Proof Nets and Explicit Substitutions
25 Mar 2000-pp 63-81
TL;DR: The simulation technique introduced in [10] is refined to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets and a version of typed λl with named variables is proposed which helps to better understand the complex mechanism of the explicit weakening notation.
Abstract: We refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the λl-calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed λl to proof nets. Finally, we propose a version of typed λl with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the λl-calculus with de Bruijn indices [8].
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Posted Content•
TL;DR: The rationalization consists of the elimination of a practically inconsequential flexibility in the unravelling of substitutions that has the inadvertent side effect of losing contextual information in terms; the modified calculus has a structure that naturally supports logical analyses, such as ones related to the assignment of types, over lambda terms.
Abstract: This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this calculus provides a cumbersome encoding of substitution composition, an operation that is important to the efficient realization of reduction. This encoding is simplified here, resulting in a treatment that is easy to use directly in applications. The rationalization consists of the elimination of a practically inconsequential flexibility in the unravelling of substitutions that has the inadvertent side effect of losing contextual information in terms; the modified calculus now has a structure that naturally supports logical analyses, such as ones related to the assignment of types, over lambda terms. The overall calculus is shown to have pleasing theoretical properties such as a strongly terminating sub-calculus for substitution and confluence even in the presence of term meta variables that are accorded a grafting interpretation. Another contribution of the paper is the identification of a broad set of properties that are desirable for explicit substitution calculi to support and a classification of a variety of proposed systems based on these. The suspension calculus is used as a tool in this study. In particular, mappings are described between it and the other calculi towards understanding the characteristics of the latter.
1 citations
Cites background or methods from "Proof Nets and Explicit Substitutio..."
...In establishing this property, we adopt the method used in [Curien et al. 1996] to demonstrate that the λσ-calculus is confluent....
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...However, straightforward additions to the rule set suffice to regain this property [Curien et al. 1996]; see also [Dowek et al. 2000] for a system closer in form to the one discussed in this paper....
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...The following lemma, proved in [Curien et al. 1996], is a critical part of the argument....
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Dissertation•
01 Jan 2009
TL;DR: The final section of the static theory for kind bigraphs defines some operations and decompositions of pureBigraphs for kindbigraphs and defines an ion as in the pure theory except that the site can contain all controls that the node can.
Abstract: kind bigraphs An abstract kind bigraph is a lean-support equivalence class of concrete kind bigraphs. We can define the category Big(ΣK) (resp. Bigh(ΣK)) of abstract kind (hard) bigraphs as having the same objects as B́ig(ΣK) (resp. B́igh(ΣK)), where its arrows are leansupport equivalence classes of concrete kind bigraphs. For any signature K, we have quotient functors [[·]] : B́ig(ΣK) → Big(ΣK) and [[·]] : B́igh(ΣK) → Bigh(ΣK) sending a concrete (hard) kind bigraph to its lean-support equivalence class. ground bigraphs A ground bigraph is one with inner face ǫ = 〈0, 〈〉, ∅〉. prime interfaces, bigraphs An interface I = 〈m, ~ θ,X〉 is prime if it has width m = 1. A prime kind bigraph has no inner names and a prime outer face. merge The definition of the prime mergem is modified to exactly satisfy K2. It is defined as mergem : 〈m, ~ θ, ∅〉 → 〈1, 〈θ〉, ∅〉 where θ = ⋃ i∈m θi. mergem has no nodes and maps m sites to a single root which can contain the union of the controls that its children can contain. mergem is opcartesian. wirings and discreteness Wirings and discreteness are as before and pertain only to link graphs. ions, atoms, molecules We define an ion as in the pure theory except that the site can contain all controls that the node can. For any non-atomic control K with arity k and sequence ~x of k distinct names, we define the discrete ion Kv,~x : 〈1, 〈kind(K)〉, ∅〉 → 〈1, 〈{K}〉, ~x〉 to have a single K-node v, whose ports are severally linked to ~x. The site is a child of the node. For atomic K, a discrete atom is Kv,~x : ǫ → 〈1, 〈{K}〉, ~x〉, again containing a single K-node v whose ports are severally linked to ~x. For any prime discrete P with outer face 〈1, 〈θ〉, Y 〉 = 〈1, 〈kind(K)〉, Y 〉 we call (Kv,~x⊗ idY )◦P a discrete molecule. Ions, atoms, and molecules are defined to be discrete and opcartesian. Arbitrary (non-discrete, nearly opcartesian) ions, molecules and atoms are constructed by precomposing by ω⊗J↑I, where ω is a wiring, with a discrete ion, molecule, or atom. Our final section of the static theory for kind bigraphs defines some operations and decompositions of pure bigraphs for kind bigraphs.
1 citations
Cites background from "Proof Nets and Explicit Substitutio..."
...On the basis of this observation, close relations have been established between normalisation of some ES calculi and cut elimination in Linear Logic’s proof nets by translating typable terms from the former into nets of the latter [52, 80, 78]....
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...It has yielded normalisation proofs for existing explicit substitution calculi [51, 52] as well as aided the development of new calculi with some or all of the properties of preservation of strong normalisation (PSN), open confluence, and full composition of substitutions [80, 78]....
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Posted Content•
TL;DR: The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic, and shows that the LSC is a complexity-preserving abstraction of abstract machines.
Abstract: It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between big-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.
1 citations
Posted Content•
TL;DR: A general proof technique of SN is formalized which consists in expanding substitutions into "pure" lambda-terms and to inherit SN of the whole calculus and by PSN, allowing to establish SN or, at least, to trace back the failure of SN to that of PSN.
Abstract: In the framework of explicit substitutions there is two termination properties: preservation of strong normalization (PSN), and strong normalization (SN). Since there are not easily proved, only one of them is usually established (and sometimes none). We propose here a connection between them which helps to get SN when one already has PSN. For this purpose, we formalize a general proof technique of SN which consists in expanding substitutions into "pure" lambda-terms and to inherit SN of the whole calculus by SN of the "pure" calculus and by PSN. We apply it successfully to a large set of calculi with explicit substitutions, allowing us to establish SN, or, at least, to trace back the failure of SN to that of PSN.
Cites background from "Proof Nets and Explicit Substitutio..."
...5 λwsn-calculus In [8] a named version of λws was proposed....
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..., see for example [8, 10]), sometimes SN proofs uses PSN (see for example [4])....
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TL;DR: The purpose of this paper is to identify programs with control operators whose reduction semantics are in exact correspondence by introducing a relation $\simeq$, defined over a revised presentation of Parigot's $\lambda\mu$-calculus the authors dub $\Lambda M$.
Abstract: The purpose of this paper is to identify programs with control operators whose reduction semantics are in exact correspondence. This is achieved by introducing a relation $\simeq$, defined over a revised presentation of Parigot's $\lambda\mu$-calculus we dub $\Lambda M$. Our result builds on two fundamental ingredients: (1) factorization of $\lambda\mu$-reduction into multiplicative and exponential steps by means of explicit term operators of $\Lambda M$, and (2) translation of $\Lambda M$-terms into Laurent's polarized proof-nets (PPN) such that cut-elimination in PPN simulates our calculus. Our proposed relation $\simeq$ is shown to characterize structural equivalence in PPN. Most notably, $\simeq$ is shown to be a strong bisimulation with respect to reduction in $\Lambda M$, i.e. two $\simeq$-equivalent terms have the exact same reduction semantics, a result which fails for Regnier's $\sigma$-equivalence in $\lambda$-calculus as well as for Laurent's $\sigma$-equivalence in $\lambda\mu$.
References
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TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.
Abstract: Linear logic was introduced by Girard in 1987 [11] . Since then many results have supported Girard' s statement, \"Linear logic is a resource conscious logic,\" and related slogans . Increasingly, computer scientists have come to recognize linear logic as an expressive and powerful logic with connection s to a variety of topics in computer science . This column presents a.n intuitive overview of linear logic, some recent theoretical results, an d summarizes several applications of linear logic to computer science . Other introductions to linear logic may be found in [12, 361 .
2,304 citations
"Proof Nets and Explicit Substitutio..." refers background or methods in this paper
...In this paper we refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets [13]....
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...While we refer the interested reader to [13] for more details on linear logic in general, we give here a one-sided presentation of the sequent calculus for MELL:...
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TL;DR: This contribution was made possible only by the miraculous fact that the first members of the Editorial Board were sharing the same conviction about the necessity of Theoretical Computer Science.
1,480 citations
01 Dec 1989
TL;DR: The λ&sgr;-calculus is a refinement of the λ-Calculus where substitutions are manipulated explicitly, and provides a setting for studying the theory of substitutions, with pleasant mathematical properties.
Abstract: The ls-calculus is a refinement of the l-calculus where substitutions are manipulated explicitly. The ls-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical l-calculus and concrete implementations.
577 citations
"Proof Nets and Explicit Substitutio..." refers background in this paper
...The pioneer calculus with explicit substitutions, λσ, was introduced in [1] as a bridge between the classical λ-calculus and concrete implementations of functional programming languages....
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TL;DR: The chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism and subjectivism and that a representative class of algorithms can be modelized by means of standard mathematics.
Abstract: Publisher Summary This chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism (that leads to static modification) and subjectivism (that leads to syntactical abuses, in other terms, bureaucracy). The new approach initiated in this chapter rests on the use of a specific C*-algebra Λ* that has the distinguished property of bearing a (non associative) inner tensor product. The chapter describes that a representative class of algorithms can be modelized by means of standard mathematics.
321 citations
"Proof Nets and Explicit Substitutio..." refers methods in this paper
...1 Using various translations of the λ-calculus into proof nets, new abstract machines have been proposed, exploiting the Geometry of Interaction and the Dynamic Algebras [14, 2, 5], leading to the works on optimal reduction [15, 17]....
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