## On categorical models of classical logic and the geometry of interaction (2005)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Führmann05oncategorical,

author = {Carsten Führmann and David Pym},

title = { On categorical models of classical logic and the geometry of interaction },

year = {2005}

}

### OpenURL

### Abstract

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarized briefly herein, we have provided a class of models called classical categories which is sound and complete and avoids this collapse by interpreting cut-reduction by a poset-enrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product. In this article, which is self-contained, we present an improved axiomatization of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negation-free models called Dummett categories. Examples include, besides the classical categories above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from hom-semilattices which have a straightforward proof-theoretic definition. Moreover, we show that the Geometry-of-Interaction construction can be extended from multiplicative linear logic to classical logic, by applying it to obtain a classical

### Citations

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Citation Context ...scapes from this denotational collapse: first, we might simply abandon classical logic and adopt, for example, intuitionistic logic or linear logic instead. As explained in Gentzen’s seminal article (=-=Gentzen 1934-=-), intuitionistic logic can an be obtained by restricting the classical sequent calculus is such a way that the succedent ∆ contains at most one formula. As is widely known, intuitionistic logic can b... |

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Citation Context ... classical provability. Second, insisting to keep classical logic, we might move to “classical natural deduction” systems (Prawitz 1965), where proofs may be represented as terms of the λµν-calculus (=-=Parigot 1992-=-, Pym & Ritter 2001). But such systems do not admit all cut-reductions: as it turns out, the call-by-name version of λµν admits only the reduction to Φ2, while the call-by-value version admits only th... |

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Citation Context ...to a different choice of ¬¬-translations (a.k.a. “continuation-passing-style transforms” in programming-language jargon) of classical logic into intuitionistic logic (Troelstra & Schwichtenberg 1996, =-=Plotkin 1975-=-). 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... |

166 | Traced monoidal categories - Joyal, Street, et al. - 1996 |

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Citation Context ... to Φ1. Each version corresponds to a different choice of ¬¬-translations (a.k.a. “continuation-passing-style transforms” in programming-language jargon) of classical logic into intuitionistic logic (=-=Troelstra & Schwichtenberg 1996-=-, Plotkin 1975). 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 ... |

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Citation Context ...ion-free term calculus for Dummett categories. Such a calculus might be based on the circuit expressions in (Blute et al. 1996), on term calculi for the classical sequent calculus along the lines of (=-=Curien & Herbelin 2000-=-, Wadler 2003). Expressions in such calculi can be seen as functional programs with an unspecified evaluation strategy, while MIX introduces an element of parallelism. Lafont’s example corresponds to ... |

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Citation Context ...ting categories are models of the negation-free fragment of the classical sequent calculus. We call them Dummett categories (inspired by Dummett’s extensive discussion, in “Elements of Intuitionism” (=-=Dummett 1977-=-), of multi-succedent intuitionistic sequent calculi). Second, we introduce classical categories as Dummett categories with the property of having negation in the sense of Cockett and Seely. We then e... |

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Citation Context ...ightforward. Not all proof structures are the images of derivations; those that are called proof nets. (When a graph is a proof net can also be characterized by the switching criterion introduced in (=-=Danos & Regnier 1989-=-), which requires that certain subgraphs of the proof structure be connected and acyclic.) Robinson’s nets, with minor notational changes, were used in (Führmann & Pym 2004b). However, in this article... |

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Citation Context ...posed to C ⌊Φ⌋ = C ⌊Ψ⌋), where C ⌊Φ⌋ and C ⌊Ψ⌋ are the morphisms denoted by Φ and Ψ in the classical category C. Classical categories are a special case of symmetric linearly distributive categories (=-=Cockett & Seely 1997-=-b): they have symmetric monoidal products ⊗ and ⊕ for modelling conjunction and disjunction, respectively. To model contraction and 3sweakening on the right, every object A is endowed with a symmetric... |

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Citation Context ...and δ is the associativity map. 18 λ⊕sRemark 2.5. Alternatively, one could define a compact closed category to be a symmetric monoidal category with, for every object A, an assigned left adjoint A ⊥ (=-=Kelly & Laplaza 1980-=-). The degenerate versions of the two equational laws for γ and τ are the triangular identities of that adjunction. 2.4 Categorical semantics of MLL In this section, we recall the semantics of MLL in ... |

101 |
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Citation Context ...o f is a parametrized copointed homomorphism if and only if fΓ∆ = 0. 4 Geometry of interaction in the presence of weakening and contraction The Geometry of Interaction (GoI) was introduced by Girard (=-=Girard 1989-=-, Girard 1990, Girard 1995) in the late 1980s in the context of modelling the dynamics of cut elimination in (classical) linear logic (Girard 1987). The aim was to capture the essential structure of t... |

83 | Control categories and duality: on the categorical semantics of the lambda-mu calculus
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Citation Context ...eduction and in which dualizing negation is interpreted as certain maps between the fibres (Ong 1996, Pym & Ritter 2001). Alternative models are given by control categories and co-control categories (=-=Selinger 2001-=-). In our companion paper (Führmann & Pym 2004b), we presented a solution that, unlike classical natural deduction, models all cut-reductions: we introduced a kind of poset-enriched category called cl... |

75 |
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Citation Context ...al logic with respect to provable sequents, and we do not wish to depart from classical provability. Second, insisting to keep classical logic, we might move to “classical natural deduction” systems (=-=Prawitz 1965-=-), where proofs may be represented as terms of the λµν-calculus (Parigot 1992, Pym & Ritter 2001). But such systems do not admit all cut-reductions: as it turns out, the call-by-name version of λµν ad... |

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Citation Context ... semantically inessential aspects of the syntax. A categorical approach to GoI, based on domain theory and arising from the construction of a categorical model of linear logic, has been described in (=-=Abramsky & Jagadeesan 1994-=-). Some years later, Abramsky et al. presented what can be seen as a general form of the Geometry of Interaction: a compact closed category is constructed from a traced symmetric monoidal category (Ab... |

67 |
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Citation Context ...help here. Other starting points for a term-language for classical categories might be Filinski’s symmetric lambda calculus (Filinski 1989) and the symmetric lambda calculus by Barbanera and Berardi (=-=Barbanera & Berardi 1996-=-). Extending Dummett categories to first-order logic Finally, we should like to mention the possibility of extending our categorical semantics to firstorder classical logic. This can be achieved using... |

67 | Natural Deduction and Coherence for Weakly Distributive Categories
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Citation Context ...us by constructing the free classical category from proof nets (Theorem 3.32). (This extends the construction of the free symmetric linearly distributive category from MLL proof nets in the sense of (=-=Blute et al. 1996-=-).) From a logical point of view, the result means that classical categories are sound and complete (in order-theoretic sense explained above) with respect to a certain super-relation of cut-reduction... |

66 | Retracing Some Paths in Process Algebra
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Citation Context ...94). Some years later, Abramsky et al. presented what can be seen as a general form of the Geometry of Interaction: a compact closed category is constructed from a traced symmetric monoidal category (=-=Abramsky 1996-=-, Abramsky, Haghverdi & Scott 2001). This construction also appeared in (Joyal, Street & Verity 1996). Many of the ideas contributing to these developments have also been described by Hyland. Beginnin... |

43 | Geometry of interaction and linear combinatory algebras - Abramsky, Haghverdi, et al. |

38 |
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Citation Context ...ional axioms of Σ. We call a derivation over Σ positive if all of its formulæ are positive. 2.2 Proof nets The essence of a sequent proof can be captured by a proof net, an idea introduced by Girard (=-=Girard 1987-=-) (or “net” for short). In this article, we shall need proof nets to describe equalities between proofs. The nets we use are, essentially, those from (Blute et al. 1996), extended to account for the a... |

33 |
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Citation Context ...metrized copointed homomorphism if and only if fΓ∆ = 0. 4 Geometry of interaction in the presence of weakening and contraction The Geometry of Interaction (GoI) was introduced by Girard (Girard 1989, =-=Girard 1990-=-, Girard 1995) in the late 1980s in the context of modelling the dynamics of cut elimination in (classical) linear logic (Girard 1987). The aim was to capture the essential structure of the proof theo... |

30 | Declarative continuations and categorical duality
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Citation Context ...rminism within one category. Dummett categories or something similar might help here. Other starting points for a term-language for classical categories might be Filinski’s symmetric lambda calculus (=-=Filinski 1989-=-) and the symmetric lambda calculus by Barbanera and Berardi (Barbanera & Berardi 1996). Extending Dummett categories to first-order logic Finally, we should like to mention the possibility of extendi... |

30 | Glueing and Orthogonality for Models of Linear Logic
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(Show Context)
Citation Context ...l example of a non-compact classical category with non-trivial hom-sets. Categories of games and strategies seem to be natural candidates. Also, the double gluing construction (Loader 1994, Tan 1997, =-=Hyland & Schalk 2003-=-) is known to turn compact closed categories into non-compact ∗-autonomous categories (i.e., non-compact symmetric linearly distributive categories with negation). It would be interesting to check whe... |

29 | Geometry of interaction III : accommodating the additives
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(Show Context)
Citation Context ...inted homomorphism if and only if fΓ∆ = 0. 4 Geometry of interaction in the presence of weakening and contraction The Geometry of Interaction (GoI) was introduced by Girard (Girard 1989, Girard 1990, =-=Girard 1995-=-) in the late 1980s in the context of modelling the dynamics of cut elimination in (classical) linear logic (Girard 1987). The aim was to capture the essential structure of the proof theory of cut eli... |

29 |
Proof nets for classical logic
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Citation Context ...orem states essentially that Φ � Φ is a theorem of a theory T whenever C ⌊Φ⌋ ≤ C ⌊Ψ⌋ holds for every model C ⌊−⌋ of T . Its proof uses a category built from Robinson’s proof nets for classical logic (=-=Robinson 2003-=-), which correspond directly to the classical sequent calculus. (We shall discuss these nets briefly in § 2.2.) A morphism of that category is an equivalence class of proof nets with respect to the pr... |

22 | Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories. Theory and Applications of Categories
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Citation Context ...posed to C ⌊Φ⌋ = C ⌊Ψ⌋), where C ⌊Φ⌋ and C ⌊Ψ⌋ are the morphisms denoted by Φ and Ψ in the classical category C. Classical categories are a special case of symmetric linearly distributive categories (=-=Cockett & Seely 1997-=-b): they have symmetric monoidal products ⊗ and ⊕ for modelling conjunction and disjunction, respectively. To model contraction and 3sweakening on the right, every object A is endowed with a symmetric... |

22 | Linear Distributive Functors
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Citation Context ...mark 3.35. Linearly distributive categories have a notion of complemented object: simply speaking, a complemented object of a linearly distributive category is an object whose negated version exists (=-=Cockett & Seely 1999-=-, Appendix). For a symmetric linearly distributive category C, the full subcategory S given by the complemented objects has negation, because complemented objects are closed under ⊗, ⊕, ⊤, and ⊥ (Cock... |

21 | Order-enriched categorical models of the classical sequent calculus - Führmann, Pym - 2004 |

20 |
On the semantics of classical disjunction
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(Show Context)
Citation Context ...vability. Second, insisting to keep classical logic, we might move to “classical natural deduction” systems (Prawitz 1965), where proofs may be represented as terms of the λµν-calculus (Parigot 1992, =-=Pym & Ritter 2001-=-). But such systems do not admit all cut-reductions: as it turns out, the call-by-name version of λµν admits only the reduction to Φ2, while the call-by-value version admits only the reduction to Φ1. ... |

16 |
Full Completeness for Models of Linear Logic
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(Show Context)
Citation Context ...ore natural example of a non-compact classical category with non-trivial hom-sets. Categories of games and strategies seem to be natural candidates. Also, the double gluing construction (Loader 1994, =-=Tan 1997-=-, Hyland & Schalk 2003) is known to turn compact closed categories into non-compact ∗-autonomous categories (i.e., non-compact symmetric linearly distributive categories with negation). It would be in... |

11 | Models of lambda calculi and linear logic: structural, equational and proof-theoretic characterisations
- Loader
- 1994
(Show Context)
Citation Context ...acking is a more natural example of a non-compact classical category with non-trivial hom-sets. Categories of games and strategies seem to be natural candidates. Also, the double gluing construction (=-=Loader 1994-=-, Tan 1997, Hyland & Schalk 2003) is known to turn compact closed categories into non-compact ∗-autonomous categories (i.e., non-compact symmetric linearly distributive categories with negation). It w... |

10 | On the geometry of interaction for classical logic - Führmann, Pym - 2004 |

10 | Classical linear logic of implications
- Hasegawa
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(Show Context)
Citation Context ...ions that result in hom-semilattices. In private communications, Hasegawa has suggested using a modified version of the multiplicative fragment his lambda calculus DCLL (Dual Classical Linear Logic) (=-=Hasegawa 2002-=-). To be precise, his approach is based on the lambda calculus below, which is sound and complete with respect to ∗-autonomous categories with symmetric comonoids: Types B − B + σ ::= b | ⊥ | σ → σ 64... |

9 |
Abstract interpretation of proofs: Classical propositional calculus
- Hyland
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(Show Context)
Citation Context ...aracterization of morphisms in such GoI categories with respect to their behaviour under cut-reduction (Prop. 4.5). • In § 5, we suggest some directions for future work. 1.2 Related work The article (=-=Hyland 2004-=-) introduces a notion of abstract interpretation of classical proof as a compact closed category in which every object is equipped with a symmetric monoid and a symmetric comonoid, satisfying certain ... |

7 | Morphisms and modules for poly-bicategories. Theory and Applications of Categories - Cockett, Koslowski, et al. |

7 |
Cut Elimination in Categories
- Do˘sen
- 1999
(Show Context)
Citation Context ...⊗ and ⊕ are functorial. Another difference between our work and (Bellin et al. 2004) is that we deal with modelling cut-reduction (using the poset-enrichment) whereas (Bellin et al. 2004) do not. In (=-=Dosen 1999-=-, Dosen & Petric 2004), a notion of “Boolean category” is introduced. This notion relies on the presence of products and coproducts, leading 6sto a more “collapsed” structure than ours, closely relate... |

7 |
Proof theory in the abstract
- Hyland
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(Show Context)
Citation Context ...tion of classical proof as a compact closed category in which every object is equipped with a symmetric monoid and a symmetric comonoid, satisfying certain conditions. (This work was foreshadowed in (=-=Hyland 2002-=-).) These abstract interpretations are almost the same as our classical categories in the compact case where ⊗ = ⊕. The only difference is that compact classical categories need to satisfy an extra eq... |

5 |
A semantic view of classical proofs
- Ong
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(Show Context)
Citation Context ...ibrations 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 (=-=Ong 1996-=-, Pym & Ritter 2001). Alternative models are given by control categories and co-control categories (Selinger 2001). In our companion paper (Führmann & Pym 2004b), we presented a solution that, unlike ... |

3 | Two paradigms of logical computation in affine logic? Logic for concurrency and synchronisation
- Bellin
(Show Context)
Citation Context ...ore “collapsed” structure than ours, closely related to the category of finite sets and relations. There is also some interesting work about confluent cut elimination in the presence of the MIX rule (=-=Bellin 2003-=-, Lamarche & Straßburger 2004). For example, one can remove the non-determinism of cut-reduction by allowing a reduction Φ ··· Γ ⊢ ∆ WR Γ ⊢ A, ∆ Γ, Γ ′ ⊢ ∆, ∆ ′ Φ ′ · · Γ ⊢ ′ ∆ ′ Γ ′ , A ⊢ ∆ ′ WL Cut ... |

2 |
Proof theory of classical propositional calculus
- Bellin, Hyland, et al.
- 2004
(Show Context)
Citation Context ...ly preserve identities. In contrast, our work fits into the existing framework of symmetric linearly distributive categories, in which ⊗ and ⊕ are functorial. Another difference between our work and (=-=Bellin et al. 2004-=-) is that we deal with modelling cut-reduction (using the poset-enrichment) whereas (Bellin et al. 2004) do not. In (Dosen 1999, Dosen & Petric 2004), a notion of “Boolean category” is introduced. Thi... |

1 |
Feedback for linearly distributibe categories: traces and fixpoints
- Blute, Cockett, et al.
- 2000
(Show Context)
Citation Context ...uired in this article. We also build on the discussions of MIX categories in (Blute, Cockett & Seely 2000) and (Cockett & Seely 1997a), and the notion of traced object in a MIX category presented in (=-=Blute et al. 2000-=-). We also rely on results from the GoI literature; the related work in this area is described in § 4. 2 Preliminaries 2.1 The sequent calculus The version of the sequent calculus to which we refer is... |

1 |
Naming proofs in propositional classical logic
- Lamarche, Straßburger
- 2004
(Show Context)
Citation Context ...d” structure than ours, closely related to the category of finite sets and relations. There is also some interesting work about confluent cut elimination in the presence of the MIX rule (Bellin 2003, =-=Lamarche & Straßburger 2004-=-). For example, one can remove the non-determinism of cut-reduction by allowing a reduction Φ ··· Γ ⊢ ∆ WR Γ ⊢ A, ∆ Γ, Γ ′ ⊢ ∆, ∆ ′ Φ ′ · · Γ ⊢ ′ ∆ ′ Γ ′ , A ⊢ ∆ ′ WL Cut � Φ ··· Γ ⊢ ∆ Φ ′ · · Γ ⊢ ′ ∆... |