The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems

Barto, Libor; Pinsker, Michael

doi:10.1145/2933575.2934544

Cited by 42 publications

(103 citation statements)

References 33 publications

Supporting

Mentioning

103

Contrasting

Order By: Relevance

“…It follows from several recent theorems in this area that if the polymorphism algebra of an

ω

‐categorical structure satisfies some non‐trivial finite set of h1 identities locally (that is, on every finite set), then it satisfies the 6‐ary pseudo‐Siggers identity

\begin{matrix} true \\ right u \circ s false(x, y, x, z, y, z false) \approx v \circ s false(y, x, z, x, z, y false); \end{matrix}

here,

u

and

v

are unary functional symbols, and

s

is 6‐ary. For a certain subclass of

ω

‐categorical structures, the converse implication holds as well , and it has been conjectured that satisfaction of this identity is the delineation of tractability and hardness of a large class of infinite‐domain Constraint Satisfaction Problems .…”

Section: Introductionmentioning

confidence: 99%

“…The satisfaction of the pseudo‐Siggers identity in the polymorphism algebra of an

ω

‐categorical structure

double-struckA

can be described by the existence of a pseudo‐loop , roughly a loop modulo the automorphism group of

double-struckA

, in certain graphs invariant under finite powers of the algebra. The theorem mentioned above which derives the identity has been obtained using this characterisation . Inspired by this fact, we consider the pseudo‐variant of the loop conditions of finite width: that is, we study sets of identities of the form

u_{1} \circ f false(x_{1, 1}, \dots, x_{1, n} false) \approx \dots \approx u_{m} \circ f false(x_{m, 1}, \dots, x_{m, n} false),

where

u_{1}, \dots, u_{m}

are unary function symbols, and refer to them as pseudo‐loop conditions .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Pseudo‐loop conditions

Gillibert

Jonušas

Pinsker

2019

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

About a decade ago, it was realised that the satisfaction of a given identity (or equation) of the form ffalse(x1,…,xnfalse)≈ffalse(y1,…,ynfalse) in an algebra is equivalent to the algebra forcing a loop into any graph on which it acts and which contains a certain finite subgraph associated with the identity. Such identities have since also been called loop conditions, and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a Siggers identity s(x,y,z,x)≈s(y,x,y,z) in any arbitrary non‐trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form ffalse(x1,1,…,x1,nfalse)≈⋯≈ffalse(xm,1,…,xm,nfalse), which we call loop conditions of width m. We show that their satisfaction in an algebra is equivalent to any action of the algebra on a certain type of relation forcing a constant tuple into the relation. Proving that for each fixed width m there is a weakest loop condition (that is, one entailed by all others), we obtain a new and short proof of the recent celebrated result stating that there exists a concrete loop condition of width 3 which is entailed in any non‐trivial idempotent, possibly infinite, algebra. The framework of classical (width 2) loop conditions is insufficient for such proof. We then consider pseudo‐loop conditions of finite width, a generalisation suitable for non‐idempotent algebras; they are of the form u1∘ffalse(x1,1,…,x1,nfalse)≈⋯≈um∘ffalse(xm,1,…,xm,nfalse), and of central importance for the structure of algebras associated with ω‐categorical structures. We show that for the latter, satisfaction of a pseudo‐loop condition is characterised by pseudo‐loops, that is, loops modulo the action of the automorphism group, and that a weakest pseudo‐loop condition exists (for ω‐categorical cores). This way we obtain a new and short proof of the theorem that the satisfaction of any non‐trivial identities of height 1 in such algebras implies the satisfaction of a fixed single identity.

show abstract

“…It follows from several recent theorems in this area that if the polymorphism algebra of an

ω

‐categorical structure satisfies some non‐trivial finite set of h1 identities locally (that is, on every finite set), then it satisfies the 6‐ary pseudo‐Siggers identity

\begin{matrix} true \\ right u \circ s false(x, y, x, z, y, z false) \approx v \circ s false(y, x, z, x, z, y false); \end{matrix}

here,

u

and

v

are unary functional symbols, and

s

is 6‐ary. For a certain subclass of

ω

Section: Introductionmentioning

confidence: 99%

“…The satisfaction of the pseudo‐Siggers identity in the polymorphism algebra of an

ω

‐categorical structure

double-struckA

can be described by the existence of a pseudo‐loop , roughly a loop modulo the automorphism group of

double-struckA

u_{1} \circ f false(x_{1, 1}, \dots, x_{1, n} false) \approx \dots \approx u_{m} \circ f false(x_{m, 1}, \dots, x_{m, n} false),

where

u_{1}, \dots, u_{m}

are unary function symbols, and refer to them as pseudo‐loop conditions .…”

Section: Introductionmentioning

confidence: 99%

Pseudo‐loop conditions

Gillibert

Jonušas

Pinsker

2019

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the ω-categorical case, it has been shown recently to depend only on Pol(Γ), viewed as a topological clone [BP15b]. Moreover, up to now no two ω-categorical structures with abstractly isomorphic polymorphism clones but CSPs of different (up to polynomial-time reductions) complexity are known, and it has been shown recently that at least some aspects of the CSP of an ω-categorical structure are captured by the algebraic structure of Pol(Γ) [BP16].…”

Section: Reconstruction Of ω-Categorical Structures From Their Polymomentioning

confidence: 99%

Reconstructing the topology of clones

Bodirsky

Pinsker²,

Pongrácz

2017

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Function clones are sets of functions on a fixed domain that are closed under composition and contain the projections. They carry a natural algebraic structure, provided by the laws of composition which hold in them, as well as a natural topological structure, provided by the topology of pointwise convergence, under which composition of functions becomes continuous. Inspired by recent results indicating the importance of the topological ego of function clones even for originally algebraic problems, we study questions of the following type: In which situations does the algebraic structure of a function clone determine its topological structure? We pay particular attention to function clones which contain an oligomorphic permutation group, and discuss applications of this situation in model theory and theoretical computer science.

show abstract

“…However, restricted classes of nonidempotent infinite algebras can possess weakest (at least in some sense) nontrivial conditions. Of particular importance for the infinite domain CSPs is the class of closed oligomorphic algebras and its subclasses (see, for example, for background). The following question is of interest in this context.…”

Section: Open Problemsmentioning

confidence: 99%

“…We remark that the original problem, [, Question 1.3] on the existence of a continuous projective homomorphism for closed function clones remains open. On the other hand, it was proved by Barto and Pinsker that the topological structure is indeed irrelevant in the Bodirsky–Pinsker dichotomy conjecture. An intermediate problem, a ‘loop lemma for near unanimity’ (NU), which they considered (under a finiteness assumption) while working on the result, turned out to have positive answer that requires no additional algebraic or topological assumptions.…”

Section: Introductionmentioning

confidence: 99%