Cuts for circular proofs

4 pages•Published: July 28, 2014

Abstract

One of the authors introduced in [1] a calculus of
circular proofs for studying the computability arising from the
following categorical operations: finite products and coproducts,
initial algebras, final coalgebras. The calculus of
[1] is cut-free; yet, even if sound and complete for
provability, it lacks an important property for the semantics of
proofs, namely fullness w.r.t. the class of natural categorical models
called μ-bicomplete category in [2].

We fix, with this work, this problem by adding the cut rule to the
calculus. To this goal, we need to modifying the syntactical
constraints on the cycles of proofs so to ensure soundness of the
calculus and at same time local termination of cut-elimination. The
enhanced proof system fully represents arrows of the intended model, a
free μ-bicomplete category. We also describe a cut-elimination
procedure as a model of computation arising from the above mentioned
categorical operations. The procedure constructs a cut-free
proof-tree with infinite branches out of a finite circular proof with
cuts.

[1] Luigi Santocanale. A calculus of circular proofs and its categorical semantics. In Mogens Nielsen and Uffe Engberg, editors, FoSSaCS, volume 2303 of Lecture Notes in Computer Science, pages 357–371. Springer, 2002.

[2] Luigi Santocanale. μ-bicomplete categories and parity games. Theoretical Informatics and Applications, 36:195–227, September 2002.

Keyphrases: categorical proof theory, fixpoints, Inductive and Coinductive Types, initial and final (co)algebras

In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 72--75

Links:	https://easychair.org/publications/paper/fRJ
	https://doi.org/10.29007/54ps