## Order-enriched categorical models of the classical sequent calculus (2003)

Venue: | LECTURE AT INTERNATIONAL CENTRE FOR MATHEMATICAL SCIENCES, WORKSHOP ON PROOF THEORY AND ALGORITHMS |

Citations: | 22 - 2 self |

### BibTeX

@INPROCEEDINGS{Führmann03order-enrichedcategorical,

author = {Carsten Führmann and David Pym},

title = {Order-enriched categorical models of the classical sequent calculus},

booktitle = {LECTURE AT INTERNATIONAL CENTRE FOR MATHEMATICAL SCIENCES, WORKSHOP ON PROOF THEORY AND ALGORITHMS},

year = {2003},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.

### Citations

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(Show Context)
Citation Context ...tuitionistic logic [24, 18]. Such a choice imposes a restriction on the equational theory of proofs which is most readily apparent when one considers cut-elimination in the classical sequent calculus =-=[9]-=-. To see this, consider the following example, due to Lafont [25, 12], in which the cut-redex has two possible reducts: · Φ1 ⊢ A WR ⊢ A, B · Φ2 ⊢ A WL B ⊢ A � · Cut ⊢ A, A CR ⊢ A Φ1 or · ⊢ A Φ2 ⊢ A Th... |

391 |
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(Show Context)
Citation Context ... on the equational theory of proofs which is most readily apparent when one considers cut-elimination in the classical sequent calculus [9]. To see this, consider the following example, due to Lafont =-=[25, 12]-=-, in which the cut-redex has two possible reducts: · Φ1 ⊢ A WR ⊢ A, B · Φ2 ⊢ A WL B ⊢ A � · Cut ⊢ A, A CR ⊢ A Φ1 or · ⊢ A Φ2 ⊢ A The loss of the symmetry of the sequent calculus forced by λµν’s choice... |

355 |
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(Show Context)
Citation Context ...terpret the proof theory of propositional classical logic, makes the categories collapse to Boolean algebras [15, 14]. Classical natural deduction [19] may be represented as terms of the λµν-calculus =-=[17, 20]-=-. Models of λµν can be obtained in fibrations over a base category of structural maps in which each fibre is a model of intuitionistic natural deduction and in which dualizing negation is interpreted ... |

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(Show Context)
Citation Context ...ing the interpretation of a dualizing negation, to interpret the proof theory of propositional classical logic, makes the categories collapse to Boolean algebras [15, 14]. Classical natural deduction =-=[19]-=- may be represented as terms of the λµν-calculus [17, 20]. Models of λµν can be obtained in fibrations over a base category of structural maps in which each fibre is a model of intuitionistic natural ... |

220 |
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(Show Context)
Citation Context ...[22].) Whilst these solutions provides non-trivial categorical models, with computationally significant examples, it relies on a choice of ¬¬-translations of classical logic into intuitionistic logic =-=[24, 18]-=-. Such a choice imposes a restriction on the equational theory of proofs which is most readily apparent when one considers cut-elimination in the classical sequent calculus [9]. To see this, consider ... |

173 | Basic Proof Theory
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(Show Context)
Citation Context ...[22].) Whilst these solutions provides non-trivial categorical models, with computationally significant examples, it relies on a choice of ¬¬-translations of classical logic into intuitionistic logic =-=[24, 18]-=-. Such a choice imposes a restriction on the equational theory of proofs which is most readily apparent when one considers cut-elimination in the classical sequent calculus [9]. To see this, consider ... |

149 |
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(Show Context)
Citation Context ...they are instances of the same formula. However, we still have structures which do no represent valid proofs, for example These structures are eliminated by using a technique due to Danos and Regnier =-=[5]-=-. Definition 2. A (Danos-Regnier) switching σ is the choice of one of the hypotheses for each node of the following forms: [∧L], [∨R], [CL], [CR]. We shall say that the remaining nodes are unswitched.... |

144 |
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(Show Context)
Citation Context ...le succedents (i.e., formulæ on the right side of the proof gate ⊢). Thus, the minimal setting for a semantic study of Lafont’s example seems to be the multi-succedent intuitionistic sequent calculus =-=[6]-=- without implication. We implicitly cover this minimal setting, because our setting differs only in that we add negation orthogonally. While sequent proofs are our conceptual starting point, they cont... |

144 |
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(Show Context)
Citation Context ... categories, it is well-known that adding the interpretation of a dualizing negation, to interpret the proof theory of propositional classical logic, makes the categories collapse to Boolean algebras =-=[15, 14]-=-. Classical natural deduction [19] may be represented as terms of the λµν-calculus [17, 20]. Models of λµν can be obtained in fibrations over a base category of structural maps in which each fibre is ... |

118 | Weakly distributive categories
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(Show Context)
Citation Context ... (4) introduce some delicate conditions about the interaction between the monoids, the co-monoids, and the partial order. Our chosen models of the linear fragment are linearly distributive categories =-=[4]-=- (formerly called “weakly distributive categories”). The resulting order-enriched categories will be sound and complete with respect to cut-reduction in the classical sequent calculus. It is worth not... |

104 | Proof-nets: The parallel syntax for proof-theory
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- 1996
(Show Context)
Citation Context ...n of sequent proofs; one of the lessons there is that the theory is very smooth for the multiplicative connectives, but more problematic for the additives, which require “boxes” to indicate subproofs =-=[11]-=-. We therefore adopt a multiplicative presentation of classical logic. A sequent has the form Γ ⊢ ∆, where both the precedent Γ and the succedent ∆ are finite sequences of propositional logical formul... |

99 | Control categories and duality: on the categorical semantics of the lambda-mu calculus
- Selinger
- 2001
(Show Context)
Citation Context ... simply state that the monoidal operations at compound carriers A ⊕ B are defined pointwise in terms of the operations of the carriers A and B. Equation 23 can be 1 This definition is taken from from =-=[22]-=-, except that Selinger’s paper deals with the more general case of premonoidal categories and uses the terminology “has co-diagonals” instead of “has monoids”.ORDER-ENRICHED CATEGORICAL MODELS OF THE... |

73 | Categorical Structure of Continuation Passing Style
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(Show Context)
Citation Context ...nstead, we call a morphism f : A ✲ B copyable if the diagram A ⊕ A ✲ A f ⊕ f f ❄ ❄ B ⊕ B ✲ B commutes, and discardable if the diagram 0 ✲ A [] f ❄ ❄ 0 ✲ B commutes. This terminology was introduced by =-=[23]-=- and also used in [22, 8, 13]. (However, those publications deal with the semantics of functional programming languages and natural-deduction calculi, and the premonoidal categories they use differ si... |

70 | Classical Logic and Computation
- Urban
- 2000
(Show Context)
Citation Context ... on the equational theory of proofs which is most readily apparent when one considers cut-elimination in the classical sequent calculus [9]. To see this, consider the following example, due to Lafont =-=[25, 12]-=-, in which the cut-redex has two possible reducts: · Φ1 ⊢ A WR ⊢ A, B · Φ2 ⊢ A WL B ⊢ A � · Cut ⊢ A, A CR ⊢ A Φ1 or · ⊢ A Φ2 ⊢ A The loss of the symmetry of the sequent calculus forced by λµν’s choice... |

69 | Natural deduction and coherence for weakly distributive categories
- Blute, Cockett, et al.
- 1996
(Show Context)
Citation Context ...is is well-known and one of the reasons why sequent calculi are studied via proof nets. Proof nets where introduced by Girard for studying linear logic [10]. A different kind of proof net was used in =-=[2]-=- to build initial linearly distributive categories. The connection between sequent proofs and proof nets is fairly obvious and has been repeatedly formulated in sequentialization theorems which state ... |

45 | Strong normalisation of cut-elimination in classical logic - Urban, Bierman |

32 |
Proof nets for classical logic
- Robinson
(Show Context)
Citation Context ...connection between sequent proofs and proof nets is fairly obvious and has been repeatedly formulated in sequentialization theorems which state that every proof net can be turned into a sequent proof =-=[10, 21]-=- (the converse is almost trivial). Therefore, we shall switch from sequent proofs to proof nets early on in this article. The proof nets we use where introduced by Robinson [21] and possess rule nodes... |

21 |
On the semantics of classical disjunction
- Pym, Ritter
(Show Context)
Citation Context ...terpret the proof theory of propositional classical logic, makes the categories collapse to Boolean algebras [15, 14]. Classical natural deduction [19] may be represented as terms of the λµν-calculus =-=[17, 20]-=-. Models of λµν can be obtained in fibrations over a base category of structural maps in which each fibre is a model of intuitionistic natural deduction and in which dualizing negation is interpreted ... |

16 |
Proof Nets for Multiplicative and Additive Linear Logic
- BELLIN
- 1991
(Show Context)
Citation Context ...e finally adopted them as the basis of our semantics. 2.3. Proof nets. Proof nets were introduced by Girard for the study of linear logic [10]. They have been applied to various other logical systems =-=[2, 1]-=-. In this article, they play a key rôle in the semantic analysis of the Classical Sequent Calculus. The proof nets we use are the two-sided sequent-style nets for classical logic recently introduced b... |

12 | The structure of call-by-value
- Führmann
- 2000
(Show Context)
Citation Context ...hism f : A ✲ B copyable if the diagram A ⊕ A ✲ A f ⊕ f f ❄ ❄ B ⊕ B ✲ B commutes, and discardable if the diagram 0 ✲ A [] f ❄ ❄ 0 ✲ B commutes. This terminology was introduced by [23] and also used in =-=[22, 8, 13]-=-. (However, those publications deal with the semantics of functional programming languages and natural-deduction calculi, and the premonoidal categories they use differ significantly from linearly dis... |

11 |
On the geometry of interaction for classical logic
- Führmann, Pym
- 2004
(Show Context)
Citation Context ...) ⇐⇒ f ≤ g and f ′ ≤ g ′ In particular, we have Example 5. Rel × B is a classical category for every Boolean lattice B. A more substantial model, based on the Geometry of Interaction, is presented in =-=[7]-=-. (The details are beyond the scope of the present paper.) The use of the eight inequality laws will be explained precisely in the soundness and completeness proofs. However, we shall first explain th... |

10 | Proof theory in the abstract
- Hyland
- 2002
(Show Context)
Citation Context ... categories, it is well-known that adding the interpretation of a dualizing negation, to interpret the proof theory of propositional classical logic, makes the categories collapse to Boolean algebras =-=[15, 14]-=-. Classical natural deduction [19] may be represented as terms of the λµν-calculus [17, 20]. Models of λµν can be obtained in fibrations over a base category of structural maps in which each fibre is ... |

5 |
A semantic view of classical proofs
- Ong
- 1996
(Show Context)
Citation Context ...fibrations over a base category of structural maps in which each fibre is a model of intuitionistic natural deduction and in which dualizing negation is interpreted as certain maps between the fibres =-=[16, 20]-=-. (Alternative models are given by control categories and co-control categories [22].) Whilst these solutions provides non-trivial categorical models, with computationally significant examples, it rel... |

3 |
and Yoshihiko Kakutani. Axioms for recursion in call-by-value
- Hasegawa
- 2002
(Show Context)
Citation Context ...hism f : A ✲ B copyable if the diagram A ⊕ A ✲ A f ⊕ f f ❄ ❄ B ⊕ B ✲ B commutes, and discardable if the diagram 0 ✲ A [] f ❄ ❄ 0 ✲ B commutes. This terminology was introduced by [23] and also used in =-=[22, 8, 13]-=-. (However, those publications deal with the semantics of functional programming languages and natural-deduction calculi, and the premonoidal categories they use differ significantly from linearly dis... |