## A Linearization of the Lambda-Calculus and Consequences (2000)

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### BibTeX

@MISC{Kfoury00alinearization,

author = {A.J. Kfoury},

title = {A Linearization of the Lambda-Calculus and Consequences},

year = {2000}

}

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### Abstract

We embed the standard #-calculus, denoted #, into two larger #-calculi, denoted # # and &# # . The standard notion of #-reduction for # corresponds to two new notions of reduction, # # for # # and &# # for &# # . A distinctive feature of our new calculus # # (resp., &# # ) is that, in every function application, an argument is used at most once (resp., exactly once) in the body of the function. We establish various connections between the three notions of reduction, #, # # and &# # . As a consequence, we provide an alternative framework to study the relationship between #-weak normalization and #-strong normalization, and give a new proof of the oft-mentioned equivalence between #-strong normalization of standard #-terms and typability in a system of "intersection types".

### Citations

634 |
A Mathematical Introduction To Logic
- Enderton
- 1972
(Show Context)
Citation Context ...op-down given in 2.5 when restricted to well-formed expanded #-terms. (For suggestions on how to formally prove the equivalence of the two definitions, see the section on "Induction and Recursion=-=" in [9]-=- pp. 22-30.) Definition 2.5 does not require that expanded #-terms be wellformed, while Definition 2.8 does. 9 Definition 2.9 (Nesting of parallel sets) Let M # # # be well-formed, and P = {P 1 , . . ... |

256 |
Combinatory Reduction Systems
- Klop
- 1980
(Show Context)
Citation Context ...any of its residuals in the course of evaluating an expression. But this is not the only way of preventing "information loss" in the course of an evaluation. In historical order, Nederpelt [=-=23], Klop [19]-=-, Karr [13], Khasidashvili [18], de Groote [8], Kfoury and Wells [14, 15], Srensen [27], and Xi [30], among several others, present di#erent techniques for deducing strong normalization from weak norm... |

203 |
The Lambda Calculus: its Syntax and Semantics (revised ed
- Barendregt
- 1984
(Show Context)
Citation Context ...decreasing. Part 2 implies that M has at most one # # -nf and, together with part 1, that M has exactly one # # -nf, thus proving part 3. It remains to prove part 2. In fact, by Proposition 3.1.25 in =-=[2]-=-, it su#ces to show that # # is WCR (weak Church-Rosser), i.e. M # # # # # # # # M2 M3 M1 R2 R1 where R 1 and R 2 are # # -redex occurrences in M : R 1 = { ((#x.P 1 ).Q 1,1 #s# Q 1,m1 ), . . . , ((#x.... |

161 | logic: its syntax and semantics
- Linear
- 1995
(Show Context)
Citation Context ...entary to our analysis in this report. In addition to Boudol's work, there is the vast literature on linear logic. For this kind of linearity, the reader is referred to two of Girard's seminal papers =-=[11, 12]-=- and the references therein. Even though connections between linear logic and our work in this report are not immediately perceived, it will be a useful investigation (left to others) whether results ... |

128 |
Types et Modèles
- Lambda-Calcul
- 1990
(Show Context)
Citation Context ....11. N # is the last &-list in both t and u. Hence, as M is &#-reduced to the &#-nf N # , M is &#-normalizing and, by Theorem 3.6, M is #-SN. An oft-mentioned result in the literature (e.g. see [10], =-=[20]-=-, [21], [24], [25], [28], [29], and the references cited therein) is that a standard #-term M is #-SN i# M is typable in (an appropriate formulation of) the system of intersection types. These referen... |

105 | Complete restrictions of the intersection type discipline
- Bakel
- 1992
(Show Context)
Citation Context ...ist in both t and u. Hence, as M is &#-reduced to the &#-nf N # , M is &#-normalizing and, by Theorem 3.6, M is #-SN. An oft-mentioned result in the literature (e.g. see [10], [20], [21], [24], [25], =-=[28]-=-, [29], and the references cited therein) is that a standard #-term M is #-SN i# M is typable in (an appropriate formulation of) the system of intersection types. These references in fact provide corr... |

66 | The π-calculus in direct style
- Boudol
- 1998
(Show Context)
Citation Context ...iding "better" resource-management in the evaluation of terms. 3 Particularly noteworthy is the work of Boudol and his colleagues on the lambda calculus with multiplicities. The reader is re=-=ferred to [3, 4, 5, 6]-=-, among other papers dealing with Boudol's approach to linearity. Of these 4 papers, the first 2 stress semantic rather than combinatorial aspects of the proposed calculi; nevertheless, they 3 We purp... |

56 |
Strong normalization in a typed lambda calculus with lambda structured types
- Nederpelt
- 1973
(Show Context)
Citation Context ...ression or any of its residuals in the course of evaluating an expression. But this is not the only way of preventing "information loss" in the course of an evaluation. In historical order, =-=Nederpelt [23]-=-, Klop [19], Karr [13], Khasidashvili [18], de Groote [8], Kfoury and Wells [14, 15], Srensen [27], and Xi [30], among several others, present di#erent techniques for deducing strong normalization fro... |

46 | Intersection Type Disciplines in Lambda Calculus and Applicative Term Rewriting Systems
- Bakel
- 1993
(Show Context)
Citation Context ... both t and u. Hence, as M is &#-reduced to the &#-nf N # , M is &#-normalizing and, by Theorem 3.6, M is #-SN. An oft-mentioned result in the literature (e.g. see [10], [20], [21], [24], [25], [28], =-=[29]-=-, and the references cited therein) is that a standard #-term M is #-SN i# M is typable in (an appropriate formulation of) the system of intersection types. These references in fact provide correct an... |

35 |
Typing and computation properties of lambda expressions
- Leivant
- 1986
(Show Context)
Citation Context ... # is the last &-list in both t and u. Hence, as M is &#-reduced to the &#-nf N # , M is &#-normalizing and, by Theorem 3.6, M is #-SN. An oft-mentioned result in the literature (e.g. see [10], [20], =-=[21]-=-, [24], [25], [28], [29], and the references cited therein) is that a standard #-term M is #-SN i# M is typable in (an appropriate formulation of) the system of intersection types. These references in... |

35 |
A type assignment for the strongly normalizable terms
- Pottinger
- 1980
(Show Context)
Citation Context ...the last &-list in both t and u. Hence, as M is &#-reduced to the &#-nf N # , M is &#-normalizing and, by Theorem 3.6, M is #-SN. An oft-mentioned result in the literature (e.g. see [10], [20], [21], =-=[24]-=-, [25], [28], [29], and the references cited therein) is that a standard #-term M is #-SN i# M is typable in (an appropriate formulation of) the system of intersection types. These references in fact ... |

31 | Principal type scheme and unification for intersection type discipline - Rocca - 1988 |

29 | A complete type inference algorithm for simple intersection types - Coppo, Giannini - 1992 |

26 |
Typing the use of resources in a concurrent calculus
- Boudol
(Show Context)
Citation Context ...iding "better" resource-management in the evaluation of terms. 3 Particularly noteworthy is the work of Boudol and his colleagues on the lambda calculus with multiplicities. The reader is re=-=ferred to [3, 4, 5, 6]-=-, among other papers dealing with Boudol's approach to linearity. Of these 4 papers, the first 2 stress semantic rather than combinatorial aspects of the proposed calculi; nevertheless, they 3 We purp... |

21 | Strong normalization and typability with intersection types
- Ghilezan
- 1996
(Show Context)
Citation Context ...tion 3.11. N # is the last &-list in both t and u. Hence, as M is &#-reduced to the &#-nf N # , M is &#-normalizing and, by Theorem 3.6, M is #-SN. An oft-mentioned result in the literature (e.g. see =-=[10]-=-, [20], [21], [24], [25], [28], [29], and the references cited therein) is that a standard #-term M is #-SN i# M is typable in (an appropriate formulation of) the system of intersection types. These r... |

18 | Addendum to new notions of reduction and non-semantic proofs of β-strong normalisation in typed λ-calculi - Kfoury, Wells |

15 | A semantics for lambda calculi with resources - Boudol, Curien, et al. - 1999 |

13 | Beta-reduction as unification
- Kfoury
- 1996
(Show Context)
Citation Context ... 26 #-SN #-term M in the type inferred by # for M . This is reflected in the fact that: (#) A standard #-term M is #-SN i#f M is typable in #. A discussion of this and related results can be found in =-=[17]-=-. System # is presented below again, along with 3 other type inference systems: # # , &# and &# # , which are special adaptations of # for the expanded #-calculi # # , &# and &# # , defined in Section... |

10 |
Strong normalization from weak normalization in typed -calculi
- S��rensen
- 1997
(Show Context)
Citation Context ...y way of preventing "information loss" in the course of an evaluation. In historical order, Nederpelt [23], Klop [19], Karr [13], Khasidashvili [18], de Groote [8], Kfoury and Wells [14, 15]=-=, Srensen [27]-=-, and Xi [30], among several others, present di#erent techniques for deducing strong normalization from weak normalization, all inspired by one or both of these two simple ideas: . By-pass erasing ste... |

7 |
Delayability" in proofs of strong normalisabilities in the typed lambda-calculus
- Karr
- 1985
(Show Context)
Citation Context ...residuals in the course of evaluating an expression. But this is not the only way of preventing "information loss" in the course of an evaluation. In historical order, Nederpelt [23], Klop [=-=19], Karr [13]-=-, Khasidashvili [18], de Groote [8], Kfoury and Wells [14, 15], Srensen [27], and Xi [30], among several others, present di#erent techniques for deducing strong normalization from weak normalization, ... |

7 |
Addendum to "New notions of reduction and non-semantic proofs of fi-strong normalization in typed -calculi
- Kfoury, Wells
- 1995
(Show Context)
Citation Context ...his is not the only way of preventing "information loss" in the course of an evaluation. In historical order, Nederpelt [23], Klop [19], Karr [13], Khasidashvili [18], de Groote [8], Kfoury =-=and Wells [14, 15]-=-, Srensen [27], and Xi [30], among several others, present di#erent techniques for deducing strong normalization from weak normalization, all inspired by one or both of these two simple ideas: . By-pa... |

6 |
Linear logic", in Theoretical
- Girard
- 1987
(Show Context)
Citation Context ...entary to our analysis in this report. In addition to Boudol's work, there is the vast literature on linear logic. For this kind of linearity, the reader is referred to two of Girard's seminal papers =-=[11, 12]-=- and the references therein. Even though connections between linear logic and our work in this report are not immediately perceived, it will be a useful investigation (left to others) whether results ... |

6 | J.B.: New notions of reduction and non-semantic proofs of beta-strong normalization in typed lambda-calculi
- Kfoury, Wells
- 1995
(Show Context)
Citation Context ...his is not the only way of preventing "information loss" in the course of an evaluation. In historical order, Nederpelt [23], Klop [19], Karr [13], Khasidashvili [18], de Groote [8], Kfoury =-=and Wells [14, 15]-=-, Srensen [27], and Xi [30], among several others, present di#erent techniques for deducing strong normalization from weak normalization, all inspired by one or both of these two simple ideas: . By-pa... |

5 |
Form Reduction Systems and Reductions of Contracted Forms and Lambda-Terms
- Khasidashvili
- 1988
(Show Context)
Citation Context ...rse of evaluating an expression. But this is not the only way of preventing "information loss" in the course of an evaluation. In historical order, Nederpelt [23], Klop [19], Karr [13], Khas=-=idashvili [18]-=-, de Groote [8], Kfoury and Wells [14, 15], Srensen [27], and Xi [30], among several others, present di#erent techniques for deducing strong normalization from weak normalization, all inspired by one ... |

3 | A Note on Intersection Types
- Retore
- 1995
(Show Context)
Citation Context ...st &-list in both t and u. Hence, as M is &#-reduced to the &#-nf N # , M is &#-normalizing and, by Theorem 3.6, M is #-SN. An oft-mentioned result in the literature (e.g. see [10], [20], [21], [24], =-=[25]-=-, [28], [29], and the references cited therein) is that a standard #-term M is #-SN i# M is typable in (an appropriate formulation of) the system of intersection types. These references in fact provid... |

3 |
Complete restrictions of the Intersection Type Discipline
- van
- 1992
(Show Context)
Citation Context ... last &-list in both t and u. Hence,as M is &β-reduced to the &β-nf N ′, M is &β-normalizing and,by Theorem 3.6,M is β-SN. An oft-mentioned result in the literature (e.g. see [10],[20],[21],[24],[25],=-=[28]-=-, [29],and the references cited therein) is that a standard λ-term M is β-SN iff M is typable in (an appropriate formulation of) the system of intersection types. These references in fact provide corr... |

2 |
Weak and strong beta normalisations in typed -calculi
- Xi
- 1997
(Show Context)
Citation Context ...enting "information loss" in the course of an evaluation. In historical order, Nederpelt [23], Klop [19], Karr [13], Khasidashvili [18], de Groote [8], Kfoury and Wells [14, 15], Srensen [27=-=], and Xi [30]-=-, among several others, present di#erent techniques for deducing strong normalization from weak normalization, all inspired by one or both of these two simple ideas: . By-pass erasing steps, i.e. devi... |

2 | Principal Type Schemes and Uni cation for Intersection Type Discipline - Rocca, S - 1988 |

2 | Strong Normalization from Weak Normalization in Typed Lambda Calculi", Report from - rensen, H - 1996 |

1 |
The Discriminating Power of Multiplicities in the Lambda-Calculus
- Boudol, Lavatelli
- 1996
(Show Context)
Citation Context ...iding "better" resource-management in the evaluation of terms. 3 Particularly noteworthy is the work of Boudol and his colleagues on the lambda calculus with multiplicities. The reader is re=-=ferred to [3, 4, 5, 6]-=-, among other papers dealing with Boudol's approach to linearity. Of these 4 papers, the first 2 stress semantic rather than combinatorial aspects of the proposed calculi; nevertheless, they 3 We purp... |

1 |
The Conservation Theorem Revisited", Typed Lambda Calculi and
- Groote
- 1993
(Show Context)
Citation Context ...g an expression. But this is not the only way of preventing "information loss" in the course of an evaluation. In historical order, Nederpelt [23], Klop [19], Karr [13], Khasidashvili [18], =-=de Groote [8]-=-, Kfoury and Wells [14, 15], Srensen [27], and Xi [30], among several others, present di#erent techniques for deducing strong normalization from weak normalization, all inspired by one or both of thes... |

1 |
A Linearization of the Lambda-Calculus and Consequneces
- Kfoury
- 1995
(Show Context)
Citation Context ...on steps. Proof It su#ces to prove the following simpler commutative diagram: M 1 M 3 M 0 M 2 # # # # # # This is a tedious case analysis. Details are in the Appendix of the full report of this paper =-=[16]-=-. In contrast to Lemma 2.20, it is not the case that we can always displace # # - reduction steps ahead of expansion steps. Consider for example the sequence: ((#x.xxx).I # I) ---# # ((#x.xxx).I # I #... |

1 |
available at: http://people.bu.edu/turtle/academia.html#master-thesis ftp://ftp.diku.dk/diku/semantics/papers/D-399.ps.gz
- Neerdgaard, Weak, et al.
- 1999
(Show Context)
Citation Context ...tions of the second. For a fuller account of known results and some of the alternative approaches on this topic, the reader is referred to the highly readable master's thesis by Peter Mller Neergaard =-=[22]-=-. 4 Acknowledgements Joe Wells played a crucial role in the early stages of the research, by proofreading numerous handwritten drafts and correcting many (sometimes serious) mistakes in them. Although... |

1 |
W.,Combinatory Reduction Systems,PhD thesis,Utrecht University
- Klop
- 1980
(Show Context)
Citation Context ...r any of its residuals in the course of evaluating an expression. But this is not the only way of preventing “information loss” in the course of an evaluation. In historical order,Nederpelt [23],Klop =-=[19]-=-, Karr [13],Khasidashvili [18],de Groote [8],Kfoury and Wells [14,15],Sørensen [27], and Xi [30],among several others,present different techniques for deducing strong normalization from weak normaliza... |

1 |
Types et Modèles,Masson,Paris,1990
- Krivine, Lambda-calcul
(Show Context)
Citation Context ...ion 3.11. N ′ is the last &-list in both t and u. Hence,as M is &β-reduced to the &β-nf N ′, M is &β-normalizing and,by Theorem 3.6,M is β-SN. An oft-mentioned result in the literature (e.g. see [10],=-=[20]-=-,[21],[24],[25],[28], [29],and the references cited therein) is that a standard λ-term M is β-SN iff M is typable in (an appropriate formulation of) the system of intersection types. These references ... |

1 |
Strong Beta Normalisations in Typed λ-calculi
- Xi, “Weak
- 1997
(Show Context)
Citation Context ...f preventing “information loss” in the course of an evaluation. In historical order,Nederpelt [23],Klop [19], Karr [13],Khasidashvili [18],de Groote [8],Kfoury and Wells [14,15],Sørensen [27], and Xi =-=[30]-=-,among several others,present different techniques for deducing strong normalization from weak normalization,all inspired by one or both of these two simple ideas: • By-pass erasing steps,i.e. devise ... |