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 three main ingredients which guide our semantical development: (1) factorization of Parigot's $\lambda\mu$-reduction into multiplicative and exponential steps by means of explicit operators, (2) adaptation of Laurent's original $\simeq_\sigma$-equivalence to $\Lambda M$, and (3) interpretation of $\Lambda M$ into Laurent's polarized proof-nets (PPN). More precisely, we first give a translation of $\Lambda M$-terms into PPN which simulates the reduction relation of our calculus via cut elimination of PPN. Second, we establish a precise correspondence between our relation $\simeq$ and Laurent's $\simeq_\sigma$-equivalence for $\lambda\mu$-terms. Moreover, $\simeq$-equivalent terms translate to structurally equivalent 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 $\simeq_\sigma$-equivalence in $\lambda$-calculus as well as for Laurent's $\simeq_\sigma$-equivalence in $\lambda\mu$.

中文翻译：

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通过乘法和指数归约法对控制算子进行强大的双仿真

本文的目的是识别具有简化语义完全对应的控制运算符的程序。这是通过引入一个关系$ \ simeq $来实现的，该关系是根据我们对$ \ Lambda M $配音的Parigot的$ \ lambda \ mu $微积分的修订表示而定义的。我们的结果建立在指导语义发展的三个主要因素的基础上：（1）通过显式运算符将Parigot的$ \ lambda \ mu $-约简分解为乘法和指数级步骤，（2）修改Laurent的原始$ \ simeq_ \ sigma与$ \ Lambda M $的等价性，以及（3）将$ \ Lambda M $解释为Laurent的极化证明网（PPN）。更准确地说，我们首先将$ \ Lambda M $ -terms转换为PPN，该模拟通过切消PPN消除了我们的演算的约简关系。第二，我们在关系$ \ simeq $和Laurent的$ \ simeq_ \ sigma $等价关系之间建立了精确的对应关系，即$ \ lambda \ mu $-项。此外，等效的\\ simeq $术语转换为结构等效的PPN。最值得注意的是，对于减少\\ Lambda M $，$ \ simeq $被证明是强烈的双重仿真，即两个等效于$ \ simeq $的术语具有完全相同的归约语义，结果对于Regnier的$ \ simeq_而言是失败的。 \ lambda $演算中的\ sigma $等价，以及Laurent的$ \ lambda \ mu $中的\ simeq_ \ sigma $等价。