Transducing Linear Decompositions of Tournaments

TL;DR

The paper proves that for tournaments of bounded linear clique-width, linear decompositions can be FO-transduced from the input, yielding linear clique-decompositions of bounded width. The approach leverages a semigroup view of linear decompositions, via bags and bag-types, together with Simon's Factorisation Forest Theorem to structure decompositions and obtain low cut-rank orderings. It then shows how to definably recover the full decomposition from the ordering and uses Backwards Translation to transfer CMSO definability to EMSO for these tournaments, establishing an EMSO-CMSO equivalence in this class. The results highlight a sharp contrast with the general graph setting by eliminating the need for CMSO (replacing it with FO) in producing decompositions for bounded-width tournaments, thereby enabling definability transfers with practical implications for descriptive complexity on this graph class.

Abstract

Bojańczyk, Pilipczuk, and Grohe [LICS '18] proved that for graphs of bounded linear clique-width, clique-decompositions of bounded width can be produced by a CMSO transduction. We show that in the case of tournaments, a first-order transduction suffices. This implies that the logics CMSO and existential MSO are equivalent over bounded linear clique-width tournaments.

Figures (4)

• Figure 1: Representation of a bag $B$. Input vertices and colours are drawn as empty nodes on the left and right respectively, while the internal vertices are filled. The colouring maps $\lambda_B$ and $\rho_B$ are drawn as dotted arrows, while edges in $E^{\text{int}}(B)$ and $E^{\partial}(B)$ are solid arrows.
• Figure 2: Product of two bags. The two bags are drawn on the left, with the output vertices of the first identified with the input of the second. The result is on the right.
• Figure 3: Two examples of linear decompositions.
• Figure 4: Simon's factorisation of a word $w_1 \dots w_{14}$ represented at a tree. At the leaves are letters $w_i$ in the chosen generating set $A$. Each internal node corresponds to a binary or idempotent operation. E.g. since $t$ is not binary, it must be idempotent: $\varphi(w_8w_9) = \varphi(w_{10}w_{11}) = \varphi(w_{12}w_{13}) = \varphi(w_{14}) = e$, and further this $e$ must satisfy $e \cdot e = e$.

Theorems & Definitions (54)

• Theorem 1.1: bojanczyk2016definabilityBojanczykP17
• Theorem 1.2: bojanczyk2018cliquewidth
• Theorem 1.3
• Theorem 1.4
• Definition 2.1: Linear NLC-width
• Definition 2.2: Linear rank-width
• Lemma 2.3
• proof
• Lemma 2.4
• proof