## Kripke Resource Models of a Dependently-Typed, Bunched lambda-Calculus (Extended Abstract) (1999)

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### BibTeX

@MISC{Ishtiaq99kripkeresource,

author = {Samin Ishtiaq and David J. Pym},

title = {Kripke Resource Models of a Dependently-Typed, Bunched lambda-Calculus (Extended Abstract)},

year = {1999}

}

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### Abstract

The lL-calculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Secondly, it is a presentation of the proof-objects of BI, the logic of bunched implications. BI is a logic which directly combines linear and intuitionistic implication and, in its predicate version, has both linear and intuitionistic quantifiers. The lL-calculus is the dependent type theory which generalizes both implications and quantifiers. In this paper, we describe the categorical semantics of the lL-calculus. This is given by Kripke resource models, which are monoid-indexed sets of functorial Kripke models, the monoid giving an account of resource consumption. We describe a class of concrete, set-theoretic models. The models are given by the category of families of sets, parametrized over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day's tensor product.

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Citation Context ...ing of J i ; and Msis a meta-logic term corresponding to the encoding of . The LF logical framework consists of the -calculus together with the judgements-as-types mechanism for representing logics [1=-=, 9, 28]-=-. One consequence of this method of encoding is that encoded systems inherit the structural properties of the meta-logic. Now, the structural strength of LF is determined by the structural strength of... |

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Citation Context ... element r 2 R can be seen as the resource able to realize the functorial Kripke structure it indexes. We work with indexed category theory, rather than, for example, Cartmell's contextual categories =-=[5]-=-, as the indexed approach allows us to separate certain conceptual issues and, hence, allows us to recognize the extra structure needed for studying the model theory of structurally weaker logics and ... |

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Citation Context ...ke semantic partiality of information, in which the further up the world structure one goes, the more objects have dened interpretations. We refer to Streicher [36], Pym [31], and Mitchell and Moggi [=-=21-=-] for some comments regarding these matters. The following lemma follows easily from the denition. Lemma 19 join and share are functors. Proof We need to show that both join and share preserve identit... |

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Citation Context ...inear occurrence in U :V ) A variable x linearly occurs in the expression U :V if it linearly occurs in U , in V , or in both. We remark that the above denitions are not \linear" in the Girard se=-=nse [4, -=-2]. However, they seem quite natural in the bunches setting. O'Hearn and Pym, for instance, have examples of BI terms | the -calculus is in propositions-as-types correspondence with a non-trivial frag... |

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Citation Context ...certain bifunctors [19].) The second semantics of BI is a Kripke-style semantics of formulae, which combines Kripke's semantics of intuitionistic logic [18] and Urquhart's semantics of relevant logic =-=[37]-=-. These can be understood to be given, respectively, in Set P , where P is a poset, and in Set M , where M is a commutative monoid. A semantics of BI can be obtained by working in Set C op , where, fo... |

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Citation Context ...ists but rather as bunches, in which there are two kinds of combining operation, \;", which admits weakening and contraction and \,", which does not. Bunches, which originally arose in relev=-=ant logic [33], allow th-=-e formation of two kinds of function space, the intuitionistic one !, corresponding to \;" and the linear one (, corresponding to \,". The introduction rules are given by ; A ` B ( I ` A ( B... |

29 | On bunched predicate logic
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(Show Context)
Citation Context ..., then we can form two kinds of quantiers, a linear one 8 new and an intuitionistic one 8. The Kripke-style semantics for predicate BI can be given by extending the above ideas, and are discussed in [=-=23, 32]-=-. Although presheaf DCCs are adequate for such a semantics of predicate BI, they do not yield a good interpretation of proofs. For this, we must move to an indexed orsbred setting in which the predica... |

24 | A relevant analysis of natural deduction
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(Show Context)
Citation Context ...espondence is, then, that the RLF meta-logic uses this fragment of BI, just as the LF meta-logic uses the f!; 8g-fragment of Intuitionistic Logic. A detailed account of the correspondence is given in =-=[12, 15]-=-. It remains a challenging and open problem to give a systematic analysis of the relationship between substructural logics and dependent type theories. In particular, it remains to formulate a depende... |

18 |
and Computation in General Logic
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Citation Context ...ing of J i ; and Msis a meta-logic term corresponding to the encoding of . The LF logical framework consists of the -calculus together with the judgements-as-types mechanism for representing logics [1=-=, 9, 28]-=-. One consequence of this method of encoding is that encoded systems inherit the structural properties of the meta-logic. Now, the structural strength of LF is determined by the structural strength of... |

14 |
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Citation Context ...ns: it must be preserved on the nose by any f and must behave well under quanti- cation. Details of the treatment of the A:K-fragment in the case of contextual categories are in Streicher's thesis [36]. The analogous development in our setting is similar and we omit the details. Denition 18 Let be a -calculus signature. A Kripke resource - model is a 5-tuple hfJ r :[W ; [C op ; Cat]] j r 2 ... |

11 |
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Citation Context ... Language +Representation: We remark that these components are not entirely independent of each other. One representation mechanism is that of judgements-as-types, which originates from Martin-Lof's [=-=20]-=- development of Kant's notion of judgement [17]. The methodology of judgements-as-types is that judgements are represented as the type of their proofs. A logical system is represented by a signature w... |

11 |
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Citation Context ...nce. It should be possible then to formulate a precise idea regarding the completeness of the set f&; !;(;;g with respect to all sentential operators that have explicit schematic introduction rules [2=-=6, 3-=-5]. A similar analysis can be undertaken for the corresponding elimination rule. Thus, by analysing the form of relevant natural deduction, the -calculus can be seen to arise as the language of the RL... |

10 | Resource interpretations, bunched implications and the ##-calculus, in preparation, preliminary version - O'Hearn |

9 |
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Citation Context .... It should be possible then to formulate a precise idea regarding the completeness of the set f&; !;(; \Pi; g with respect to all sentential operators that have explicit schematic introduction rules =-=[23, 31]-=-. A similar analysis can be undertaken for the corresponding elimination rule. Thus, by analysing the form of relevant natural deduction, the -calculus can be seen to arise as the language of the RLF ... |

5 | Errata and Remarks
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(Show Context)
Citation Context ...cting a class of Kripke resource models from the category of families of sets. This paper, the content of which was sketched in [14], continues the worksrst reported in Ishtiaq and Pym [16] (see also =-=[15]-=-). The reader is referred to that paper for a full syntactic study of the type theory, together with its use as a language in the logical framework RLF. The ideas presented there weresrst considered b... |

4 |
A proof of the Church-Rosser property for the Edinburgh LF with j-conversion
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Citation Context ...hurch-Rosser for -reduction. This was the main diculty in Harper et al.'s metatheoretic study of the -calculus. One solution, due to Salvesen, was to use van Daalen's technique of label-conversion [34]. We exploit that result by giving a sound and consistent translation of the -calculus into the - calculus, and appealing to the reduction properties of the latter. 2.7 The propositions-as-types c... |

3 |
A note on representation and semantics in logical frameworks
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(Show Context)
Citation Context ...the structural rules of weakening and contraction. Consequently, LF is able to uniformly represent, i.e., the encoding + is surjective on proof-objects, only logics which also admit these structurals =-=[30, 10-=-]. We illustrate the use of LF by giving a brief example of how the f^; gfragment of Intuitionistic Logic (IL) is uniformly represented in LF via judgementsas -types. The natural deduction presentatio... |

2 |
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(Show Context)
Citation Context ...the structural rules of weakening and contraction. Consequently, LF is able to uniformly represent, i.e., the encoding + is surjective on proof-objects, only logics which also admit these structurals =-=[30, 10-=-]. We illustrate the use of LF by giving a brief example of how the f^; gfragment of Intuitionistic Logic (IL) is uniformly represented in LF via judgementsas -types. The natural deduction presentatio... |

2 |
A relevant analysis of natural deduction. Lecture at Workshop, EU Espirit Basic Research Action 3245, Logical Frameworks: Design, Implementation and Experiment
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(Show Context)
Citation Context ...s referred to that paper for a full syntactic study of the type theory, together with its use as a language in the logical framework RLF. The ideas presented there weresrst considered by one of us in =-=[2-=-9]. 2 The -calculus In this section, we give a syntactic presentation of the type theory, which we henceforth refer to as System N. We brie y comment on the propositions-astypes correspondence with a ... |

2 |
Functorial kripke models of the *-calculus
- Pym
- 1995
(Show Context)
Citation Context ... see this later, in Section 3.2, when we motivate the structure needed to model the -calculus. Kripke resource models generalize, as we might expect, the functorial Kripke models of the -calculus [31]. These consist of a functor J:[W ; [C op ; Cat]], where W is a Kripke world structure, C is a category with a (; 1) cartesian monoidal structure on it and [C op ; Cat] is a strict indexed category. ... |

2 |
Generalized rules for quanti and the completeness of the intuitionistic operators &; _; ; f; 8; 9
- Schroeder-Heister
- 1983
(Show Context)
Citation Context ...nce. It should be possible then to formulate a precise idea regarding the completeness of the set f&; !;(;;g with respect to all sentential operators that have explicit schematic introduction rules [2=-=6, 3-=-5]. A similar analysis can be undertaken for the corresponding elimination rule. Thus, by analysing the form of relevant natural deduction, the -calculus can be seen to arise as the language of the RL... |

2 |
On bunched predicate logic. To appear
- Pym
- 1999
(Show Context)
Citation Context ... then we can form two kinds of quantifiers, a linear one 8 new and an intuitionistic one 8. The Kripke-style semantics for predicate BI can be given by extending the above ideas, and are discussed in =-=[20, 24]-=-. Although presheaf DCCs are adequate for such a semantics of predicate BI, they do not yield a good interpretation of proofs. For this, we must move to an indexed or fibred setting in which the predi... |

2 |
Functorial Kripke models of the \Pi-calculus
- Pym
- 1997
(Show Context)
Citation Context ...ll see this later, in Section 3.2, when we motivate the structure needed to model the -calculus. Kripke resource models generalize, as we might expect, the functorial Kripke models of the \Picalculus =-=[28]-=-. These consist of a functor J:[W ; [C op ; Cat]], where W is a Kripke world structure, C is a category with a (\Theta; 1) cartesian monoidal structure on it and [C op ; Cat] is a strict indexed categ... |

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