当前位置:
X-MOL 学术
›
arXiv.cs.LO
›
论文详情
Our official English website, www.x-mol.net, welcomes your
feedback! (Note: you will need to create a separate account there.)
Adding Negation to Lambda Mu
arXiv - CS - Logic in Computer Science Pub Date : 2021-09-21 , DOI: arxiv-2109.10447 Steffen van Bakel
arXiv - CS - Logic in Computer Science Pub Date : 2021-09-21 , DOI: arxiv-2109.10447 Steffen van Bakel
We present $\cal L$, an extension of Parigot's $\lambda\mu$-calculus by
adding negation as a type constructor, together with syntactic constructs that
represent negation introduction and elimination. We will define a notion of
reduction that extends $\lambda\mu$'s reduction system with two new reduction
rules, and show that the system satisfies subject reduction. Using Aczel's
generalisation of Tait and Martin-L\"of's notion of parallel reduction, we show
that this extended reduction is confluent. Although the notion of type
assignment has its limitations with respect to representation of proofs in
natural deduction with implication and negation, we will show that all
propositions that can be shown in there have a witness in $\cal L$. Using
Girard's approach of reducibility candidates, we show that all typeable terms
are strongly normalisable, and conclude the paper by showing that type
assignment for $\cal L$ enjoys the principal typing property.
中文翻译:
向 Lambda Mu 添加否定
我们提出了 $\cal L$,它是 Parigot 的 $\lambda\mu$ 演算的扩展,通过添加否定作为类型构造函数,以及表示否定引入和消除的句法构造。我们将定义一个归约概念,它用两个新的归约规则扩展 $\lambda\mu$ 的归约系统,并证明该系统满足主题归约。使用 Aczel 对 Tait 的推广和 Martin-L"of 的并行归约概念,我们证明了这种扩展归约是融合的。尽管类型分配的概念在用蕴涵和否定的自然演绎中的证明表示方面有其局限性,但我们将表明所有可以在那里显示的命题在 $\cal L$ 中都有一个见证。使用 Girard 的可约性候选方法,
更新日期:2021-09-23
中文翻译:
向 Lambda Mu 添加否定
我们提出了 $\cal L$,它是 Parigot 的 $\lambda\mu$ 演算的扩展,通过添加否定作为类型构造函数,以及表示否定引入和消除的句法构造。我们将定义一个归约概念,它用两个新的归约规则扩展 $\lambda\mu$ 的归约系统,并证明该系统满足主题归约。使用 Aczel 对 Tait 的推广和 Martin-L"of 的并行归约概念,我们证明了这种扩展归约是融合的。尽管类型分配的概念在用蕴涵和否定的自然演绎中的证明表示方面有其局限性,但我们将表明所有可以在那里显示的命题在 $\cal L$ 中都有一个见证。使用 Girard 的可约性候选方法,