## On Bunched Typing (2002)

Citations: | 33 - 2 self |

### BibTeX

@MISC{O'Hearn02onbunched,

author = {Peter O'Hearn},

title = {On Bunched Typing},

year = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most

### Citations

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Citation Context ...! [[]]. We intend that this is obtained from a proof that using commutative monoid laws: any such proof determines an isomorphism, using the coherent isomorphisms of symmetric monoidal categories [2=-=-=-4]. We claim that applications of symmetry morphisms are explicitly disambiguated by the use of different identiers for dierent types appearing in bunches, so that the morphism i is unique. (This appe... |

288 | Computational interpretations of linear logic
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Citation Context ...r. The analogy with temporary resources is clear in several formal interpretations of linear logic, including the original coherence space model [16] and the concurrent semantics of proof expressions =-=[-=-1]. In contrast, the sharing interpretation of concerns what is often labelled a permanent resource. Here, permanent does not literally mean permanent, but potentially long lived; examples includesle... |

278 | Local reasoning about programs that alter data structures - O’Hearn, Reynolds, et al. - 2001 |

222 | The practice of logical frameworks
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Citation Context ...ameworks enable a more exact representation of the semantics of languages with state, and it stands to reason that the same should hold true for program logics as well. Important advances include LLF =-=[10]-=-, which incorporates linear (non-dependent) function types, and the RLF system [21], which has even linear dependent function types. The problems in this area are dicult, but progress could prove espe... |

192 | The logic of bunched implications
- O’hearn, David, et al.
- 1999
(Show Context)
Citation Context ...of the material in this paper was presented, in preliminary form, at the 1999 TLCA conference [33]. Related material on the logic BI of bunched implications, the logical cousin of , may be found in [3=-=4, 38, 20]-=-. See in particular Pym's forthcoming monograph for a treatment of some of the foundational issues avoided here [39]. 2 Routes to Bunched Typing 2.1 Sharing and Contraction The work reported in this p... |

165 |
as an assertion language for mutable data structures
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- 2001
(Show Context)
Citation Context ...of the material in this paper was presented, in preliminary form, at the 1999 TLCA conference [33]. Related material on the logic BI of bunched implications, the logical cousin of , may be found in [3=-=4, 38, 20]-=-. See in particular Pym's forthcoming monograph for a treatment of some of the foundational issues avoided here [39]. 2 Routes to Bunched Typing 2.1 Sharing and Contraction The work reported in this p... |

165 |
The essence of Algol
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- 1981
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Citation Context ...ly, while at the same time exhibiting doubly closed structure. The central technique for linking the formal properties of bunched typing to sharing is the spatial approach to possible world semantics =-=[43, 28, 29, 30]-=-. In this approach, a world is viewed as corresponding to an area of memory (or, more generally, to resource), and the semantics of types and terms is parameterized by worlds. A semantic expression de... |

161 | Bi as an assertion language for mutable data structures - Ishtiaq, O’Hearn - 2001 |

135 | Syntactic control of interference - Reynolds, Jan - 1978 |

108 |
Display logic
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- 1982
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Citation Context ...nchessrst arose in the 70s for completely dierent reasons, as a result of a problem in the proof theory of relevant logics [13]. Since then, bunches have been a standard device used by relevantists' [=-=3, 40, 47]-=-. The most famous property of relevant logics is their denial of Weakening ` C ; A ` C Weakening: This denial is done in a bid to ensure that the premisses in an entailment are actually relevant to th... |

108 | Intuitionistic reasoning about shared mutable data structure - Reynolds - 2000 |

107 |
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Citation Context ...tegorical properties and resource interpretations. Historically, however, bunchessrst arose in the 70s for completely dierent reasons, as a result of a problem in the proof theory of relevant logics [=-=13]-=-. Since then, bunches have been a standard device used by relevantists' [3, 40, 47]. The most famous property of relevant logics is their denial of Weakening ` C ; A ` C Weakening: This denial is done... |

100 | A mixed linear and non-linear logic: Proofs, terms and models (preliminary report
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Citation Context ...-based control over structurals and that obtained from linear logic, or linear typing. In models of linear logic two closed categories are involved, where one is often presented as a Kleisli category =-=[4, 5, 2]-=-. For instance, in the original coherence space model there are indeed two function types, butsis closed structure in the category of linear maps, while the additive !, which can be represented as !As... |

100 |
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Citation Context ... construction cannot be used to give non-degenerate realizability models of . Indeed, the problem ofsnding a convincing realizability interpretation of remains open. 25 6.2 Day's Construction In [11], Day shows how to obtain closed structures in functor categories. Day's denitions are phrased in the language of enriched category theory, and use what is termed a premonoidal structure on a categor... |

100 |
The Semantics and Proof Theory of the Logic of Bunched Implications. Applied Logic Series
- Pym
- 2002
(Show Context)
Citation Context ...logic BI of bunched implications, the logical cousin of , may be found in [34, 38, 20]. See in particular Pym's forthcoming monograph for a treatment of some of the foundational issues avoided here [3=-=9]-=-. 2 Routes to Bunched Typing 2.1 Sharing and Contraction The work reported in this paper arose originally from a failed attempt to reconcile two substructural type systems, systems where the structura... |

98 | What is a categorical model of intuitionistic linear logic?’, Typed lambda calculi and applications
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Citation Context ... linear logic" to mean a symmetric monoidal closed category, with products and coproducts, and equipped with a \monoidal comonod"; these are the models of intuitionistic linear logic, as pre=-=sented in [6]-=-. So we explicitly dene: Denition 19 An with \!" model is a bicartesian dcc with a monoidal comonad structure (where the monoidal structure used is that for (; I)). (These are the minimum condit... |

95 |
Operating System Principles
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Citation Context ...f dozen of the other. But the distinction is actually critical in computer science. The number-of-uses explanation of linear logic calls to mind the notion of temporary resources in Operating Systems =-=[7]-=-, the canonical example of which is a message produced by one process and consumed by another. The analogy with temporary resources is clear in several formal interpretations of linear logic, includin... |

93 |
The linear abstract machine
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Citation Context ...both. Ideally, we would like a way to combine linear and bunched typing in a way that simultaneously accounts for sharing and consumption. (Examples of the consumptive aspect of linear typing include =-=[22, 51, 30-=-].) A hint of the way to proceed was given at the end of Section 6.4: we might start from a pair of categories, one to modelsand the other to model and !. Then, we would expect to decompose ! intos, ... |

92 |
Dual intuitionistic linear logic
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- 1996
(Show Context)
Citation Context ...-based control over structurals and that obtained from linear logic, or linear typing. In models of linear logic two closed categories are involved, where one is often presented as a Kleisli category =-=[4, 5, 2]-=-. For instance, in the original coherence space model there are indeed two function types, butsis closed structure in the category of linear maps, while the additive !, which can be represented as !As... |

87 |
Formal Category Theory: Adjointness for 2-Categories
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Citation Context ...gory whose objects are functors and whose morphisms are \transformations," i.e. families of maps but without naturality constraints. The symmetric monoidal structure is given by Gray's tensor pro=-=duct [18]-=- with the one object category as unit. So Cat is an ane, bicartesian dcc. These are the only symmetric monoidal closed structures on Cat [14]. A number of other naturally occurring examples arise from... |

77 | A term calculus for intuitionistic linear logic
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(Show Context)
Citation Context ...-based control over structurals and that obtained from linear logic, or linear typing. In models of linear logic two closed categories are involved, where one is often presented as a Kleisli category =-=[4, 5, 2]-=-. For instance, in the original coherence space model there are indeed two function types, butsis closed structure in the category of linear maps, while the additive !, which can be represented as !As... |

73 |
The Craft of Programming
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(Show Context)
Citation Context ...fragment, like IA without references or pointers) do in fact satisfy the disjointness policy of SCI. 32 But a problem with recursion was raised by Reynolds [45] and, indeed, one example from the book =-=[42]-=- that violates the policy illustrates the point well. For the record, it is the classic Towers of Hanoi program, where disks are moved between pegs. procedure movemany(k; a; b; c : int) if k > 0 then ... |

63 | A spatial logic for querying graphs - Cardelli, Gardner, et al. - 2002 |

55 | From Algol to polymorphic linear lambda-calculus
- O’Hearn, Reynolds
(Show Context)
Citation Context ...ly, while at the same time exhibiting doubly closed structure. The central technique for linking the formal properties of bunched typing to sharing is the spatial approach to possible world semantics =-=[43, 28, 29, 30]-=-. In this approach, a world is viewed as corresponding to an area of memory (or, more generally, to resource), and the semantics of types and terms is parameterized by worlds. A semantic expression de... |

46 |
The Geometry of Concurrency
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Citation Context ...cc. These are the only symmetric monoidal closed structures on Cat [14]. A number of other naturally occurring examples arise from Day's construction, including a model of higher-dimensional automata =-=[17-=-] and complexity models [19]. 6.4 On Adding \!" to , or BI In section 3.2 we showed how and linear -calculi mix additive and multiplicative function types in a fundamentally dierent way. We prob... |

45 |
A category-theoretic approach to the semantics of programming languages
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- 1982
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Citation Context ...nt operator for pointed domains. For this interpretation to exist we must have that each [[]]X is pointed. And for it to be natural we require that each morphism part [[]]f preserves least elements [3=-=-=-6]. These properties are satised by all the typed in SCI+, and are part of the identication of subcategory of Predom I in the following section. 9.2 Bunches, Environments and Non-interference [The pre... |

40 | Syntactic control of interference revisited
- O’Hearn, Power, et al.
- 1999
(Show Context)
Citation Context ... to resources in a computer system. As we have suggested, the main point is the emphasis on sharing, supported by a spatial view of possible world semantics which has developed over a number of years =-=[43, 26, 29, 30]-=-. Section 2 charted the initial route that led us to bunched typing. This began with an analogy between syntactic control of interference and linear logic, where both systems limit the use of Contract... |

36 | Semantics of local variables - O’Hearn, Tennent - 1992 |

36 | Pomset logic: a non-commutative extension of classical linear logic
- Retore
- 1997
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Citation Context ...an be given using three categories, with appropriate mappings between them, just as DILL arises from models based on a pair of categories. The relationship to the system of Ruet and Fages [46] (also, =-=[41]-=-) is less obvious. Their system uses bunches to combine two multiplicative fragments, the 13 non-commutative and the commutative. However, it does not use bunches to treat the additives, instead relyi... |

34 |
A reflection theorem for closed categories
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Citation Context ...part of a syntactic conservativity result, while fullness says that adding such structure does not cause any new maps to added, when we focus on just ccc or smcc types being embedded. Proposition 15 (=-=[12]-=-) Suppose C is a (small) symmetric monoidal closed category. Then the Yoneda embedding takes the symmetric monoidal closed structure of C to that on Set C op , as given by Day's construction. Since Se... |

34 | Natural deduction for intuitionistic noncommutative linear logic
- Polakow, Pfenning
- 1999
(Show Context)
Citation Context ...N : A ` MN : B E (We will not attempt to syntactically disambiguate the various forms of application.) The way that this system mixes its three fragments is dierent from Polakow and Pfenning's [37] three-zone, non-commutative variant of DILL, similarly to how DILL and are dierent, as discussed above. It appears that models of their system can be given using three categories, with appropriat... |

33 |
modal logics for mobile ambients
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- 2000
(Show Context)
Citation Context ...ved from SCI, but on the other hand conceptually very important: there are a number of interesting models that make good sense from a truth-based perspective, independently of type-theoretic concerns =-=[35, 20, 9-=-]. So, although related to the interpretations given here, this logical reading should be considered genuinely dierent, because of its declarative character. 11.2 Further Directions A number of questi... |

33 |
Polymorphic eect systems
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- 1988
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Citation Context ...where a reference may hold a procedure or another reference as its contents. This problem has been addressed using dierent methods, and with some success, by the eect systems of Giord and Lucassen [23=-=-=-]. The types used in eect systes are, however, very detailed, and they seem more suitable to an intermediate language used in a compiler (where their complexity can be hidden from the programmer) than... |

32 | Relevant Logic. A Philosophical Examination of Inference - Read - 1988 |

29 | On bunched predicate logic
- Pym
- 1999
(Show Context)
Citation Context ...of the material in this paper was presented, in preliminary form, at the 1999 TLCA conference [33]. Related material on the logic BI of bunched implications, the logical cousin of , may be found in [3=-=4, 38, 20]-=-. See in particular Pym's forthcoming monograph for a treatment of some of the foundational issues avoided here [39]. 2 Routes to Bunched Typing 2.1 Sharing and Contraction The work reported in this p... |

27 |
Syntactic control of interference, part 2
- Reynolds
- 1989
(Show Context)
Citation Context ...idate properties relating typing to substitution and reduction. Typing and reduction are areas where substructural type systems, which are surprisingly delicate, have encountered problems in the past =-=[45, 44, 27, 50]-=-, so it is appropriate that they be be explored early. We then spell out how the calculus can be interpreted in any cartesian dcc. 4.1 Substitution and Reduction Before tackling reduction, we need tha... |

24 | A relevant analysis of natural deduction
- Ishtiaq, Pym
- 1998
(Show Context)
Citation Context ...e, and it stands to reason that the same should hold true for program logics as well. Important advances include LLF [10], which incorporates linear (non-dependent) function types, and the RLF system =-=[21-=-], which has even linear dependent function types. The problems in this area are dicult, but progress could prove especially useful in controlling sharing relationships between dierent parts of a spec... |

24 | Algol-like Languages (Two volumes - O’Hearn, Tennent - 1997 |

19 | A model for syntactic control of interference
- O’Hearn
(Show Context)
Citation Context ...ather than allowing sharing. So limiting Contraction gives control over the number of times a piece of data can be used. This conceptual discrepancy between LL and SCI was clear to the author in 1990 =-=[26, 27, 28-=-], but there was a central question left unresolved then: is the dierence merely one of having two semantic interpretations of the same system (say, LL), or is a separate formal structure appropriate ... |

17 |
Linear logic and interference control
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(Show Context)
Citation Context ...ring and Contraction The work reported in this paper arose originally from a failed attempt to reconcile two substructural type systems, systems where the structural rule of Contraction is restricted =-=[26, 27]-=-. The background is that in 1978 Reynolds proposed syntactic control of interference, or SCI, a type theoretic method of controlling aliasing and other shared variable interference in imperative progr... |

16 | Objects, interference, and the Yoneda embedding - O’Hearn, Reddy - 1995 |

12 |
The semantics of non-interference: A natural approach
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(Show Context)
Citation Context ...ring and Contraction The work reported in this paper arose originally from a failed attempt to reconcile two substructural type systems, systems where the structural rule of Contraction is restricted =-=[26, 27]-=-. The background is that in 1978 Reynolds proposed syntactic control of interference, or SCI, a type theoretic method of controlling aliasing and other shared variable interference in imperative progr... |

12 | Linear *-Calculus and Categorical Models Revisited - Benton, Bierman, et al. |

11 |
Algebraic categories with few monoidal biclosed structures or none
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- 1980
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Citation Context ...For uniqueness of the map D ! I we make use of the following two facts. (i) [15] If C is a well pointed category then there is only one natural endomorphism on the identity functor id C : C ! C. (ii) =-=[14-=-] If (C; ; I) is a monoidal category then there is an injective function from C[I ; I ] into C C [id C ; id C ]. It follows that the identity is the only endomorphism on I in C. Now, suppose (toward c... |

10 |
Resource interpretations, bunched implications and the ##-calculus, in preparation, preliminary version
- O'Hearn
(Show Context)
Citation Context ...aders who prefer a less leisurely approach can skip forward directly to the synopsis in Section 2.5. Some of the material in this paper was presented, in preliminary form, at the 1999 TLCA conference =-=[3-=-3]. Related material on the logic BI of bunched implications, the logical cousin of , may be found in [34, 38, 20]. See in particular Pym's forthcoming monograph for a treatment of some of the foundat... |

6 | 29{April 1 - Brookes, Main, et al. |

4 | A linear logic model of state - Reddy - 1993 |

3 |
Type Systems for Polynomial Time Complexity. Habilitation thesis, Darmstadt. Available as Edinburgh report ECS-LFCS-99-406
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- 1999
(Show Context)
Citation Context ...claim that in a well-pointed dcc and would collapse to ^ and ! as well. However, Martin Hofmann has constructed well-pointed ane dcc's in which the products and function types are indeed distinct [1=-=9]-=-. Nonetheless, the proposition does establish an obstacle to the search for models by a number of standard techniques. For example, realizability models are often given using partial equivalence relat... |

3 | Possible worlds and resources: the semantics of BI. Submitted - O’Hearn, Pym, et al. - 2000 |

3 | Linear continuationpassing - Berdine, O'Hearn, et al. - 2002 |