Validating Back-links of FOLID Cyclic Pre-proofs

TL;DR

This paper introduces a new method for validating back-links in cyclic pre-proofs within first-order logic with inductive definitions, eliminating the need to process minimal cycles and improving efficiency.

Contribution

It proposes a novel approach that simplifies validation by focusing on root-bud paths, avoiding redundant cycle processing in cyclic pre-proof validation.

Findings

The new method avoids processing minimal cycles, reducing computational complexity.

Validation conditions are based on root-bud paths from strongly connected components.

The approach enhances efficiency in verifying soundness of cyclic pre-proofs.

Abstract

Cyclic pre-proofs can be represented as sets of finite tree derivations with back-links. In the frame of the first-order logic with inductive definitions, the nodes of the tree derivations are labelled by sequents and the back-links connect particular terminal nodes, referred to as buds, to other nodes labelled by a same sequent. However, only some back-links can constitute sound pre-proofs. Previously, it has been shown that special ordering and derivability conditions, defined along the minimal cycles of the digraph representing a particular normal form of the cyclic pre-proof, are sufficient for validating the back-links. In that approach, a same constraint could be checked several times when processing different minimal cycles, hence one may require additional recording mechanisms to avoid redundant computation in order to downgrade the time complexity to polynomial. We present a new approach that does not need to process minimal cycles. It based on a normal form that allows to define the validation conditions by taking into account only the root-bud paths from the non-singleton strongly connected components of its digraph.