# Quantitative Types for the Linear Substitution Calculus

##### Citations

53 citations

### Cites background or methods from "Quantitative Types for the Linear S..."

...Non-idempotent IT systems were used in [40, 41] to characterize different normalization properties of higher-order calculi....

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...For example, linear-head, head, weak and strong normalization are characterized in [40] by means of appropriate non-idempotent types in the framework of the linear substitution calculus [1], a calculus with explicit substitution at a distance....

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34 citations

### Cites background from "Quantitative Types for the Linear S..."

...The second one, called A (for answers), characterizes the set of need-normalizing terms; it extends the one in [24] which characterizes the set of head-linear normalizing terms in λ-calculus with explicit substitutions....

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...The same remark applies to the rule (ans) in system A with respect to the one in [24]....

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...Different assignment systems with nonidempotent intersection types have been studied in the literature for different purposes [8, 10, 18, 19, 24, 26, 27, 30, 35, 34]....

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31 citations

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##### References

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139,059 citations

### "Quantitative Types for the Linear S..." refers background in this paper

...[21]) which allows to obtain partial typing inference algorithms [40, 39, 31] and exact bounds for termination (cf....

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2,304 citations

### "Quantitative Types for the Linear S..." refers background or methods in this paper

...Relationship with Linear Logic [24] and Relevant Logic [23, 18] provides an insight on the information refinement as-...

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...Another key subrelation studied in this paper is linear-head reduction [19, 35], a strategy related to abstract machines [19] and linear logic [24]....

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...Relationship with Linear Logic [24] and Relevant Logic [23, 18] provides an insight on the information refinement aspect of non-idempotent intersection types....

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...But calculi with ES can also be interpreted in Linear Logic [22, 28, 26, 5] by implementing another kind of operational semantics: their dynamics is defined using contexts (i.e. terms with holes) that allows the ES to act directly at a distance on single variable occurrences, with no need to commute with any other constructor in between....

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1,480 citations

##### Related Papers (5)

##### Frequently Asked Questions (18)

###### Q2. What are the future works mentioned in the paper "Quantitative types for the linear substitution calculus" ?

This remains as future work.

###### Q3. What is the key subrelation studied in this paper?

Another key subrelation studied in this paper is linear-head reduction [19, 35], a strategy related to abstract machines [19] and linear logic [24].

###### Q4. What is the type of a term that is not a terminal term?

The HW-type system is known to type also some non weakly M-normalizing terms: for instance, if Ω is any non-terminating term, then x:[ ]→σ `HW xΩ:σ.

###### Q5. What makes the type systems for strong normalization particularly natural?

the type systems for strong normalization make use of a special notion of witness derivation for the arguments (of applications and explicit substitutions) which makes them particularly natural.

###### Q6. What is the IE property for the termination properties of the linear substitution calculus?

Although type inference is undecidable for any system characterizing termination properties, semi-decidable restrictions are expected to hold.

###### Q7. How can the authors obtain the pushed out list context L in the rule dB?

The pushed out list context L in rule dB can be obtained by using an equivalence related to Regnier’s σ-equivalence [38]: L[λx.t]u ∼σ L[(λx.t)u] →dB L[t[x/u]].

###### Q8. What is the way to solve the inhabitation problem for non-idempotent type?

An interesting challenge would be relax the notion of linear types in order to gain expressivity while staying in a different class.

###### Q9. What is the purpose of this paper?

This paper focuses on functional programs specified – via the Curry-Howard isomorphism – by intuitionistic logic, in natural deduction style.

###### Q10. What is the inhabitation property for -calculus?

The HW-system also enjoys the inhabitation property for λ-calculus [12], which is a proper sub-calculus of the linear substitution calculus.

###### Q11. What is the simplest way to recover a type of term?

Types of terms can also be recovered by means of Subject Expansion (SE), a property which will be particularly useful in Sec. 3.1 and 3.2.Lemma 3 (SE I).

###### Q12. What is the rewriting rule for tt?

The M-calculus is given by the set of terms TM and the reduction relation→dB∪c∪w, the union of→dB,→c, and→w, denoted by→M, which are, respectively, the closure by term contexts C of the following rewriting rules:L[λx.t]u 7→dB L[t[x/u]] C[[x]][x/u] 7→c C[[u]][x/u] t[x/u] 7→w t if |t|x =

###### Q13. What is the inhabitation problem for non-idempotent type systems?

Last but not least, the inhabitation problem for idempotent intersection types in the λ-calculus is known to be undecidable [41], while the problem was recently shown to be decidable in the non-idempotent case [12].

###### Q14. what is the smallest set of positive subtypes of a type?

the set of positive (resp. negative) subtypes of a type is the smallest set satisfying the following conditions (cf.[13]).– A ∈ P(A). – A ∈ P([σi]i∈I) if ∃i A ∈ P(σi); A ∈ N ([σi]i∈I) if ∃i A ∈ N (σi).

###### Q15. What is the definition of intersection types?

In particular, intersection types allow to characterize head/weakly/strongly normalizing terms, i.e. a term t is typable in an intersection type system iff t is head/weakly/strongly normalizing; quantitative information about the behaviour of programs can also be obtained if the intersection types enjoy non-idempotence.

###### Q16. What is the connection between the linear-head reduction and the -calculus?

This is related to a recent result [3] stating that linear-head reduction is standard for the M-calculus, exactly as left-to-right reduction is standard for the λ-calculus.

###### Q17. What is the dB-redex occurrence of t?

given p ∈ pos(t), p is said to be a dB-redex occurrence of t if t|p = L[λx.t]u, p is a w-redex occurrence of t if t|p = v[x/u] with |v|x = 0, and p is a c-redex occurrence of t if p = p1p2, t|p1 = C[[x]][x/u] and C|p2 = 2.

###### Q18. What is the difference between calculi with es and intersection types?

But calculi with ES can also be interpreted in Linear Logic [22, 28, 26, 5] by implementing another kind of operational semantics: their dynamics is defined using contexts (i.e. terms with holes) that allows the ES to act directly at a distance on single variable occurrences, with no need to commute with any other constructor in between.