## On Bunched Predicate Logic (1999)

Venue: | Proceedings of the IEEE Symposium on Logic in Computer Science |

Citations: | 29 - 17 self |

### BibTeX

@INPROCEEDINGS{Pym99onbunched,

author = {David J. Pym},

title = {On Bunched Predicate Logic},

booktitle = {Proceedings of the IEEE Symposium on Logic in Computer Science},

year = {1999},

pages = {183--192},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication, and may be viewed as a merging of intuitionistic logic and multiplicative, intuitionistic linear logic. The predicate version of BI includes, in addition to usual additive quantifiers, multiplicative (or intensional) quantifiers 8new and 9new , which arise from observing restrictions on structural rules on the level of terms as well as propositions. Moreover, these restrictions naturally allow the distinction between additive predication and multiplicative predication for each propositional connective. We provide a natural deduction system, a sequent calculus, a Kripke semantics and a BHK semantics for BI. We mention computational interpretations, based on locality and sharing, at both the propositiona...

### Citations

615 | Linear logic
- Girard
- 1987
(Show Context)
Citation Context ...nectives consistent with the existence of \Gamma -typed functions using their arguments multiple times. Bunches are not similar to the zoned contexts used in some presentations of linear logic (e.g., =-=[12, 5]). In particular, ";" and "," -=-can be nested in a bunch, and ";" (just like ", ") is internalized as a connective in the logic, while the stoup ";" does not internalize in this way. Linear logic admits... |

373 | A.: Uniform proofs as a foundation for logic programming
- Miller, Nadathur, et al.
- 1991
(Show Context)
Citation Context ...lution, which is available in BI: (X )\Gamma ` G Id (Aoe = A 0 oe) (Y )Aoe ` A 0 oe Resolution; (X ; Y )\Gamma; 8new y:G ! A ` A 0 where A and A 0 are atoms and G is an hereditary Harrop goal-formula =-=[26, 18]-=-. Here the rule must be read as a reduction (or search) operator, from conclusion to premisses. The idea is that the program is given by the antecendent, i.e., the bunch (X ; Y ) \Gamma; 8new y:G ! A,... |

303 | D.: Logic programming in a fragment of intuitionistic linear logic
- Hodas, Miller
- 1994
(Show Context)
Citation Context ...ading, based on the original coherence space model; an eager and lazy evaluation reading, based on a strict function model; a concurrency reading of proofs [1]; and a logic programming interpretation =-=[26, 13]-=-. In these early days for BI, we have just two related computational readings. At the propositional level, BI's - calculus, ff, can be used to characterize local variables and local state in imperativ... |

281 | Computational interpretations of linear logic
- Abramsky
- 1993
(Show Context)
Citation Context ...l readings, including: the number-of-uses reading, based on the original coherence space model; an eager and lazy evaluation reading, based on a strict function model; a concurrency reading of proofs =-=[1]-=-; and a logic programming interpretation [26, 13]. In these early days for BI, we have just two related computational readings. At the propositional level, BI's - calculus, ff, can be used to characte... |

125 |
On the meanings of the logical constants and the justifications of the logical laws
- Martin-Löf
- 1996
(Show Context)
Citation Context ...m of labelled trees. JUDGEMENTS We consider terms- and propositions-in-context, with a syntax of the form X ` t : Term and X ` OE : Prop asserting that a term or predicate is well-formed in context X =-=[17]. Constants and pred-=-icate letters can be considered to be given by schematic judgements and, as in the bunched logic itself, Contraction and Weakening are allowed for ";" but not for ",". We omit a fo... |

105 |
Display logic
- Belnap
- 1982
(Show Context)
Citation Context ...mma / and \Gamma ; OE ` / \Gamma ` OE ! / The antecedents are no longer sequences; rather, they are trees with propositions as leaves and internal nodes labelled with "," or ";", o=-=r in short, bunches [10, 6, 27]-=-. It is all very well to postulate proof rules in this way, but what meaning or significance, if any, does the resulting logic have ? We argue herein that BI possesses two very natural semantics. The ... |

101 |
Semantical analysis of intuitionistic logic I, Formal Systems and Recursive
- Kripke
- 1963
(Show Context)
Citation Context ... to denote propositions in the established sense. Recall that Kripke gave a semantics of IL based on possible worlds, which can be understood as using functor categories Set P , where P is a preorder =-=[16]-=-. Similarly, Urquhart's semantics of MILL uses Set M , where M is a commutative monoid [29]. We obtain a semantics of BI, which combines these two semantics directly, by working in categories Set C op... |

99 | What is a categorical model of intuitionistic linear logic
- Bierman
- 1995
(Show Context)
Citation Context ...t distribution, and so is incomplete for Kripke Resource Semantics. Categorical models of linear logic are based on two closed categories (often one is a Kleisli category), with functors between them =-=[8, 5]. Fo-=-r BI, we use a semantics of proofs based on a single category with two closed structures. The Tarski-style models of BI, which explain formul�� in terms of a notion of truth, are obtained by combi... |

98 |
The Semantics and Proof Theory of the Logic of Bunched Implications
- Pym
(Show Context)
Citation Context ... propositions. The quantifiers reflect this, giving rise to a form of "resource-sensitivity" over individuals or term denotations in a predicate logic. The details of propositional BI can be=-= found in [24]-=- and the details of predicate BI can be found in [25]. Computational interpretations of BI, at both the propositional predicate levels, are briefly discussed in x 10; more details can be found in [20,... |

97 |
On closed categories of functors
- Day
- 1974
(Show Context)
Citation Context ...ows that BI and MILL, with !, treat intuitionistic implication in essentially different ways. 5 Day's construction We can generate a rich class of models of BI using a general construction due to Day =-=[9]-=-. He shows that any monoidal (not necessarily closed) category (C; ; I) induces a monoidal closed structure on the functor category Set C op , and that if (C; ; I) is symmetric monoidal, then so is Se... |

89 |
Dual intuitionistic linear logic
- Barber
- 1996
(Show Context)
Citation Context ...nectives consistent with the existence of \Gamma -typed functions using their arguments multiple times. Bunches are not similar to the zoned contexts used in some presentations of linear logic (e.g., =-=[12, 5]). In particular, ";" and "," -=-can be nested in a bunch, and ";" (just like ", ") is internalized as a connective in the logic, while the stoup ";" does not internalize in this way. Linear logic admits... |

80 |
Intuitionistic logic
- Dalen
- 2002
(Show Context)
Citation Context ...efined at n \Delta m Y and that (u X ; v) be defined at mX \Delta n. One way to enforce these conditions would be to require that the domain functor D be constant, so that we should have a Beth model =-=[30]-=-. The pairing of the additive propositional implication, !, with multiplicative variable maintenance is more problematic; its discussion is deferred to another occasion. 9 Quantifiers The predicate ve... |

69 | A Uniform Proof-Theoretic Investigation of Linear Logic Programming
- Harland, Pym
- 1994
(Show Context)
Citation Context ...ading, based on the original coherence space model; an eager and lazy evaluation reading, based on a strict function model; a concurrency reading of proofs [1]; and a logic programming interpretation =-=[26, 13]-=-. In these early days for BI, we have just two related computational readings. At the propositional level, BI's - calculus, ff, can be used to characterize local variables and local state in imperativ... |

32 |
Semantics for Relevant Logics
- Urquhart
- 1972
(Show Context)
Citation Context ... based on possible worlds, which can be understood as using functor categories Set P , where P is a preorder [16]. Similarly, Urquhart's semantics of MILL uses Set M , where M is a commutative monoid =-=[29]-=-. We obtain a semantics of BI, which combines these two semantics directly, by working in categories Set C op , where C is a symmetric monoidal category. For simplicity, we describe this for the speci... |

30 |
Relevant logic: A philosophical examination of inference
- Read
- 1988
(Show Context)
Citation Context ...mma / and \Gamma ; OE ` / \Gamma ` OE ! / The antecedents are no longer sequences; rather, they are trees with propositions as leaves and internal nodes labelled with "," or ";", o=-=r in short, bunches [10, 6, 27]-=-. It is all very well to postulate proof rules in this way, but what meaning or significance, if any, does the resulting logic have ? We argue herein that BI possesses two very natural semantics. The ... |

23 | A relevant analysis of natural deduction
- Ishtiaq, Pym
- 1998
(Show Context)
Citation Context ...ication \Gamma , although a version allied to multiplicative conjunction is possible under restricted circumstances. It would be interesting to formulate a dependent function type, along the lines of =-=[15]-=-, which generalizes both of them. For the additive universal quantifier, one might expect the elimination rule to be (X) \Gamma ` 8x : OE X ` t : Term (X) \Gamma ` OE[t=x] 8E Such a rule is admissible... |

18 | Entailment: The Logic of Relevance and Necessity, volume I - Anderson, Belnap - 1975 |

12 |
Life in the undistributed middle
- Belnap
- 1993
(Show Context)
Citation Context ... proof theory of relevant logics [27] and Schroeder-Heister's work on structural frameworks [28]. Distribution ofsovershas been a source of debate within and without the relevant logic community (see =-=[7]-=- for a lively account). But for our purposes the debate is not a serious issue. Simply, distribution is valid in Kripke resource semantics, which we regard as a natural semantics, and we must accept i... |

11 | First order linear logic in symmetric monoidal closed categories
- Ambler
- 1991
(Show Context)
Citation Context ... Despite many rumblings on this topic over the years, we have not been able to locate a worked-out predicate logic with multiplicative quantifiers, apart from Ambler's system for the existential only =-=[2]-=-, the formulation of which is somewhat more complex than ours. Also relevant is the theory of multiplicative dependent function types in [15], together with its fibrational semantics [14]; it can be r... |

8 | Kripke resource models of a dependently-typed, bunched #-calculus, in preparation, available at http://www.dcs.qmw.ac.uk/ ~ pym
- Ishtiaq, Pym
(Show Context)
Citation Context ...stential only [2], the formulation of which is somewhat more complex than ours. Also relevant is the theory of multiplicative dependent function types in [15], together with its fibrational semantics =-=[14]-=-; it can be regarded as relying on a version of bunches appropriate to dependent type theory. Linear logic has a number of computational readings, including: the number-of-uses reading, based on the o... |

8 |
Structural frameworks, substructural logics, and the role of elimination inferences
- Schroeder-Heister
- 1991
(Show Context)
Citation Context ...s. Since then bunches have been a standard device in relevant logic. See, for example, Read's account of the proof theory of relevant logics [27] and Schroeder-Heister's work on structural frameworks =-=[28]-=-. Distribution ofsovershas been a source of debate within and without the relevant logic community (see [7] for a lively account). But for our purposes the debate is not a serious issue. Simply, distr... |

4 |
Entailment: The Logic of Relevance and Necessity, Volume II
- Anderson, Belnap, et al.
- 1992
(Show Context)
Citation Context ...chniques, such as bunches and resource semantics, have not spread further into theoretical computer science. The study of bunches arose from Dunn's work on a sequent calculus for the relevant logic R =-=[4]-=-. The motivation was not implication but distribution of additive conjunction over disjunction. The difficulty is that distribution fails in standard sequent calculi for logics, such as linear logic, ... |

4 |
Doubly closed categories , resource interpretations, and the αλ-calculus
- O'Hearn
- 1999
(Show Context)
Citation Context ...[24] and the details of predicate BI can be found in [25]. Computational interpretations of BI, at both the propositional predicate levels, are briefly discussed in x 10; more details can be found in =-=[20, 19, 23, 22]-=-. We compare BI with intuitionistic logic, linear logic and other relevant logics, also in x 10. A similar discussion is in [20] (joint work with Peter O'Hearn). 2 A Proof-theoretic introduction Impli... |

3 |
Conseqution formulation of positive R with co-tenability and t
- Dunn
(Show Context)
Citation Context ...mma / and \Gamma ; OE ` / \Gamma ` OE ! / The antecedents are no longer sequences; rather, they are trees with propositions as leaves and internal nodes labelled with "," or ";", o=-=r in short, bunches [10, 6, 27]-=-. It is all very well to postulate proof rules in this way, but what meaning or significance, if any, does the resulting logic have ? We argue herein that BI possesses two very natural semantics. The ... |

3 |
Logic programming with bunched implications
- Pym
- 1998
(Show Context)
Citation Context ...[24] and the details of predicate BI can be found in [25]. Computational interpretations of BI, at both the propositional predicate levels, are briefly discussed in x 10; more details can be found in =-=[20, 19, 23, 22]-=-. We compare BI with intuitionistic logic, linear logic and other relevant logics, also in x 10. A similar discussion is in [20] (joint work with Peter O'Hearn). 2 A Proof-theoretic introduction Impli... |

2 | The logic of bunched implications. To appear - O'Hearn, Pym - 1999 |

2 | Logic programming with bunched implications (extended abstract
- Pym
- 1998
(Show Context)
Citation Context ...[24] and the details of predicate BI can be found in [25]. Computational interpretations of BI, at both the propositional predicate levels, are briefly discussed in x 10; more details can be found in =-=[20, 19, 23, 22]-=-. We compare BI with intuitionistic logic, linear logic and other relevant logics, also in x 10. A similar discussion is in [20] (joint work with Peter O'Hearn). 2 A Proof-theoretic introduction Impli... |

1 |
type theory and recursion
- Linear
- 1993
(Show Context)
Citation Context ...e fact that Es(\Gamma) and E(\Gamma) are left adjoints. This is not to suggest that one wants distribution in all circumstances; the use of linear typing to provide a type structure for domain theory =-=[21] is a prim-=-e example of where one does not. Turning to linear logic, we briefly compare BI and intuitionistic linear logic, with "!", in logical (including categorical) and computational terms. The lin... |