Flipping and Forking
Wojciech Przybyszewski, Szymon Toruńczyk
TL;DR
We show that flips and flip independence provide a combinatorial bridge between model-theoretic forking independence and structural tameness in monadically stable classes, extending the graph-focused toolkit to arbitrary relational structures. By defining definable flips for relational structures and establishing that forking over models coincides with $r$-flip independence for all radii, the work unifies stability theory with finite-model theory and algorithmic model checking. It yields a chain of equivalences linking nowhere density, monadic stability, and weak sparsity, and introduces flip-flatness and a separation game as practical tools with implications for model-checking algorithms. The results generalize Ivanov’s forking characterizations to higher-arity relational structures and provide refined insights in nowhere dense contexts, with potential algorithmic relevance for FO model checking and sparse-dense decompositions.
Abstract
Monadic stability and the more general monadic dependence (or NIP) are tameness conditions for classes of logical structures, studied in the 80's in Shelah's classification program in model theory. They recently emerged in algorithmic and structural graph theory and finite model theory as central notions in relation with the model checking problem for first-order logic: the problem was shown to be fixed-parameter tractable for inputs which come from a fixed class of graphs which is monadically stable, and is conjectured to be tractable in all monadically dependent classes. Several combinatorial characterizations of such graph classes turned out to be essential in their algorithmic treatment; they are all based on the fundamental operation of "flipping" a graph. We introduce the notions of $\textit{flips}$ and $\textit{flip independence}$ in arbitrary relational structures. We lift prior combinatorial characterizations of monadically stable graph classes to monadically stable classes of relational structures. We show the equivalence of flip independence with $\textit{forking independence}$ (over models) -- a logical notion of paramount importance in stability theory -- in monadically stable structures, shedding new light on the relevance of flips, also characterizing forking independence (over models) combinatorially. We give more precise descriptions of forking independence in the case of monadically stable graphs, and relational structures with a nowhere dense Gaifman graph.