An Abstract Framework for the Analysis of Cyclic Derivations
Dominik Wehr
Abstract:
Cyclic derivations are finite graphs in which adjacent nodes are labeled by sequents according to locally sound derivation rules. These graphs serve as a finite representation of the infinite derivation trees obtained by unraveling them. To be considered proofs, such derivations need to satisfy an additional soundness condition called the global trace condition. Intuitively, any given cyclic derivation system can be separated into two different aspects: (i) its logical content, captured by the derivation rules and by how the global trace condition relates to the sequents being derived, and (ii) its cyclic content, which is concerned with the combinatorial aspects of the global trace condition and which kinds of graphs can serve as derivations. When comparing cyclic derivation systems, even ones with wildly different logical content, one often observes that their cyclic content is very similar. A general theory of the cyclic content of cyclic derivation systems could prove useful when designing, studying and comparing concrete cyclic derivation systems.
In this thesis, we formalize the cyclic content described above in an abstract manner easily applicable to most cyclic derivation systems. We do this by abstracting infinite branches through derivations to infinite sequences of morphisms through trace categories, categories which are equipped with a condition on such sequences of morphisms which is invariant under certain transformations. Cyclic derivations can then be represented as functors mapping cyclic graphs into trace categories. Notably, this representation discards all of the logical content of a cyclic derivation. Indeed, a handful of different trace categories are sufficient to capture the cyclic content of all cyclic derivation systems in the literature we have examined so far.
We demonstrate the viability of these abstract notions of cyclic content in two ways: First, we use the framework to derive uniform proofs of properties of cyclic derivation systems. Among them the regularization theorem for infinite proofs in finite derivation systems and a well-known application of Ramseyâ€™s theorem which reduces proof checking to the examination of certain periodic paths. Second, we demonstrate that various cyclic derivation systems from the literature fit within our framework, allowing the uniform proofs to be applied to them.