Quasi Directed Jonsson Operations Imply Bounded Width (For fo-expansions of symmetric binary cores with free amalgamation)

TL;DR

The paper addresses infinite-domain CSPs arising from first-order expansions of finitely bounded homogeneous symmetric binary cores with free amalgamation, aiming to understand algebraic tractability via chains of quasi directed Jónsson operations. It develops a width-based framework, showing that if the template is preserved by such a chain, then the CSP enjoys relational width $$(2, \mathbb{L}_{\mathbb{A}})$$ and is solvable in polynomial time by local consistency methods; a corollary applies when a $(k+1)$-ary quasi near-unanimity polymorphism is present, implying bounded or strict width. The authors establish a dichotomy: implicationally uniform relational clones yield bounded width, while implicationally non-uniform clones cannot be preserved by chains of quasi directed Jónsson operations. This advances the infinite-domain algebraic tractability program by identifying a broad, natural class of infinite-domain CSPs for which tractability follows from a chain of quasi directed Jónsson operations and bounded width.

Abstract

Every CSP(B) for a finite structure B is either in P or it is NP-complete but the proofs of the finite-domain CSP dichotomy by Andrei Bulatov and Dimitryi Zhuk not only show the computational complexity separation but also confirm the algebraic tractability conjecture stating that tractability origins from a certain system of operations preserving B. The establishment of the dichotomy was in fact preceded by a number of similar results for stronger conditions of this type, i.e. for system of operations covering not necessarily all tractable finite-domain CSPs. A similar, infinite-domain algebraic tractability conjecture is known for first-order reducts of countably infinite finitely bounded homogeneous structures and is currently wide open. In particular, with an exception of a quasi near-unanimity operation there are no known systems of operations implying tractability in this regime. This paper changes the state-of-the-art and provides a proof that a chain of quasi directed Jonsson operations imply tractability and bounded width for a large and natural class of infinite structures.

Figures (11)

• Figure 1: A structure in $\textrm{Age}(\mathbb{A})$ needed in the proof of the base case of Claim \ref{['claim:bnbnoJonsson']}. If both $\mathbf{A}$ and $\mathbf{B}$ are anti reflexive, then $\mathbf{O}$ and $\mathbf{P}$ are $\mathbf{N}$. If $\mathbf{A}$ is $=$, then $\mathbf{O}$ and $\mathbf{P}$ are $\mathbf{B}$. If $\mathbf{B}$ is $=$, then $\mathbf{O}$ is $\mathbf{A}$ and $\mathbf{P}$ is $\mathbf{N}$.
• Figure 2: Since the age of $\mathbb{A}$ has free amalgamation over $\mathbf{N}$, it is straightforward to show that there are four vectors $s,t,u,v \in A^3$ and a single additional element $u'[2]$ described by the diagram above. By homogeneity of $\mathbb{A}$ we may assume that $u,v \in A^3$ are the vectors from the proof of Claim \ref{['claim:bnbnoJonsson']}. For edges not depicted in the picture the label either follows by the depicted ones and the semi-transitivity of equality or it is $\mathbf{N}$.
• Figure 3: Triples $s,t,u,v$ and $u'$ used for a final step in the proof of Lemma \ref{['lem:bnbnoJonsson']}. If it does not follows from the labels in the diagram and the semi-transitivity of equality, all omitted edges are labelled with $\mathbf{N}$.
• Figure 4: A substructure of $\mathbb{A}$ needed for the proof of the base case of induction in the proof of Claim \ref{['claim:beqbnoJonsson']}.
• Figure 5: Triples $u,v,s,t$ and $u'$ which play a role in the proof of the the base case of induction in Claim \ref{['claim:beqbnoJonsson']}.

Theorems & Definitions (60)

• Conjecture 1
• Definition 1
• Theorem 1
• Theorem 2
• Corollary 1
• proof
• Definition 2
• Example 1
• Definition 3
• Example 2