The wonderland of reflections

Barto, Libor; Opršal, Jakub; Pinsker, Michael

doi:10.1007/s11856-017-1621-9

Cited by 97 publications

(189 citation statements)

References 38 publications

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“…Theorem 5.1 and Corollary 5.2 demonstrate that our reduction preserves almost all Maltsev conditions corresponding (or conjectured to be equivalent) to important algorithmic properties of decision CSPs. The linearity assumption in Theorem 5.1 is not limiting: Barto, Opršal and Pinsker recently improved on the algebraic approach to the CSP by showing that, if A is a core, the complexity of CSP(A) depends only on linear idempotent Maltsev conditions satisfied by A [BOP15]. Still, we were not able to extend our result to include all linear idempotent Maltsev conditions (in particular, nonbalanced identities in more than two variables).…”

Section: Discussionmentioning

confidence: 82%

A finer reduction of constraint problems to digraphs

Bulín¹,

Delić²,

Jackson³

et al. 2015

Logical Methods in Computer Science

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Abstract. It is well known that the constraint satisfaction problem over a general relational structure A is polynomial time equivalent to the constraint problem over some associated digraph. We present a variant of this construction and show that the corresponding constraint satisfaction problem is logspace equivalent to that over A. Moreover, we show that almost all of the commonly encountered polymorphism properties are held equivalently on the A and the constructed digraph. As a consequence, the Algebraic CSP dichotomy conjecture as well as the conjectures characterizing CSPs solvable in logspace and in nondeterministic logspace are equivalent to their restriction to digraphs.

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Section: Discussionmentioning

confidence: 82%

A finer reduction of constraint problems to digraphs

Bulín¹,

Delić²,

Jackson³

et al. 2015

Logical Methods in Computer Science

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“…It follows from several recent theorems in this area [6][7][8] 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…”

Section: Pseudo-loop Conditions Of Finite Widthmentioning

confidence: 99%

“…The satisfaction of the pseudo-Siggers identity in the polymorphism algebra of an ωcategorical structure A can be described by the existence of a pseudo-loop, roughly a loop modulo the automorphism group of A, in certain graphs invariant under finite powers of the algebra. The theorem mentioned above which derives the identity (6) has been obtained using this characterisation [7,8]. 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…”

Section: Pseudo-loop Conditions Of Finite Widthmentioning

confidence: 99%

“…In this case, smallness or regularity assumptions on the relational structure

double-struckA

which come from outside the field of universal algebra appear naturally. Such conditions might be, for example, model‐theoretic (for example,

ω

‐categoricity , homogeneity , finite boundedness , model‐completeness , or coreness ), combinatorial (for example, Ramsey‐type properties ), or group‐theoretic (for examnple, oligomorphicity or a certain orbit growth of the automorphism group of

double-struckA

) — see, for example, . One recent prime source of algebras satisfying such conditions are infinite domain Constraint Satisfaction Problems, where the understanding of the structure of polymorphism algebras has led to a great number of results of classifying the computational complexity for various classes of

ω

‐categorical relational structures (for example, in ).…”

Section: Introductionmentioning

confidence: 99%

“…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%

See 2 more Smart Citations

Pseudo‐loop conditions

Gillibert

Jonušas

Pinsker

2019

Bull. London Math. Soc.

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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.

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Finite Relation Algebras with Normal Representations

Bodirsky

2018

Relational and Algebraic Methods in Computer Science

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We study the computational complexity of the general network satisfaction problem for a finite relation algebra A with a normal representation B. If B contains a non-trivial equivalence relation with a finite number of equivalence classes, then the network satisfaction problem for A is NP-hard. As a second result, we prove hardness if B has domain size at least three and contains no non-trivial equivalence relations but a symmetric atom a with a forbidden triple (a, a, a), that is, a ≤ a • a. We illustrate how to apply our conditions on two small relation algebras.

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The wonderland of reflections

Cited by 97 publications

References 38 publications

A finer reduction of constraint problems to digraphs

A finer reduction of constraint problems to digraphs

Pseudo‐loop conditions

Finite Relation Algebras with Normal Representations

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