Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of unicity of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence which is shown to imply the desired unicity (even for infinite sequences of reduction) and further properties. We then carry over several criteria from the clas...

We extend the well-known results about the process of confluence for the Gauss hypergeometric differential equation to the case of general hypergeometric systems. We see that the process of confluence comes from the geometry of the set of regular elements of the Lie algebra of complex general linear group. As a consequence, we give a geometric and group-theoretic view on the process of confluen...

Abstract Convergent rewriting systems on algebraic structures give methods to solve decision problems, prove coherence results, and compute homological invariants. These are based higher-dimensional extensions of the critical branching lemma that proves local confluence from branchings. The analysis structures, such as groups or linear algebras, is complicated because underlying axioms. This ar...

In 2016, Hopkins, McConville, and Propp proved that labeled chip-firing on a line always leaves the chips in sorted order provided initial number of is even. We present novel proof this result. then apply our methods to resolve related conjectures concerning confluence systems.