## Linear Läuchli semantics (1996)

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Venue: | ANNALS PURE APPL. LOGIC |

Citations: | 19 - 7 self |

### BibTeX

@ARTICLE{Blute96linearläuchli,

author = {R. F. Blute and P. J. Scott},

title = {Linear Läuchli semantics},

journal = {ANNALS PURE APPL. LOGIC},

year = {1996},

pages = {77--101}

}

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### Citations

424 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...nt (i.e. an "abstract proof"). Lauchli's semantics also has a categorical interpretation. The category of Z-sets is a cartesian closed category (= ccc), and so interprets simply-typed -calcu=-=lus as in [29]-=-, or equivalently deductions in a fragment of intuitionistic logic. A categorical presentation can be found in [23]. While Lauchli's semantics is a semantics of proofs, Lauchli's theorem is finally ab... |

213 | Intensional Interpretations of Functionals of Finite Type - Tait - 1967 |

210 | Closed categories - Eilenberg, Kelly - 1966 |

183 | Topos Theory - Johnstone - 1977 |

173 |
Hopf algebras
- Sweedler
- 1969
(Show Context)
Citation Context ...ralize the previous results to the noncommutative, braided and cyclic settings. We briefly review the basic theory before stating our conservativity result. For a more complete discussion, see [1] or =-=[45]-=-. 11.1 Definition and Categorical Structure Definition 11.1 A Hopf algebra is a vector space, H, equipped with an algebra structure, a compatible coalgebra structure and an antipode. These must satisf... |

132 |
The structure of multiplicatives
- Danos, Regnier
- 1989
(Show Context)
Citation Context ...process. It is this interaction which makes nets useful in analyzing coherence problems in -autonomous categories [9]. The version of proof net we present is a simplification due to Danos and Regnier =-=[14]-=-. We first define the notion of proof structure. These are certain graphs whose nodes are labelled by formulas (or better, formula occurrences). Proof structures are constructed inductively from four ... |

121 |
Hopf algebras
- Abe
- 2004
(Show Context)
Citation Context ...to generalize the previous results to the noncommutative, braided and cyclic settings. We briefly review the basic theory before stating our conservativity result. For a more complete discussion, see =-=[1]-=- or [45]. 11.1 Definition and Categorical Structure Definition 11.1 A Hopf algebra is a vector space, H, equipped with an algebra structure, a compatible coalgebra structure and an antipode. These mus... |

115 |
Autonomous Categories
- Barr
- 1979
(Show Context)
Citation Context ...0 o g) (5) A generalization of this example to nested evaluations is in Lemma 10.2 below. 7.2 Interpreting MLL Sequents Functorial polymorphism can be extended to handle Barr's -autonomous categories =-=[7]-=-, i.e. smc categories C equipped with an involution functor ( ) ? : C op ! C given by a dualising object. Such categories interpret the multiplicative fragment of classical linear logic [42, 9]. We mo... |

112 | Introduction to distributive categories
- Cockett
- 1993
(Show Context)
Citation Context ...derivable for i = 1; 2. The set of sequents mentioned in the above theorem is then obtained by using three canonical morphisms which exist in any model of MLL + MIX. These are the weak distributivitys=-=[13]-=- and the MIX morphism [15]: ffi: A\Omega (B . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... |

109 |
Fully abstract models of the typed lambda calculus
- Milner
- 1977
(Show Context)
Citation Context ...[2] for multiplicative linear logic (= MLL) using game semantics recently led to a solution of the full abstraction problem for PCF [3], a fundamental problem in denotational semantics for many years =-=[34]-=-. Finally, from a categorical viewpoint, full completeness theorems are asking for a full representation of a certain kind of free category, say C, into a model category E . In this sense, a full comp... |

107 |
Coherence for compact closed categories
- Kelly, Laplaza
- 1980
(Show Context)
Citation Context ...here are elements which are fixed by arbitrary actions. These are the scalar multiples of the trace element, and arise because the category of finite dimensional representations of a group is compact =-=[26]-=-. Thus, while the category of finite-dimensional vector spaces is a model of multiplicative linear logic, it will not satisfy the appropriate full completeness theorem. Notice also that this lemma con... |

85 |
Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic
- Abrusci
- 1991
(Show Context)
Citation Context ...ear logic, and considers the representations of groups and Hopf algebras in such spaces. This theory leads to a large class of new models of commutative linear logic [19], noncommutative linear logic =-=[4, 28]-=-, and braided linear logic [11]. Proofs of all of the following results can be found in [6] and [10]. Definition 5.1 Let V be a vector space over a discrete field k. A topology,s, on V is linear if it... |

83 |
Quantales and (noncommutative) linear logic
- Yetter
- 1990
(Show Context)
Citation Context ...eserves the - autonomous structure. 2 These results suggest that general Hopf algebras may be useful in deriving full completeness theorems for noncommutative logics, such as those studied in [4] and =-=[48]-=-. 12 Conclusion We now discuss possible extensions of this work. While Theorem 11.7 is straightforward to prove, it is an extremely suggestive result. In [10], it is observed that Hopf algebras provid... |

80 |
Quasitriangular Hopf algebras and Yang-Baxter equations
- Majid
- 1990
(Show Context)
Citation Context ...]. While MOD(H) and T MOD(H) may not be symmetric, they will always be closed. For the rest of the section, we will assume H is cocommutative, although the next result holds in a more general setting =-=[34]-=-. Theorem 11.5 Let H be a cocommutative Hopf algebra. symmetric monoidal closed categories. MOD(H) and T MOD(H) are Proof. In the case of MOD(H) the internal HOM is given by the space of k-linear maps... |

67 | Natural Deduction and Coherence for Weakly Distributive Categories
- Blute, Cockett, et al.
(Show Context)
Citation Context ... that one may only introduce axiom links for which the conclusions are literals, i.e. atoms or negations of atoms. This has no effect on expressive power and allows us to avoid the expansion rules of =-=[12]-=-. There is a straightforward translation from sequent deductions to proof structures. We wish to identify those structures which correspond to derivable sequents. One of the ad5 vantages of this syste... |

59 |
Lambda definability in the full type hierarchy
- Plotkin
- 1980
(Show Context)
Citation Context ...ly faithful representation for ccc's, but it fails 3 to yield an independent model E in our sense: Set C op depends too much on C. The first full completeness result that we know of is due to Plotkin =-=[39]-=-. Inspired by Lauchli's work, Plotkin attempted to characterize lambda definability of set-theoretic functions in the full type hierarchy (= a full sub-ccc of Sets) generated from an infinite atomic s... |

47 |
Algebraic Topology
- Lefschetz
- 1942
(Show Context)
Citation Context ...closed category, objects for whichsis an isomorphism are called reflexive, or more precisely reflexive with respect to ?. 14 5.1 Linear Topology The approach we use goes back to the work of Lefschetz =-=[31]-=-, and has been studied by Barr [6] and the first author [10]. The idea is to add to the linear structure an additional topological structure, and then define the dual space to be the linear continuous... |

44 | Kripke-style models for typed lambda calculus
- Mitchell, Moggi
- 1991
(Show Context)
Citation Context ...arbitrary rank, by moving to the category of Kripke Logical Relations rather than Set-based logical relations. For a categorical reformulation, in terms of toposes of the form Sets P , P a poset, see =-=[36, 37]-=-. Finally, we also mention recent work of R. Loader [32]. Loader proves the undecidability of the Plotkin-Statman problem: in any model of simply typed lambda calculus over a finite base type, is it d... |

44 |
Logical relations and the typed lambda calculus
- Statman
- 1985
(Show Context)
Citation Context ...h the three canonical morphisms described above. 3 Logical Relations and Logical Permutations Logical relations play an important role in the recent proof theory and semantics of typed lambda calculi =-=[35, 39, 40, 43]-=-. We begin with logical relations on Henkin models; for further developments see [5, 35, 37, 38]. 9 3.1 Definitions and Examples Consider a simply typed lambda calculus with product types. A Henkin mo... |

39 |
Linear logic and lazy computation
- Girard, Lafont
- 1987
(Show Context)
Citation Context ...ss Theorem" below. 7.1 Interpreting\Omega ; \Gammaffi We shall first work in the theory of symmetric monoidal closed (= smc) categories without units, equivalently in intuitionistic MLL without u=-=nits [20, 9]-=-. Thus formulas are built from atoms, using the connectives\Omega ; \Gammaffi. Following the lead of functorial polymorphism (loc cit), we interpret formulas as multivariant functors over an smc categ... |

39 |
Topological vector spaces and distributions
- Horváth
- 1966
(Show Context)
Citation Context ...ld k is given the discrete topology. fflsis hausdorff. ffl 0 2 V has a neighborhood basis of open linear subspaces. The first requirement means that we have a topological vector space in the sense of =-=[24]-=- (except that most texts take the field to be the real or complex numbers with its usual topology). The third requirement is quite stringent. For example, it implies that the only linear topology on a... |

28 | Full abstraction for - Abramsky, Jagadeesan, et al. - 2000 |

24 | Notes on sconing and relators
- Mitchell, Scedrov
- 1993
(Show Context)
Citation Context ...ical relations play an important role in the recent proof theory and semantics of typed lambda calculi [35, 39, 40, 43]. We begin with logical relations on Henkin models; for further developments see =-=[5, 35, 37, 38]-=-. 9 3.1 Definitions and Examples Consider a simply typed lambda calculus with product types. A Henkin model is a wellpointed cartesian closed category (ccc). Equivalently, a Henkin model is a type-ind... |

17 |
An Abstract Notion of Realizability for which Intuitionistic Predicate Calculus is Complete
- Läuchli
- 1970
(Show Context)
Citation Context ...ereditary permutations: a closed term M : 1 ! oe corresponds to an invariant lambda term in Sets G (now for the extended lambda calculus with binary coproduct types.) This is the viewpoint of Lauchli =-=[30]-=-. The Lauchli Completeness Theorem is a converse to Soundness, for the case G = Z: Theorem 4.4 (Lauchli [30]) A formula oe of intuitionistic propositional calculus is provable if and only if for every... |

17 |
Typed lambda models and cartesian closed categories
- Mitchell, Scott
- 1989
(Show Context)
Citation Context ...ical relations play an important role in the recent proof theory and semantics of typed lambda calculi [35, 39, 40, 43]. We begin with logical relations on Henkin models; for further developments see =-=[5, 35, 37, 38]-=-. 9 3.1 Definitions and Examples Consider a simply typed lambda calculus with product types. A Henkin model is a wellpointed cartesian closed category (ccc). Equivalently, a Henkin model is a type-ind... |

16 | logic, Theoretical Computer Science 50 - Girard, Linear - 1987 |

15 |
Proofs and Types. Cambridge Tracts in Theoretical Computer Science
- Girard, Lafont, et al.
- 1989
(Show Context)
Citation Context ...al framework for Heyting's ideas led the way to many fundamental discoveries, for example Kleene's Realizability, Godel's Dialectica Interpretation , and (more recently) the Curry-Howard Isomorphism (=-=[21]-=-). However somewhat less familiar is Lauchli's seminal work in the 1960's [30]: this was the rst attempt to give both an abstract model of \proof" for intuitionistic logic and a Completeness Theorem f... |

13 |
Bilinear logic in algebra and linguistics
- Lambek
(Show Context)
Citation Context ...ear logic, and considers the representations of groups and Hopf algebras in such spaces. This theory leads to a large class of new models of commutative linear logic [19], noncommutative linear logic =-=[4, 28]-=-, and braided linear logic [11]. Proofs of all of the following results can be found in [6] and [10]. Definition 5.1 Let V be a vector space over a discrete field k. A topology,s, on V is linear if it... |

12 | Normal forms and cutfree proofs as natural transformations
- Girard, Scedrov, et al.
- 1989
(Show Context)
Citation Context ... this paper may be viewed as the beginnings of a theory of logical relations for linear logic and concurrency. In this paper, we present a semantics based upon an extension of functorial polymorphism =-=[5, 9, 22]-=- to the linear setting. In this setting, types are definable multivariant functors on a category of topological vector spaces. We then interpret terms, i.e. deductions in the theory MLL+MIX as certain... |

12 | Linear Logic, Totality and Full Completeness - Loader |

11 |
Type systems for programming languages, Handbook of theoretical computer science
- Mitchell
- 1990
(Show Context)
Citation Context ...h the three canonical morphisms described above. 3 Logical Relations and Logical Permutations Logical relations play an important role in the recent proof theory and semantics of typed lambda calculi =-=[35, 39, 40, 43]-=-. We begin with logical relations on Henkin models; for further developments see [5, 35, 37, 38]. 9 3.1 Definitions and Examples Consider a simply typed lambda calculus with product types. A Henkin mo... |

8 | Mechanizing logical relations - Stoughton - 1994 |

6 |
Lambek’s categorical proof theory and Läuchli’s abstract realizability
- Harnik, Makkai
- 1992
(Show Context)
Citation Context ...th a distinguished permutation may be identi ed with a Z-set (a set with an action of the free cyclic group Z). Thus, from this viewpoint, Lauchli's abstract models are nothing more than Z-set models =-=[23]-=-. Lauchli's Completeness Theorem says: a formula is provable if and only if its interpretation in every abstract model contains an invariant element (i.e. an \abstract proof"). Lauchli's semantics als... |

5 |
Games and Full Completeness for Multiplicative Linear
- Abramsky, Jagadeesan
- 1994
(Show Context)
Citation Context ...t provability, rather than genuine proofs. Thus, we might ask for a better result: can one find a notion of abstract model which characterizes proofs themselves? This is the full completeness problem =-=[2]-=-. From the Curry-Howard viewpoint, which identifies formulas with types and (natural deduction) proofs with typed \Gammaterms, we are asking for a typed lambda model E with a surjective interpretation... |

5 |
Dinaturality for Free, in
- Freyd, Robinson, et al.
- 1992
(Show Context)
Citation Context ... 7 Functorial Polymorphism We shall give a further development of the theory of Functorial Polymorphism applied to linear logic, following [5, 9, 22]. For other developments of the general theory, cf =-=[16, 17]-=-. Recall that in functorial polymorphism, the types of a -calculus are interpreted directly as certain multivariant functors, while terms are an appropriate multivariant version of natural transformat... |

5 |
Topological Vector Spaces, Springer Graduate Texts
- Schaefer
- 1970
(Show Context)
Citation Context ...h an identity. This point is discussed in [10]. 5.2 Quotients and Direct Sums We now discuss quotients and direct sums of topological vector spaces. More complete discussions can be found in [24] and =-=[41]-=-. Given a topological vector space V and an arbitrary linear subspace U , it is readily seen that the quotient topology on the quotient space V=U gives a topological vector space. It is not generally ... |

4 |
Types, Abstraction and Parametric
- Reynolds
- 1983
(Show Context)
Citation Context ...ly compose [5]; however, in certain known cases they do. When composition is well-defined, the dinatural calculus permits interesting "parametric" interpretations of the relevant lambda calc=-=ulus (cf. [40]). For exa-=-mple: 1. In [5] it was shown that certain uniform dinaturals between "logically definable" functors over Per (the category of partial equivalence relations on the natural numbers) do compose... |

3 | Duality of Vector Spaces, Cahiers de Top. et Geom - Barr - 1976 |

2 |
Representation Theory, Springer-Verlag Graduate Texts
- Fulton, Harris
- 1991
(Show Context)
Citation Context ... model of intuitionistic logic. To model linear logic, it is natural to replace sets with vector spaces. This leads to the classical subject of group representation theory as described for example in =-=[18]-=-. However, we must build a -autonomous category of vector spaces, in order to be able to model the involutive negation of classical linear logic. Recall that a symmetric monoidal closed category is -a... |

2 | Lambek's Categorical Proof Theory and L"auchli's Abstract Realizability - Harnik, Makkai - 1992 |

1 |
Separability of tensor in Chu categories of vector spaces, Appendix to [10
- Barr
(Show Context)
Citation Context ... monoidal structure exists for abstract reasons, it is possible to prove that the underlying vector space of V\Omega LT W is the usual algebraic tensor product. This issue is discussed in Barr's note =-=[8]-=-, which is an appendix to [10]. We now define duality for this category. Given an object V in T VEC we define V ? to be V \Gammaffi LT k where the base field k is topologized discretely. Lefschetz pro... |

1 |
Hopf Algebras and Linear Logic, to appear
- Blute
- 1995
(Show Context)
Citation Context ... logic. Hopf algebras are viewed as a unifying structure for these different logics, and the structure of the logic you are modeling is reflected in the structure of the algebra. This is discussed in =-=[10]-=-. While MOD(H) and T MOD(H) may not be symmetric, they will always be closed. For the rest of the section, we will assume H is cocommutative, although the next result holds in a more general setting [... |

1 |
Braided Proof Nets and Categories
- Blute
- 1995
(Show Context)
Citation Context ...sentations of groups and Hopf algebras in such spaces. This theory leads to a large class of new models of commutative linear logic [19], noncommutative linear logic [4, 28], and braided linear logic =-=[11]-=-. Proofs of all of the following results can be found in [6] and [10]. Definition 5.1 Let V be a vector space over a discrete field k. A topology,s, on V is linear if it satisfies the following three ... |

1 |
The Undecidability of *-definability, manuscript
- Loader
- 1993
(Show Context)
Citation Context ...Relations rather than Set-based logical relations. For a categorical reformulation, in terms of toposes of the form Sets P , P a poset, see [36, 37]. Finally, we also mention recent work of R. Loader =-=[32]-=-. Loader proves the undecidability of the Plotkin-Statman problem: in any model of simply typed lambda calculus over a finite base type, is it decidable whether a function is lambda definable or not? ... |

1 | Braided Proof Nets and Categories, in preparation - Blute - 1995 |

1 | The M IX Rule, preprint - Fleury, R'etor'e - 1991 |

1 | Bilinear Logic in Algebra and Linguistics, preprint - Lambek - 1993 |

1 | 47] W.W. Tait, Intensional Interpretation of Functionals of Finite Type - Sweedler, Algebras, et al. - 1969 |

1 | and Full Abstraction for PCF, Parts I, II. Research Announcements - Abramsky, Jagadeesan, et al. - 1993 |

1 |
The Undecidability of -de nability, manuscript
- Loader
- 1993
(Show Context)
Citation Context ... Relations rather than Set-based logical relations. For a categorical reformulation, in terms of toposes of the form SetsP , P a poset, see [37, 38]. Finally, we also mention recent work of R. Loader =-=[32]-=-. Loader proves the undecidability of the Plotkin-Statman problem: in any model of simply typed lambda calculus over a nite base type, is it decidable whether a function is lambda de nable or not? In ... |