How To Play The Accordion. On the (Non-)Conservativity of the Reduction Induced by the Taylor Approximation of λ-Terms
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The Taylor expansion, which stems from Linear Logic and its differential extensions, is an approximation framework for the λ-calculus (and many of its variants). The reduction of the approximants of a λ-term induces a reduction on the λ-term itself, which enjoys a simulation property: whenever a term reduces to another, the approximants reduce accordingly. In recent work, we extended this result to an infinitary λ-calculus (namely, Λ_∞^001). This short paper solves the question whether the converse property also holds: if the approximants of some term reduce to the approximants of another term, is there a β-reduction between these terms? This happens to be true for the λ-calculus, as we show, but our proof fails in the infinitary case. We exhibit a counter-example, refuting the conservativity for Λ_∞^001.
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