Statman's Hierarchy Theorem
In the Simply Typed $\lambda$-calculus Statman investigates the reducibility relation $\leq_{\beta\eta}$ between types: for $A,B \in \mathbb{T}^0$, types freely generated using $\rightarrow$ and a single ground type $0$, define $A \leq_{\beta\eta} B$ if there exists a $\lambda$-definable injection f...
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| Veröffentlicht in: | Logical methods in computer science Jg. 13, Issue 4 |
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| Hauptverfasser: | , , , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Logical Methods in Computer Science e.V
01.01.2017
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| Schlagworte: | |
| ISSN: | 1860-5974, 1860-5974 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | In the Simply Typed $\lambda$-calculus Statman investigates the reducibility relation $\leq_{\beta\eta}$ between types: for $A,B \in \mathbb{T}^0$, types freely generated using $\rightarrow$ and a single ground type $0$, define $A \leq_{\beta\eta} B$ if there exists a $\lambda$-definable injection from the closed terms of type $A$ into those of type $B$. Unexpectedly, the induced partial order is the (linear) well-ordering (of order type) $\omega + 4$. In the proof a finer relation $\leq_{h}$ is used, where the above injection is required to be a B\"ohm transformation, and an (a posteriori) coarser relation $\leq_{h^+}$, requiring a finite family of B\"ohm transformations that is jointly injective. We present this result in a self-contained, syntactic, constructive and simplified manner. En route similar results for $\leq_h$ (order type $\omega + 5$) and $\leq_{h^+}$ (order type $8$) are obtained. Five of the equivalence classes of $\leq_{h^+}$ correspond to canonical term models of Statman, one to the trivial term model collapsing all elements of the same type, and one does not even form a model by the lack of closed terms of many types. |
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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.23638/LMCS-13(4:19)2017 |