We present Algeo, a functional logic programming language based on the theory of infinite dimensional modules. Algeo is reversible in the sense that every function has a generalised inverse, an adjoint, which can be thought of as an inverse execution of the forward function. In particular, when the given function is invertible, the adjoint is guaranteed to coincide with the inverse. Algeo generalises “ordinary” forward-backward deterministic reversible programming by permitting relational and probabilistic features. This allows functions to be defined in a multitude of ways, which we summarise by the motto that “all definitions are extensional characterisations; all extensional characterisations are definitions.” We describe the syntax, type system, and the axiomatic semantics of Algeo, and showcase novel features of the language through examples.