arXiv subjects

Antoine Mottet

Publications and source records attributed to Antoine Mottet.

At least 19 recordsLinked to original sources

arXiv2025

Quantum Polymorphisms and the Complexity of Quantum Constraint Satisfaction

We introduce the concept of quantum polymorphisms to the complexity theory of quantum constraint satisfaction. Via this notion, we build an algebraic framework of reductions between quantum CSPs, and we establish a Galois connection between quantum polymorphism minions and quantum relational constructions. By leveraging a contextuality property of quantum polymorphisms, we fully characterise the existence of commutativity gadgets for relational structures, introduced by Ji as a method for achieving quantum soundness of classical CSP reductions. Prior to our work, only a partial classification was known for a subclass of Boolean languages and for non-Boolean languages meeting specific structural conditions [Culf--Mastel, FOCS'25]. As an application of our framework, we prove that the quantum CSPs parameterised by odd cycles and the quantum CSP expressing quantum satisfiability of Siggers clauses are undecidable.

quant-ph↗

arXiv2025

Edge-coloring problems with forbidden patterns and planted colors

Edge-coloring problems with forbidden patterns are decision problems asking to find an edge-coloring of the input graph which avoids a homomorphism from a fixed forbidden family of edge-colored graphs. In the precolored version of these problems, some of the edges of the input graph are already colored, and the goal is to find an extension of this coloring which omits a homomorphism from a forbidden graph. The existence of a complexity classification for such problems is an open question of Bienvenu, ten Cate, Lutz, and Wolter (ACM TODS'14) and we answer it for certain forbidden families consisting of odd cycles and cliques. The proof consists of two main stages. First, we combine the techniques from infinite constraint satisfaction and finite Ramsey theory in order to show that the edge-coloring problem is poly-time equivalent to its precolored version. After that, we show that the precolored version is poly-time equivalent to a finite constraint satisfaction problem, which has a P vs.\ NP-complete dichotomy by the seminal results of Bulatov (FOCS'17) and Zhuk (FOCS'17).

cs.CC↗

arXiv2025

Equations over Finite Monoids with Infinite Promises

Larrauri and \v{Z}ivn\'y [ICALP'25/ACM ToCL'24] recently established a complete complexity classification of the problem of solving a system of equations over a monoid $N$ assuming that a solution exists over a monoid $M$, where both monoids are finite and $M$ admits a homomorphism to $N$. Using the algebraic approach to promise constraint satisfaction problems, we extend their complexity classification in two directions: we obtain a complexity dichotomy in the case where arbitrary relations are added to the monoids, and we moreover allow the monoid $M$ to be finitely generated.

cs.CC↗

arXiv2025

Algebraic and algorithmic synergies between promise and infinite-domain CSPs

We establish a framework that allows us to transfer results between some constraint satisfaction problems with infinite templates and promise constraint satisfaction problems. On the one hand, we obtain new algebraic results for infinite-domain CSPs giving new criteria for NP-hardness. On the other hand, we show the existence of promise CSPs with finite templates that reduce naturally to tractable infinite-domain CSPs within the scope of the Bodirsky-Pinsker conjecture, but that are not finitely tractable, thereby showing a non-trivial connection between those two fields of research. In an important part of our proof, we also obtain uniform polynomial-time algorithms solving temporal constraint satisfaction problems.

cs.LO↗

arXiv2024

Collapsing the bounded width hierarchy for infinite-domain CSPs: when symmetries are enough

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities $n \geq 3$) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu, ten Cate, Lutz, and Wolter. In particular, the bounded width hierarchy collapses in those cases as well. Our results extend the scope of theorems of Barto and Kozik characterizing bounded width for finite structures, and show the applicability of infinite-domain CSPs to other fields.

cs.LO↗

arXiv2024

Complexity Classification Transfer for CSPs via Algebraic Products

We study the complexity of infinite-domain constraint satisfaction problems: our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure $\mathfrak A$ can be transferred to a classification of the CSPs of first-order expansions of another structure $\mathfrak B$. We exploit a product of structures (the algebraic product) that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSPs for first-order expansions of the $n$-fold algebraic power of $(\mathbb{Q};<)$. This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of $(\mathbb{Q};<)$ and explicit descriptions of the expressible relations in terms of syntactically restricted first-order formulas. By combining our classification result with general classification transfer techniques, we obtain surprisingly strong new classification results for highly relevant formalisms such as Allen's Interval Algebra, the $n$-dimensional Block Algebra, and the Cardinal Direction Calculus, even if higher-arity relations are allowed. Our results confirm the infinite-domain tractability conjecture for classes of structures that have been difficult to analyse with older methods. For the special case of structures with binary signatures, the results can be substantially strengthened and tightly connected to Ord-Horn formulas; this solves several longstanding open problems from the AI literature.

math.LO↗

arXiv2023

Finite Algebras with Hom-Sets of Polynomial Size

We provide an internal characterization of those finite algebras (i.e., algebraic structures) $\mathbf A$ such that the number of homomorphisms from any finite algebra $\mathbf X$ to $\mathbf A$ is bounded from above by a polynomial in the size of $\mathbf X$. Namely, an algebra $\mathbf A$ has this property if, and only if, no subalgebra of $\mathbf A$ has a nontrivial strongly abelian congruence. We also show that the property can be decided in polynomial time for algebras in finite signatures. Moreover, if $\mathbf A$ is such an algebra, the set of all homomorphisms from $\mathbf X$ to $\mathbf A$ can be computed in polynomial time given $\mathbf X$ as input. As an application of our results to the field of computational complexity, we characterize inherently tractable constraint satisfaction problems over fixed finite structures, i.e., those that are tractable and remain tractable after expanding the fixed structure by arbitrary relations or functions.

math.RA↗

arXiv2023

Symmetries of structures that fail to interpret something finite

We investigate structural implications arising from the condition that a given directed graph does not interpret, in the sense of primitive positive interpretation with parameters or orbits, every finite structure. Our results generalize several theorems from the literature and yield further algebraic invariance properties that must be satisfied in every such graph. Algebraic properties of this kind are tightly connected to the tractability of constraint satisfaction problems, and we obtain new such properties even for infinite countably categorical graphs. We balance these positive results by showing the existence of a countably categorical hypergraph that fails to interpret some finite structure, while still lacking some of the most essential algebraic invariance properties known to hold for finite structures.

cs.LO↗

arXiv2023

An order out of nowhere: a new algorithm for infinite-domain CSPs

We consider the problem of satisfiability of sets of constraints in a given set of finite uniform hypergraphs. While the problem under consideration is similar in nature to the problem of satisfiability of constraints in graphs, the classical complexity reduction to finite-domain CSPs that was used in the proof of the complexity dichotomy for such problems cannot be used as a black box in our case. We therefore introduce an algorithmic technique inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based on symmetries that depend on a linear order that is external to the structures under consideration. Our second main result is a P/NP-complete complexity dichotomy for such problems over many sets of uniform hypergraphs. The proof is based on the translation of the problem into the framework of constraint satisfaction problems (CSPs) over infinite uniform hypergraphs. Our result confirms in particular the Bodirsky-Pinsker conjecture for CSPs of first-order reducts of many homogeneous hypergraphs including the random hypergraphs and hypergraphs omitting a generalised clique. This forms a vast generalisation of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongr\'acz (ICALP'16) on graph satisfiability.

cs.LO↗

arXiv2021

Smooth approximations and CSPs over finitely bounded homogeneous structures

We develop the novel machinery of smooth approximations, and apply it to confirm the CSP dichotomy conjecture for first-order reducts of the random tournament, various homogeneous graphs including the random graph, and for expansions of the order of the rationals. Apart from obtaining these dichotomy results, we show how our new proof technique allows to unify and significantly simplify the previous results from the literature. For all but the last structure, we moreover characterize those CSPs which are solvable by local consistency methods, again using the same machinery.

cs.LO↗

arXiv2021

New Techniques for Universality in Unambiguous Register Automata

Register automata are finite automata equipped with a finite set of registers ranging over the domain of some relational structure like $(\mathbb N;=)$ or $(\mathbb Q;<)$. Register automata process words over the domain, and along a run of the automaton, the registers can store data from the input word for later comparisons. It is long known that the universality problem, i.e., the problem to decide whether a given register automaton accepts all words over the domain, is undecidable. Recently, we proved the problem to be decidable in 2-ExpSpace if the register automaton under study is over $(\mathbb N;=)$ and unambiguous, i.e., every input word has at most one accepting run; this result was shortly after improved to 2-ExpTime by Barloy and Clemente. In this paper, we go one step further and prove that the problem is in ExpSpace, and in PSpace if the number of registers is fixed. Our proof is based on new techniques that additionally allow us to show that the problem is in PSpace for single-register automata over $(\mathbb Q;<)$. As a third technical contribution we prove that the problem is decidable (in ExpSpace) for a more expressive model of unambiguous register automata, where the registers can take values nondeterministically, if defined over $(\mathbb N;=)$ and only one register is used.

cs.FL↗

arXiv2021

Constraint Satisfaction Problems over Finite Structures

We initiate a systematic study of the computational complexity of the Constraint Satisfaction Problem (CSP) over finite structures that may contain both relations and operations. We show the close connection between this problem and a natural algebraic question: which finite algebras admit only polynomially many homomorphisms into them? We give some sufficient and some necessary conditions for a finite algebra to have this property. In particular, we show that every finite equationally nontrivial algebra has this property which gives us, as a simple consequence, a complete complexity classification of CSPs over two-element structures, thus extending the classification for two-element relational structures by Schaefer (STOC'78). We also present examples of two-element structures that have bounded width but do not have relational width (2,3), thus demonstrating that, from a descriptive complexity perspective, allowing operations leads to a richer theory.

cs.LO↗

arXiv2021

ω-categorical structures avoiding height 1 identities

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 if the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to the dichotomy 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 within the range of the conjecture 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 doubly exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.

math.LO↗

arXiv2021

When symmetries are not enough: a hierarchy of hard Constraint Satisfaction Problems

We produce a class of $ω$-categorical structures with finite signature by applying a model-theoretic construction -- a refinement of the Hrushosvki-encoding -- to $ω$-categorical structures in a possibly infinite signature. We show that the encoded structures retain desirable algebraic properties of the original structures, but that the constraint satisfaction problems (CSPs) associated with these structures can be badly behaved in terms of computational complexity. This method allows us to systematically generate $ω$-categorical templates whose CSPs are complete for a variety of complexity classes of arbitrarily high complexity, and $ω$-categorical templates that show that membership in any given complexity class cannot be expressed by a set of identities on the polymorphisms. It moreover enables us to prove that recent results about the relevance of topology on polymorphism clones of $ω$-categorical structures also apply for CSP templates, i.e., structures in a finite language. Finally, we obtain a concrete algebraic criterion which could constitute a description of the delineation between tractability and NP-hardness in the dichotomy conjecture for first-order reducts of finitely bounded homogeneous structures.

cs.LO↗

arXiv2021

Cores over Ramsey structures

We prove that if an $ω$-categorical structure has an $ω$-categorical homogeneous Ramsey expansion, then so does its model-complete core.

math.LO↗

arXiv2020

A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

The logic MMSNP is a restricted fragment of existential second-order logic which allows to express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and Vardi who showed that every MMSNP sentence is computationally equivalent to a finite-domain constraint satisfaction problem (CSP); the involved probabilistic reductions were derandomized by Kun using explicit constructions of expander structures. We present a new proof of the reduction to finite-domain CSPs which does not rely on the results of Kun. This new proof allows us to obtain a stronger statement and to verify the more general Bodirsky-Pinsker dichotomy conjecture for CSPs in MMSNP. Our approach uses the fact that every MMSNP sentence describes a finite union of CSPs for countably infinite $ω$-categorical structures; moreover, by a recent result of Hubička and Nešetřil, these structures can be expanded to homogeneous structures with finite relational signature and the Ramsey property. This allows us to use the universal-algebraic approach to study the computational complexity of MMSNP.

cs.CC↗

arXiv2019

On the Containment Problem for Unambiguous Single-Register Automata with Guessing

Register automata extend classical finite automata with a finite set of registers that can store data from an infinite data domain for later equality comparisons with data from an input data word. While the registers in the original model of register automata, introduced in 1994 by Kaminski and Francez, can only store data occurring in the data word processed so far, we study here the more expressive class of register automata with guessing, where registers can nondeterministically take any value from the infinite data domain, even if this data does not occur in the input data word. It is well known that the containment problem, i.e., the problem of deciding for two given register automata with guessing A and B, whether the language L(A) accepted by A is contained in the language L(B) accepted by B, is undecidable, even if B only uses a single register. We prove that the problem is decidable if B is unambiguous and uses a single register.

cs.FL↗

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↗