## A terminating and confluent linear lambda calculus (2006)

Venue: | PROC. OF 17TH INT. CONFERENCE RTA 2006, VOLUME 4098 OF LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Ohta06aterminating,

author = {Yo Ohta and Masahito Hasegawa},

title = {A terminating and confluent linear lambda calculus},

booktitle = {PROC. OF 17TH INT. CONFERENCE RTA 2006, VOLUME 4098 OF LECTURE NOTES IN COMPUTER SCIENCE},

year = {2006},

pages = {166--180},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}-fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and Church-Rosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the equational theory of the linear lambda calculus. As an application we prove the strong normalization of the simply typed computational lambda calculus by giving a reduction-preserving translation into the linear lambda calculus.

### Citations

440 | Computational lambda-calculus and monads
- Moggi
- 1989
(Show Context)
Citation Context ...Translation from the Computational Lambda Calculus 8.1 The Simply Typed Computational Lambda Calculus λc The simply typed computational lambda calculus λc (its untyped version was introduced by Moggi =-=[17]-=-) has the same syntax as the simply typed lambda calculus plus the let-binding Γ ⊢ M : σ Γ,x: σ ⊢ N : τ Γ ⊢ let x σ be M in N : τ It is a call-by-value calculus however, and its rewriting / equational... |

364 |
Confluent reductions: Abstract properties and applications to term rewriting systems: Abstract properties and applications to term rewriting systems
- Huet
- 1980
(Show Context)
Citation Context ...mply typed lambda calculus via Girard’s translation as σ → τ = !σ ⊸ τ) by a classical rewriting-theoretic method. Specifically, we appeal to the seminal result by Huet on reduction modulo equivalence =-=[12]-=-. We provide a rewriting system ≻ together with a (trivially) decidable equational theory ∼ ∗ generated by the symmetric commuting conversion ∼ on linear lambda terms such that (following the terminol... |

140 |
P.J.: Introduction to Higher-Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...studied the equality between terms (or proofs) seriously. Just like the simply typed lambda calculus with the βηequality is sound and complete for semantic models given by cartesian closed categories =-=[13,5]-=-, it is desirable for a linear lambda calculus to be equipped with an equational theory which is sound and complete for the now well-established categorical models of linear logic [19,3,4,16]. Barber ... |

102 | Linear logic, *-autonomous categories and cofree coalgebras
- Seely
- 1989
(Show Context)
Citation Context ...ed categories [13,5], it is desirable for a linear lambda calculus to be equipped with an equational theory which is sound and complete for the now well-established categorical models of linear logic =-=[19,3,4,16]-=-. Barber and Plotkin’s Dual Intuitionistic Linear Logic (DILL) [1,2] is one of such calculi: its equational theory, determined by the standard βη-axioms and a few axioms for commuting conversions (for... |

99 | What is a categorical model of intuitionistic linear logic
- Bierman
- 1995
(Show Context)
Citation Context ...ed categories [13,5], it is desirable for a linear lambda calculus to be equipped with an equational theory which is sound and complete for the now well-established categorical models of linear logic =-=[19,3,4,16]-=-. Barber and Plotkin’s Dual Intuitionistic Linear Logic (DILL) [1,2] is one of such calculi: its equational theory, determined by the standard βη-axioms and a few axioms for commuting conversions (for... |

87 |
Autonomous categories and linear logic
- Barr
- 1991
(Show Context)
Citation Context ...ed categories [13,5], it is desirable for a linear lambda calculus to be equipped with an equational theory which is sound and complete for the now well-established categorical models of linear logic =-=[19,3,4,16]-=-. Barber and Plotkin’s Dual Intuitionistic Linear Logic (DILL) [1,2] is one of such calculi: its equational theory, determined by the standard βη-axioms and a few axioms for commuting conversions (for... |

80 |
Categories for Types
- Crole
- 1993
(Show Context)
Citation Context ...studied the equality between terms (or proofs) seriously. Just like the simply typed lambda calculus with the βηequality is sound and complete for semantic models given by cartesian closed categories =-=[13,5]-=-, it is desirable for a linear lambda calculus to be equipped with an equational theory which is sound and complete for the now well-established categorical models of linear logic [19,3,4,16]. Barber ... |

60 | A Reflection on Call-by-Value
- Sabry, Wadler
- 1997
(Show Context)
Citation Context ...giving a reduction-preserving translation into the linear lambda calculus. Together with this result, our work can be considered as a follow-up of the work by Maraist et al. [15] and Sabry and Wadler =-=[18]-=-. The rest of this paper is organized as follows. We introduce the linear lambda calculus in Section 2, and our rewriting system in Section 3. Section 4 is a quick reminder of the classical definition... |

38 |
Linear logic,” Theoretical
- Girard
- 1987
(Show Context)
Citation Context ...duction-preserving translation into the linear lambda calculus. 1 Introduction In the literature, there exist many proposals of linearly typed lambda calculi which correspond to Girard’s linear logic =-=[7]-=- via the Curry-Howard correspondence. However, only a few of them have studied the equality between terms (or proofs) seriously. Just like the simply typed lambda calculus with the βηequality is sound... |

37 |
Linear Type Theories, Semantics and Action Calculi
- Barber
- 1997
(Show Context)
Citation Context ...quipped with an equational theory which is sound and complete for the now well-established categorical models of linear logic [19,3,4,16]. Barber and Plotkin’s Dual Intuitionistic Linear Logic (DILL) =-=[1,2]-=- is one of such calculi: its equational theory, determined by the standard βη-axioms and a few axioms for commuting conversions (for identifying the terms representing the same proof modulo trivial pr... |

28 | Call-by-name, call-by-value, call-by-need and the linear lambda calculus
- Maraist, Ordesky, et al.
- 1995
(Show Context)
Citation Context ...tional lambda calculus by giving a reduction-preserving translation into the linear lambda calculus. Together with this result, our work can be considered as a follow-up of the work by Maraist et al. =-=[15]-=- and Sabry and Wadler [18]. The rest of this paper is organized as follows. We introduce the linear lambda calculus in Section 2, and our rewriting system in Section 3. Section 4 is a quick reminder o... |

23 | Categorical models of linear logic revisited, to appear
- Melliès
- 2009
(Show Context)
Citation Context |

15 | Reducibility and ⊤⊤-lifting for computation types
- Lindley, Stark
(Show Context)
Citation Context ... lambda calculus. ⊓⊔ Corollary 3 (strong normalization). λc is strongly normalizing. ⊓⊔ We note that a different proof of this result via the reducibility argument has been given by Lindley and Stark =-=[14]-=-. Remark 5. For reasoning about commutative effects like non-termination and non-determinism, it makes sense to add the commutativity axiom com let x be L in let y be M in N = let y be M in let x be L... |

13 | Linearly used effects: Monadic and CPS transformations into the linear lambda calculus
- Hasegawa
- 2002
(Show Context)
Citation Context ...ew, this work does not present much new idea. However, it does give an interesting case motivated by the study on the semantic and logical foundations of functional programming languages. Recent work =-=[8,9,10]-=- suggest that there exist many interesting translations of various calculi into this linear lambda calculus, including monadic and CPS translations. As an interesting example, we prove the strong norm... |

10 | Classical linear logic of implications
- Hasegawa
- 2007
(Show Context)
Citation Context ...we only consider the equality on the well-typed terms under the same typing contexts. Thus, for example, in com3, y cannot be free in M;andincom5, x and y cannotbefreeinL and M. Remark 2. As noted in =-=[11]-=-, this linear lambda calculus allows a yet simpler axiomatization: β⊸ (λx.M)N = M[N/x] η⊸ λx.M x = M β! let !x be !M in N = N[M/x] η ′ ! let !x be M in L (!x)=LM While this is very compact, it does no... |

6 |
Dual intuitionistic linear logic (manuscript). An earlier version available as
- Barber, Plotkin
- 1997
(Show Context)
Citation Context ...quipped with an equational theory which is sound and complete for the now well-established categorical models of linear logic [19,3,4,16]. Barber and Plotkin’s Dual Intuitionistic Linear Logic (DILL) =-=[1,2]-=- is one of such calculi: its equational theory, determined by the standard βη-axioms and a few axioms for commuting conversions (for identifying the terms representing the same proof modulo trivial pr... |

6 | Semantics of linear continuation-passing in call-by-name
- Hasegawa
(Show Context)
Citation Context ...ew, this work does not present much new idea. However, it does give an interesting case motivated by the study on the semantic and logical foundations of functional programming languages. Recent work =-=[8,9,10]-=- suggest that there exist many interesting translations of various calculi into this linear lambda calculus, including monadic and CPS translations. As an interesting example, we prove the strong norm... |

4 | Girard translation and logical predicates
- Hasegawa
- 2000
(Show Context)
Citation Context ...ew, this work does not present much new idea. However, it does give an interesting case motivated by the study on the semantic and logical foundations of functional programming languages. Recent work =-=[8,9,10]-=- suggest that there exist many interesting translations of various calculi into this linear lambda calculus, including monadic and CPS translations. As an interesting example, we prove the strong norm... |

3 | Adjoint rewriting and the !-type constructor
- Ghani
- 1998
(Show Context)
Citation Context ... on a rewriting system, while it is the case for many of the standard typed lambda calculi. Regarding decidability, the answer is actually known: Barber [1] in his PhD thesis, and independently Ghani =-=[6]-=- in an unpublished manuscript, have shown that the equational theory of DILL is decidable. However, their proofs are long and complicated, using some new notations and/or advanced techniques which F. ... |