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Common lambda calculus reduction strategies include: [33] [34] [35] Normal order The leftmost outermost redex is reduced first. That is, whenever possible, arguments are substituted into the body of an abstraction before the arguments are reduced. If a term has a beta-normal form, normal order reduction will always reach that normal form ...
Any other reduction is an internal beta reduction. A normal form is a term that does not contain any beta redex, [3] [5] i.e. that cannot be further reduced. A head normal form is a term that does not contain a beta redex in head position, i.e. that cannot be further reduced by a head reduction. When considering the simple lambda calculus (viz ...
The meaning of lambda expressions is defined by how expressions can be transformed or reduced. [6] There are three kinds of transformation: α-conversion: changing bound variables (alpha); β-reduction: applying functions to their arguments (beta), calling functions; η-reduction: which captures a notion of extensionality (eta).
Viewing the lambda calculus as an abstract rewriting system, the Church–Rosser theorem states that the reduction rules of the lambda calculus are confluent. As a consequence of the theorem, a term in the lambda calculus has at most one normal form, justifying reference to "the normal form" of a given normalizable term.
Optimal reduction is not a reduction strategy for the lambda calculus in a narrow sense because performing β-reduction loses the information about the substituted redexes being shared. Instead it is defined for the labelled lambda calculus, an annotated lambda calculus which captures a precise notion of the work that should be shared.
In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by Michel Parigot. [1] It introduces two new operators: the μ operator (which is completely different both from the μ operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operator.
Such an essentially semantic, reduction-free, approach differs from the more traditional syntactic, reduction-based, description of normalisation as reductions in a term rewrite system where β-reductions are allowed deep inside λ-terms. NBE was first described for the simply typed lambda calculus. [1]
In sequent calculus, the completeness of atomic initial sequents states that initial sequents A ⊢ A (where A is an arbitrary formula) can be derived from only atomic initial sequents p ⊢ p (where p is an atomic formula). This theorem plays a role analogous to eta expansion in lambda calculus, and dual to cut elimination and beta reduction.