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...esign of indexed versions of known named calculi. 1 Introduction This article is devoted to explicit substitutions (ES, for short), a formalism that has attracted attention since the appearance of λσ =-=[1]-=- and, later, of Melliès’ counterexample [17], showing the lack of the preservation of β -strong normalization property (PSN, for short) in λσ . One of the main motivations behind the field of ES is s...

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... such a task hard. In this direction, a mix of ideas from λ rex and calculi with n-ary substitutions (i.e., λσ -styled calculi) may lead to the solution of both issues. Particularly, a swap-based λσ⇑ =-=[7]-=- could be an option. This comes from the following observation: in λσ⇑, the (Lamb) rule is: (Lamb) (λa)[s]→ λa[⇑(s)] where the intuitive semantics of ⇑(s) is: 1 ·(s ◦↑). We observe here that this is n...

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...pendently discovered, the swapping mechanism introduced in this article was first depicted by de Bruijn for his ES calculus Cλξφ [6], and, later, updated w.r.t. notation – λξφ – and compared to λυ in =-=[2]-=-. We will now briefly discuss the main differences between these calculi and our swapping-based approach. Firstly, neither Cλξφ nor λξφ have composition of substitutions nor Garbage Collection, two ke...

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...less notion makes sense in the classical λdB-calculus (because the substitution operator is located in the metalevel), it seems not to be the case in certain ES calculi derived from λdB, such as: λ s =-=[11]-=-, λ se [12] or λ t [13]. These calculi have constructions of the form a[i := b] to denote ES (notations vary). Here, even though i is not a name per se, it plays a similar role: i indicates which free...

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... first devised λ r, and then made substitutions explicit orienting the definition for λ r’s meta-substitution. At that point, we got a calculus we called λ re, which turned out to be isomorphic to λx =-=[4, 3]-=-. Encouraged by this result, we added Garbage Collection to λ re, obtaining a calculus isomorphic to λxgc [4]: λ regc. Finally, we added composition of substitutions in the style of λex to λ regc, obt...

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... first devised λ r, and then made substitutions explicit orienting the definition for λ r’s meta-substitution. At that point, we got a calculus we called λ re, which turned out to be isomorphic to λx =-=[4, 3]-=-. Encouraged by this result, we added Garbage Collection to λ re, obtaining a calculus isomorphic to λxgc [4]: λ regc. Finally, we added composition of substitutions in the style of λex to λ regc, obt...

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...le to λ rex should be fairly simple using the decrement meta-operator. Finally, studying the possible relation between these swapping-based formalisms and nominal logic or nominal rewriting (see e.g. =-=[10, 9]-=-) could be an interesting approach in gathering a deeper understanding of λ r’s underlying logic. 10 Swapping: a natural bridge between named and indexed explicit substitution calculi Acknowledgements...

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...culi. 1 Introduction This article is devoted to explicit substitutions (ES, for short), a formalism that has attracted attention since the appearance of λσ [1] and, later, of Melliès’ counterexample =-=[17]-=-, showing the lack of the preservation of β -strong normalization property (PSN, for short) in λσ . One of the main motivations behind the field of ES is studying how substitution behaves when interna...

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... makes sense in the classical λdB-calculus (because the substitution operator is located in the metalevel), it seems not to be the case in certain ES calculi derived from λdB, such as: λ s [11], λ se =-=[12]-=- or λ t [13]. These calculi have constructions of the form a[i := b] to denote ES (notations vary). Here, even though i is not a name per se, it plays a similar role: i indicates which free variable s...

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... a priority, because the main merit of λ rex is evidencing the accessory nature of index updates. From a different perspective, an attempt to use λ rex in proof assistants or higher order unification =-=[8]-=- implementations may be taken into account. In such a case, a typed version of λ rex should be developed as well. Also, adding an η rule to λ rex should be fairly simple using the decrement meta-opera...

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... in the classical λdB-calculus (because the substitution operator is located in the metalevel), it seems not to be the case in certain ES calculi derived from λdB, such as: λ s [11], λ se [12] or λ t =-=[13]-=-. These calculi have constructions of the form a[i := b] to denote ES (notations vary). Here, even though i is not a name per se, it plays a similar role: i indicates which free variable should be sub...

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...le to λ rex should be fairly simple using the decrement meta-operator. Finally, studying the possible relation between these swapping-based formalisms and nominal logic or nominal rewriting (see e.g. =-=[10, 9]-=-) could be an interesting approach in gathering a deeper understanding of λ r’s underlying logic. 10 Swapping: a natural bridge between named and indexed explicit substitution calculi Acknowledgements...

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...en shown to have a whole set of desirable properties: simulation of β -reduction, PSN, meta-confluence, full composition, etc. For a detailed introduction to the ES field, we refer the reader to e.g. =-=[16, 15, 20]-=-. In 2008, D. Kesner proposed λex [14, 15], a formalism with variable names that has the entire set of properties expected from an ES calculus. As Kesner points in [15], for implementation purposes a ...

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...d variable renaming (i.e., working modulo α-equivalence) is known to be error-prone and computationally expensive. Among others, one of the ways this problem is tackled is by using de Bruijn notation =-=[5]-=-, which is a technique that simply avoids the need of working modulo α-equivalence. As far as we know, no ES calculus with de Bruijn indexes and the whole set of properties enjoyed by λex exists to da...

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...erties: simulation of β -reduction, PSN, meta-confluence, full composition, etc. For a detailed introduction to the ES field, we refer the reader to e.g. [16, 15, 20]. In 2008, D. Kesner proposed λex =-=[14, 15]-=-, a formalism with variable names that has the entire set of properties expected from an ES calculus. As Kesner points in [15], for implementation purposes a different approach to variable names shoul...

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...SNλex. 5 Related work It is important to mention that, even though independently discovered, the swapping mechanism introduced in this article was first depicted by de Bruijn for his ES calculus Cλξφ =-=[6]-=-, and, later, updated w.r.t. notation – λξφ – and compared to λυ in [2]. We will now briefly discuss the main differences between these calculi and our swapping-based approach. Firstly, neither Cλξφ n...