## Computational Types from a Logical Perspective I (1995)

### Cached

### Download Links

Citations: | 56 - 6 self |

### BibTeX

@MISC{Benton95computationaltypes,

author = {P.N. Benton and G.M. Bierman and V.C.V. de Paiva},

title = {Computational Types from a Logical Perspective I},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

Moggi's computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus also arises naturally as the term calculus corresponding (by the Curry-Howard correspondence) to a novel intuitionistic modal propositional logic. We give natural deduction, sequent calculus and Hilbert-style presentations of this logic and prove a strong normalisation result. 1 Introduction The computational lambda calculus was introduced by Moggi as a metalanguage for denotational semantics which more faithfully models real programming language features such as non-termination, differing evaluation strategies, non-determinism and side-effects than does the ordinary simply typed lambda calculus [17, 18]. The starting point for Moggi's work is an explicit semantic distinction between compu...

### Citations

762 | Notions of Computation and Monads - Moggi - 1991 |

466 | Combining Monads - King, Wadler - 1993 |

462 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ... of the same logic and will sometimes subscript turnstiles with one of N,S or H to indicate which system is meant. 3.1 Natural deduction formulation of CL-logic Using the Curry-Howard correspondence (=-=Howard, 1980-=-), we can simply take Moggi’s original presentation (given in Figure 1) and erase the terms to produce a logic. Each type constructor corresponds to a logical connective as follows: Constructor Connec... |

455 | Computational lambda-calculus and monads - Moggi - 1989 |

225 | Coherence in closed categories - Kelly, Lane - 1971 |

221 |
Intensional interpretations of functionals of finite type I
- Tait
- 1967
(Show Context)
Citation Context ...n. We will find it convenient to work with the term calculus λMLT , rather than the logic, simply for reasons of space. Strong normalisation proofs usually use variants of Tait’s reducibility method (=-=Tait, 1967-=-); the extension of Tait’s method to commuting conversions as well as βreductions is due to Prawitz (1971). It is possible to use Prawitz’s technique to give a proof of strong normalisation for λMLT (... |

148 | Proofs and Types, volume 7 of Cambridge Tracts - Girard, Lafont, et al. - 1989 |

129 | Ideas and results in proof theory - Prawitz - 1971 |

109 | Proofs and Types. Cambridge Tracts - Girard, Lafont, et al. - 1989 |

108 | Functional Programming and Input/Output - Gordon - 1994 |

100 | A mixed linear and non-linear logic: Proofs, terms and models (preliminary report
- Benton
- 1995
(Show Context)
Citation Context ...the monad part of the16 Benton, Bierman and de Paiva model is always a commutative strong monad. More discussion of the relationship between intuitionistic linear logic and CL-logic may be found in (=-=Benton, 1995-=-a; Benton & Wadler, 1996), but there is still scope for further work looking at whether, for example, CL-logic is more closely associated with a non-commutative variant of intuitionistic linear logic.... |

98 | What is a categorical model of intuitionistic linear logic?’, Typed lambda calculi and applications
- Bierman
- 1995
(Show Context)
Citation Context ...ierman & de Paiva, 1996). In fact, there is a close relationship between CLlogic and intuitionistic linear logic. Any linear category (model for intuitionistic linear logic, see (Benton et al., 1992; =-=Bierman, 1995-=-)), gives rise to a CL-model as a subcategory of the category of algebras for the ! comonad. Whilst this is interesting, not all CL-models arise in this way because the monad part of the16 Benton, Bi... |

91 | Categorical logic - Pitts - 1996 |

82 | Categories for Types - Crole - 1994 |

60 | Constructive logics. Part I: A tutorial on proof systems and typed λ-calculi - Gallier - 1991 |

55 | The essence of functional programming (Invited talk), in - Wadler - 1992 |

54 | Term assignment for intuitionistic linear logic. Announced on the Types electronic mailing list - Benton, Bierman, et al. - 1992 |

41 | A functional theory of exceptions - Spivey - 1990 |

40 | Elementary predicate logic - Hodges - 2001 |

34 | Linear logic, monads and the lambda calculus
- Benton, Wadler
- 1996
(Show Context)
Citation Context ...of the16 Benton, Bierman and de Paiva model is always a commutative strong monad. More discussion of the relationship between intuitionistic linear logic and CL-logic may be found in (Benton, 1995a; =-=Benton & Wadler, 1996-=-), but there is still scope for further work looking at whether, for example, CL-logic is more closely associated with a non-commutative variant of intuitionistic linear logic. Acknowledgements We are... |

23 | Intuitionistic necessity revisited - Bierman, Paiva - 1996 |

20 | The elimination theorem when modality is present - Curry - 1952 |

20 | Cut-free Sequent and Tableau Systems for Propositional Normal Modal Logics - Goré - 1992 |

18 | An intuitionistic modal logic with applications to the formal verification of hardware - Fairtlough, Mendler - 1995 |

12 |
Strong normalisation for the linear term calculus
- Benton
- 1995
(Show Context)
Citation Context ...the monad part of the16 Benton, Bierman and de Paiva model is always a commutative strong monad. More discussion of the relationship between intuitionistic linear logic and CL-logic may be found in (=-=Benton, 1995-=-a; Benton & Wadler, 1996), but there is still scope for further work looking at whether, for example, CL-logic is more closely associated with a non-commutative variant of intuitionistic linear logic.... |

12 | Programming Metalogics with a Fixpoint Type - Crole - 1991 |

7 | A functional theory of exceptions. Science of computer programming - Spivey - 1990 |

7 |
Comprehending monads. Pages 61--78 of
- Wadler
- 1990
(Show Context)
Citation Context ... shown that monads provide an elegant way to structure functional programs which perform naturally imperative operations, such as dealing with updatable state or engaging in interactive input/output (=-=Wadler, 1990-=-; Wadler, 1992; Gordon, 1994). This paper looks at (an extension of) Moggi’s computational lambda calculus from a logical perspective. Using the Curry-Howard correspondence ‘the other way round’ we de... |

7 | Three monads for continuations - Kieburtz, Agapiev, et al. - 1992 |

5 |
Computational lambda-calculus and monads. Pages 14–23 of
- Moggi
- 1989
(Show Context)
Citation Context ...ons from η-rules, particularly as the more general equational rules are non-local. 6 Categorical Models Since the computational lambda calculus was originally derived from categorical considerations (=-=Moggi, 1989-=-), we already know that a categorical model is a cartesian closed category (CCC) with a strong monad. For completeness we shall sketch these categorical definitions. Definition 1 A monad over a catego... |

4 |
Functional programming and input/output. Distinguished dissertations in computer science
- Gordon
- 1993
(Show Context)
Citation Context ...n elegant way to structure functional programs which perform naturally imperative operations, such as dealing with updatable state or engaging in interactive input/output (Wadler, 1990; Wadler, 1992; =-=Gordon, 1994-=-). This paper looks at (an extension of) Moggi’s computational lambda calculus from a logical perspective. Using the Curry-Howard correspondence ‘the other way round’ we derive a logic which we term C... |

3 | Relating categorical and Kripke semantics for intuitionistic modal logics - Alechina, Paiva, et al. - 1998 |

3 | natural deduction and the beck condition - Hyperdoctrines |

2 |
Term rewriting systems. Pages 1--116 of: Abramsky
- Klop
- 1992
(Show Context)
Citation Context ...ons 9 and 6.Computational Types from a Logical Perspective 11 4.4 Confluence Given the property of strong normalisation it is relatively straightforward to show confluence. We employ Newman’s lemma (=-=Klop, 1992-=-), which states that if a reduction system is weakly confluent and strongly normalising then it is confluent. We first need the following simple facts. Lemma 11 1. If u →βc u ′ then u[v/x] → ∗ βc u′ [... |

2 | Intuitionistic Logic, volume 3 - Dalen - 1986 |

1 | Computational Types from a Logical Perspective 17 - Hodges - 1983 |

1 | Ideas and results in proof theory. Pages 235--307 of: Fenstad, J.E. (ed), Proceedings of second scandinavian logic symposium - Prawitz - 1971 |

1 | Intuitionistic Logic. Chap - Dalen - 1986 |