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Antoine Mottet

Publications and source records attributed to Antoine Mottet.

25 records · Page 2Linked to original sources
arXiv2019

Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems)

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable when the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to this conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of non-trivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of non-trivial height 1 identities differ for $ω$-categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.

cs.LO↗
arXiv2019

The Containment Problem for Unambiguous Register Automata

We investigate the complexity of the containment problem "Does $L(A)\subseteq L(B)$ hold?", where $B$ is an unambiguous register automaton and $A$ is an arbitrary register automaton. We prove that the problem is decidable and give upper bounds on the computational complexity in the general case, and when $B$ is restricted to have a fixed number of registers.

cs.FL↗
arXiv2018

A Dichotomy for First-Order Reducts of Unary Structures

Many natural decision problems can be formulated as constraint satisfaction problems for reducts $\mathbb{A}$ of finitely bounded homogeneous structures. This class of problems is a large generalisation of the class of CSPs over finite domains. Our first result is a general polynomial-time reduction from such infinite-domain CSPs to finite-domain CSPs. We use this reduction to obtain new powerful polynomial-time tractability conditions that can be expressed in terms of the topological polymorphism clone of $\mathbb{A}$. Moreover, we study the subclass $\mathcal{C}$ of CSPs for structures $\mathbb{A}$ that are reducts of a structure with a unary language. Also this class $\mathcal{C}$ properly extends the class of all finite-domain CSPs. We apply our new tractability conditions to prove the general tractability conjecture of Bodirsky and Pinsker for reducts of finitely bounded homogeneous structures for the class $\mathcal{C}$.

math.LO↗
arXiv2018

Classification transfer for qualitative reasoning problems

We study formalisms for temporal and spatial reasoning in the modern context of Constraint Satisfaction Problems (CSPs). We show how questions on the complexity of their subclasses can be solved using existing results via the powerful use of primitive positive (pp) interpretations and pp-homotopy. We demonstrate the methodology by giving a full complexity classification of all constraint languages that are first-order definable in Allen's Interval Algebra and contain the basic relations (s) and (f). In the case of the Rectangle Algebra we answer in the affirmative the old open question as to whether ORD-Horn is a maximally tractable subset among the (disjunctive, binary) relations. We then generalise our results for the Rectangle Algebra to the r-dimensional Block Algebra.

cs.LO↗
arXiv2016

Discrete Temporal Constraint Satisfaction Problems

A discrete temporal constraint satisfaction problem is a constraint satisfaction problem (CSP) whose constraint language consists of relations that are first-order definable over $(\Bbb Z,<)$. Our main result says that every distance CSP is in Ptime or NP-complete, unless it can be formulated as a finite domain CSP in which case the computational complexity is not known in general.

math.LO↗
arXiv2016

Distance Constraint Satisfaction Problems

We study the complexity of constraint satisfaction problems for templates $Γ$ that are first-order definable in $(\Bbb Z; succ)$, the integers with the successor relation. Assuming a widely believed conjecture from finite domain constraint satisfaction (we require the tractability conjecture by Bulatov, Jeavons and Krokhin in the special case of transitive finite templates), we provide a full classification for the case that Gamma is locally finite (i.e., the Gaifman graph of $Γ$ has finite degree). We show that one of the following is true: The structure Gamma is homomorphically equivalent to a structure with a d-modular maximum or minimum polymorphism and $\mathrm{CSP}(Γ)$ can be solved in polynomial time, or $Γ$ is homomorphically equivalent to a finite transitive structure, or $\mathrm{CSP}(Γ)$ is NP-complete.

cs.CC↗