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We develop a mathematical abstraction for uniformly exploiting symmetries in programming and reasoning based on the mathematical proof principle `without-loss-of-generality' (w.l.o.g.). Given: group actions that describe symmetries of the input and the output; and a canoniser, which captures how arbitrary inputs get sent to canonical ones, our construction extends any function from the canonical inputs to the whole domain. We characterise the conditions on the canonisers which guarantee that the construction gives the unique equivariant (i.e. symmetry-preserving) extension. Furthermore, we adapt the construction to a dependently-typed setting, where it captures traditional symmetric w.l.o.g. arguments via the Curry-Howard correspondence. Finally, we extend the construction to algebraic data types, where the symmetries of the constructors both uniquely determine the symmetries of the data type and pave the way for efficient execution.

The Arthur-Nimue-Merlin degrees are a generalization of the Turing degrees introduced by Kihara as a tangible description of the partially ordered set of Lawvere-Tierney topologies on the effective topos (equivalently, subtoposes of the effective topos). They are defined in terms of a three-player game that introduces both angelic and demonic non-determinism into oracle queries. We construct an order embedding of the Turing degrees with their order reversed into the Arthur-Nimue-Merlin degrees, whose image we call the "co-Turing degrees"; we then study the order relationship of these co-Turing degrees with the (naturally embedded) Turing degrees within the Arthur-Nimue-Merlin degrees. This is a short version of arXiv:2603.19946, focused on the main results.

Monadically dependent graph classes are those in which first-order logic cannot encode all graphs. They include all nowhere dense classes and all classes of bounded twin-width. We show that first-order model checking is fixed-parameter tractable on every monadically dependent graph class. Together with the known hardness result for hereditary classes that are not monadically dependent, this gives a complete characterization of the hereditary graph classes with fixed-parameter tractable first-order model checking. This resolves a central open problem in algorithmic model theory. Our proof combines ideas from structural graph theory (graph decompositions), finite model theory (locality of first-order logic), and model theory (counting types). In particular, we show that in monadically dependent classes, tuples of fixed length have only $|U|^{O(1)}$ types of fixed quantifier rank over any set of parameters $U$.

A graph class $\mathcal{C}$ is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from $\mathcal{C}$ using a fixed formula of first-order logic. We prove that every monadically dependent graph class has almost bounded merge-width, thereby confirming a conjecture of the first and the last author [STOC '25]. Our proof is algorithmic and the witnessing decompositions in the form of merge sequences can be computed in nearly-cubic time. This is a step towards the proof of fixed-parameter tractability of model checking first-order logic on monadically dependent graph classes, which is achieved in a companion paper.