsbxwifchtmakjek22qxm6u

Characterizations of monadic NIP

Samuel Braunfeld, Michael Laskowski
2021 Transactions of the American Mathematical Society. Series B  
We give several characterizations 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.  ... 
doi:10.1090/btran/94 fatcat:euv3sy6xkrcrpedn2sgvhk3zcu

Front Matter, Table of Contents, Preface, Conference Organization

Maribel Fernández, Anca Muscholl, Michael Wagner
2020 Annual Conference for Computer Science Logic  
It further proves 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.  ... 
doi:10.4230/lipics.csl.2020.0 dblp:conf/csl/X20 fatcat:xrswfikjybh4pjsl7h6fd2vt24

Characterizations of monadic NIP [article]

Samuel Braunfeld, Michael C. Laskowski
2021 arXiv   pre-print
We give several characterizations 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.  ... 
arXiv:2104.12989v2 fatcat:vvtbbz7k4baqbj5z7i4e3qgw4y

Dichotomies for classes of homomorphism problems involving unary functions

Tomás Feder, Florent Madelaine, Iain A. Stewart
2004 Theoretical Computer Science  
We prove 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.  ... 
doi:10.1016/j.tcs.2003.12.015 fatcat:tm3uyu76jfesda735fsrv4x4um

Tractable Triangles [chapter]

Martin C. Cooper, Stanislav Živný
2011 Lecture Notes in Computer Science  
We study the computational complexity 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  ... 
doi:10.1007/978-3-642-23786-7_17 fatcat:c35lgphdprg7ddlgavzt5cxgiu
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