## Abstract

We study the decidability of k-provability in PA —the relation ‘being provable in PA with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of PA is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.

Original language | English |
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Pages (from-to) | 477-516 |

Journal | Logica Universalis |

Volume | 15 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2021 |

## Keywords

- Decidability
- k-provability
- Peano arithmetic
- Proof-skeleton problem