## Complexity of strongly normalising λ-terms via non-idempotent intersection types

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@MISC{Bernadet_complexityof,

author = {Alexis Bernadet and Stéphane Lengrand},

title = {Complexity of strongly normalising λ-terms via non-idempotent intersection types},

year = {}

}

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### Abstract

We present a typing system for the λ-calculus, with non-idempotent intersection types. As it is the case in (some) systems with idempotent intersections, a λ-term is typable if and only if it is strongly normalising. Nonidempotency brings some further information into typing trees, such as a bound on the longest β-reduction sequence reducing a term to its normal form. We actually present these results in Klop’s extension of λ-calculus, where the bound that is read in the typing tree of a term is refined into an exact measure of the longest reduction sequence. This complexity result is, for longest reduction sequences, the counterpart of de Carvalho’s result for linear head-reduction sequences.

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Citation Context ...ng system that we are going to use. 2.1 Lambda Calculus with Klop’s extension As said in the introduction, the language that we are going to type is the pure λ-calculus extended with Klop’s construct =-=[Klo80]-=-:Definition 1 (Syntax and reduction rules). – Terms are defined by the following grammar M, N ::= x | λx.M | MN | [M, N] Unless otherwise stated, we do not consider any restriction on this syntax. So... |

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Citation Context ...sequence. This complexity result is, for longest reduction sequences, the counterpart of de Carvalho’s result for linear head-reduction sequences. 1 Introduction Intersection types were introduced in =-=[CD78]-=-, extending the simply-typed λcalculus with a notion of finite polymorphism. This is achieved by a new construct A ∩ B in the syntax of types and new typing rules such as: M : A M : B M : A ∩ B where ... |

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Citation Context ... intersection types have been used to provide model-based proofs of strong normalisation for well-known typing systems (simple types, system F, system Fω,. . . ,). Such model constructions (I-filters =-=[CS07]-=-, orthogonality) can also be done with non-idempotent intersection types with no increased difficulty, and with the extra advantage that the strong normalisation of terms in the models is much simpler... |

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Citation Context ...ot bound by binders in M. Remark 2. If M is a λI term then every reduction is in ↩−−→ɛ. 4 This motivates our choice of sticking to the reduction rules of [Klo80] rather than opting for the variant in =-=[Bou03]-=- where β-reduction can generate new instances of Klop’s construct.2.2 Intersection types and contexts Definition 4 (Intersection types). Our intersection types are defined by the following grammar : ... |

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Citation Context ...re λ-calculus the longest reduction sequences, and show that their lengths are exactly those that can be predicted in λI. For the longest reduction sequences we simply use the perpetual strategy from =-=[vRSSX99]-=-, shown in Fig. 4. x ∈ fv(t) or t ′ is a β-normal form t ′ � t ′′ x /∈ fv(t) (λx.t) t ′ −→ tj � t{x := t ′ } −→ tj t � t ′ x −→ tj t −→ pj � x −→ tj t ′ −→ pj (λx.t) t ′ −→ tj � (λx.t) t ′′ −→ tj t � ... |

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Citation Context ...o [Sør97,Xi97] for a survey on different techniques based on the λI-calculus to infer normalisation properties. Intersection types in the framework of Church-Klop’s calculus have been studied in e.g. =-=[DCT07]-=-, but, to our knowledge, they were always considered idempotent. Hence, the quantitative analysis provided by non-idempotency was not carried out. In order to obtain the complexity result we want, we ... |

1 | Induction principles as the foundation of the the - Lengrand |

1 |
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Citation Context ... would have to deal with two or three cases instead of one. – Notice that, by construction, if A → B is a type then B is not an intersection. This limitation, which corresponds to the strict types of =-=[vB92]-=-, is useful for the key property of separation (Lemma 1). Definition 5 (Type inclusion). We define inclusion on types as follows: U ⊆ V if V = ω, or U = V , or U = A and V = B with A = B ∩ C for some ... |