## 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".

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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 , . . ... |

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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... |

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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.... |

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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 ... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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 ... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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, ... |

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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... |

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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 | New notions of reduction and non-semantic proofs of beta-strong normalisation in typed lambda-calculi
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- 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... |

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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 ... |

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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 |
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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 |
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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... |

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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 |
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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
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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
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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 |
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(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
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(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
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(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 ... |