Non-well-founded trees in categories

Non-well-founded trees in categories

Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call M-types. We derive existence results for M-types in locally cartesian closed pretoposes with a natural numbers object, using their inter...

Tree-Automatic Well-Founded Trees

We investigate tree-automatic well-founded trees. Using Delhommé’s decomposition technique for tree-automatic structures, we show that the (ordinal) rank of a tree-automatic well-founded tree is strictly below ω. Moreover, we make a step towards proving that the ranks of tree-automatic well-founded partial orders are bounded by ω ω : we prove this bound for what we call upwards linear partial o...

Well- and Non-Well-Founded Fregean Extensions

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege’s inconsistent Axiom V. We build on Boolos’s interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of the principle of limitation of size. After providing a complete structural description of all...

Language Constructs for Non-Well-Founded Computation

Recursive functions defined on a coalgebraic datatype C may not converge if there are cycles in the input, that is, if the input object is not well-founded. Even so, there is often a useful solution. Unfortunately, current functional programming languages provide no support for specifying alternative solution methods. In this paper we give numerous examples in which it would be useful to do so:...

Non-Ground Stable and Well-Founded Semantics