Combinatory reduction systems with explicit substitution that preserve strong normalisation

Bloo, Roel; Rose, Kristoffer H.

doi:10.1007/3-540-61464-8_51

Cited by 20 publications

(13 citation statements)

References 3 publications

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“…But it is confluent on closed terms (of explicit substitutions), which are sufficient to represent all λ-terms. Later several calculi were designed with full confluence [11], or with both properties by suppressing some of the operations of explicit substitutions such as the composition of substitutions [4,23,7]. Until very recently no fully expressive calculus existed with both properties of confluence and strong normalization.…”

Section: Introductionmentioning

confidence: 99%

Explicit Substitutions and Programming Languages

Lévy

Maranget

1999

Lecture Notes in Computer Science

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Abstract. The λ-calculus has been much used to study the theory of substitution in logical systems and programming languages. However, with explicit substitutions, it is possible to get finer properties with respect to gradual implementations of substitutions as effectively done in runtimes of programming languages. But the theory of explicit substitutions has some defects such as non-confluence or the non-termination of the typed case. In this paper, we stress on the sub-theory of weak substitutions, which is sufficient to analyze most of the properties of programming languages, and which preserves many of the nice theorems of the λ-calculus.

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Section: Introductionmentioning

confidence: 99%

Explicit Substitutions and Programming Languages

Lévy

Maranget

1999

Lecture Notes in Computer Science

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“…These two last approaches study the logical meaning of a given reduction system with explicit substitutions rather than the computational meaning of a given logical cut elimination process. Finally, it is worth noticing that the T P C ES -calculus is not a particular case of the higher order calculi with explicit substitutions proposed in [8], which do not allow to make abstractions on patterns, neither to have different forms of explicit substitutions.…”

Section: Resultsmentioning

confidence: 99%

“…Indeed, reduction rules for let terms will allow to express the pattern matching process inside the T P C ES calculus, while reduction rules for explicit substitutions terms will allow to express the behavior of explicit substitutions as is classically done in the literature [1,5,8,26]. Therefore, both pattern matching and explicit substitution computations can be modeled via the cut elimination process for sequent proofs.…”

Section: Typing Rulesmentioning

confidence: 99%

“…In such a calculus, the substitution is a meta-level operation, executed by operators that are external to the language. In -calculi with explicit substitutions, as, for instance, [1,5,8,26], and the pattern calculus T P C ES that we propose here, the substitution is internalized and handled by symbols and reduction rules belonging to the syntax of the calculus itself. Explicit substitutions, by decomposing the rule into finer steps, closer to what happens in environment machines, allow a better understanding of the execution models.…”

Section: Introductionmentioning

confidence: 99%

“…,`M :A ,; x :A`x:A proj ,`x x=M : A sub = ,`M :A which suggests that the process of cut elimination consists in reducing the term x x=M to the term M, exactly as in the V a r 1 reduction rule in [1,5,8,26] defined by:…”

Section: Introductionmentioning

confidence: 99%

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Pattern matching as cut elimination

Cerrito¹,

Kesner²

Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158)

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We present a typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for minimal logic. Our calculus is inspired by the Curry-Howard Isomorphism, in the sense that types, both for patterns and terms, correspond to propositions, terms correspond to proofs, and term reduction corresponds to sequent proof normalization performed by cut elimination. The calculus enjoys subject reduction, confluence, preservation of strong normalization w.r.t a system with meta-level substitutions and strong normalization for well-typed terms. As a consequence, it can be seen as an implementation calculus for functional formalisms defined with meta-level operations for pattern matching and substitutions.

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