Adding Negation to Lambda Mu
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...
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| Vydané v: | Logical methods in computer science Ročník 19, Issue 2 |
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| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Logical Methods in Computer Science e.V
25.05.2023
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| Predmet: | |
| ISSN: | 1860-5974, 1860-5974 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | 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. |
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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.46298/lmcs-19(2:12)2023 |