## The three dimensions of proofs (2006)

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Venue: | Annals of Pure and Applied Logic |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Guiraud06thethree,

author = {Yves Guiraud},

title = {The three dimensions of proofs},

journal = {Annals of Pure and Applied Logic},

year = {2006},

pages = {1}

}

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### Abstract

Abstract: In this document, we study a 3-polygraphic translation for the proofs of SKS, a formal system for classical propositional logic. We prove that the free 3-category generated by this 3-polygraph describes the proofs of classical propositional logic modulo structural bureaucracy. We give a 3-dimensional generalization of Penrose diagrams and use it to provide several pictures of a proof. We sketch how local transformations of proofs yield a non contrived example of 4-dimensional rewriting. Outline In the first section of this paper, we give a 2-dimensional translation of the formulas of system SKS, a formal system for propositional classical logic [Brünnler 2004] expressed in the style of the calculus of structures [Guglielmi 2004]. The idea consists in the replacement of formulas by circuit-like objects organized in a 2-polygraph [Burroni 1993]. This construction is formalized in theorem 1.4.16. We proceed to section 2, whose purpose is to translate the proofs of SKS into 3-dimensional objects that form a 3-polygraph. There we note that every inference rule can be interpreted as a directed 3-cell between two circuits. We prove theorem 2.4.3 stating that the 3-polygraph we have built can be equipped with a proof theory which is the same as the SKS one. Section 3 is where the 3-dimensional nature of

### Citations

938 |
Term Rewriting and All That
- Baader, Nipkow
- 1999
(Show Context)
Citation Context ...exactly the same way one builds rewriting paths from rewriting rules: each inference rule α generates a proof step from C[s(α) · σ] to C[t(α) · σ], for every context C and every substitution σ - see (=-=Baader and Nipkow 1998-=-) or (Guiraud 2004a) for more information about these notions. Then one can compose proof steps, provided the target of each one matches the source of the next one. Such a path going from a formula u ... |

615 | Linear logic - Girard - 1987 |

474 |
Categories for the working mathematician
- MacLane
- 1998
(Show Context)
Citation Context ... 2-category generated by the 2-polygraph Σ; the circuits are the 2-arrows of 〈Σ〉. If f is one of them, then s1(f) and t1(f) respectively denote the source and target 1-arrows of f. � One can find in (=-=MacLane 1998-=-) more information about 2-categories and (Chang and Lauda 2004) is a really comprehensive survey about the zoology of n-categories. Now, we sketch how SKS formulas are related to the 2-arrows of 〈Σ〉.... |

38 | Higher-dimensional word problems with applications to equational logic
- Burroni
- 1993
(Show Context)
Citation Context ... of arrows between parallel paths. They are pictured as directed surfaces, like in the following examples: y c a b ⇓ α x ⇓ β x y x x c x b x b c ⇓ γ x x c For a formal definition of 2-polygraph, see (=-=Burroni 1993-=-), where these objects were introduced, or the first pages of (Métayer 2003). There is another, more useful representation of 2-cells: Penrose diagrams, already used in (Lafont 2003; Guiraud 2004a, 20... |

26 | Towards an algebraic theory of Boolean circuits - Lafont |

22 |
Higher-dimensional categories: an illustrated guide book
- Cheng, Lauda
- 2004
(Show Context)
Citation Context ...s are the 2-arrows of 〈Σ〉. If f is one of them, then s1(f) and t1(f) respectively denote the source and target 1-arrows of f. � One can find in (MacLane 1998) more information about 2-categories and (=-=Chang and Lauda 2004-=-) is a really comprehensive survey about the zoology of n-categories. Now, we sketch how SKS formulas are related to the 2-arrows of 〈Σ〉. First, let us build a map from 2-arrows to (families) of formu... |

7 |
Présentations d’opérades et systèmes de réécriture
- Guiraud
(Show Context)
Citation Context ...ctorisation results. 1 The two dimensions of formulas This section is about a 2-dimensional translation of SKS formulas, heavily inspired by the one already known for terms, described and studied in (=-=Guiraud 2004-=-a, 2004b). 1.1 The formulas of system SKS System SKS is a deep inference formal system for proofs of propositional logic (Brünnler 2004). It is one of the formalisms expressed in the calculus of struc... |

7 | Resolutions by polygraphs, Theory and Applications of Categories 11 - Métayer - 2003 |

5 |
Deep Inference and Symmetry in Classical Proofs. Logos
- Brünnler
- 2004
(Show Context)
Citation Context ...red by the one already known for terms, described and studied in (Guiraud 2004a, 2004b). 1.1 The formulas of system SKS System SKS is a deep inference formal system for proofs of propositional logic (=-=Brünnler 2004-=-). It is one of the formalisms expressed in the calculus of structures style, an alternative to sequent calculus where inference rules can be applied at any depth inside formulas (Guglielmi 2005). Her... |

5 | MACLANE Categories for the working mathematician - SAUNDERS - 1998 |

3 | The problem of bureaucracy and identity of proofs from the perspective of deep inference - Guglielmi |

2 |
Deep inference and the calculus of structures. Project web page: Available at http://alessio. guglielmi.name/res/cos/index.html
- Guglielmi
(Show Context)
Citation Context ...logic (Brünnler 2004). It is one of the formalisms expressed in the calculus of structures style, an alternative to sequent calculus where inference rules can be applied at any depth inside formulas (=-=Guglielmi 2005-=-). Here a slightly alternative definition of SKS is used. Definition 1.1.1. Let us consider a countable set A. The set A of SKS atoms is the set of terms built on the variables in A and the signature ... |

2 | Linear logic and noncommutativity in the calculus of structures - STRASSBURGER - 2003 |

1 |
Categorification, ArXiv preprint
- Baez, Dolan
- 1998
(Show Context)
Citation Context ...e replacement of equalities by computations proving these equalities; this point of view is more in adequation with the computational flavour of the considered objects. Then, both (Burroni 1993) and (=-=Baez and Dolan 1998-=-) tell us that this calculus can be well-represented by objects in the dimension above. 2 The three dimensions of proofs 2.1 The proofs of system SKS We give here a slightly different definition than ... |

1 | Termination orders for 3-dimensional rewriting, submitted preprint - Guiraud - 2004 |