On Berry's conjectures about the stable order in PCF
PCF is a sequential simply typed lambda calculus language. There is a unique order-extensional fully abstract cpo model of PCF, built up from equivalence classes of terms. In 1979, G\'erard Berry defined the stable order in this model and proved that the extensional and the stable order togethe...
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| Format: | Article |
| Language: | English |
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Logical Methods in Computer Science e.V.
2012-10-01
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| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/925/pdf |
| _version_ | 1797268726122283008 |
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| author | Fritz Müller |
| author_facet | Fritz Müller |
| author_sort | Fritz Müller |
| collection | DOAJ |
| description | PCF is a sequential simply typed lambda calculus language. There is a unique
order-extensional fully abstract cpo model of PCF, built up from equivalence
classes of terms. In 1979, G\'erard Berry defined the stable order in this
model and proved that the extensional and the stable order together form a
bicpo. He made the following two conjectures: 1) "Extensional and stable order
form not only a bicpo, but a bidomain." We refute this conjecture by showing
that the stable order is not bounded complete, already for finitary PCF of
second-order types. 2) "The stable order of the model has the syntactic order
as its image: If a is less than b in the stable order of the model, for finite
a and b, then there are normal form terms A and B with the semantics a, resp.
b, such that A is less than B in the syntactic order." We give counter-examples
to this conjecture, again in finitary PCF of second-order types, and also
refute an improved conjecture: There seems to be no simple syntactic
characterization of the stable order. But we show that Berry's conjecture is
true for unary PCF. For the preliminaries, we explain the basic fully abstract
semantics of PCF in the general setting of (not-necessarily complete) partial
order models (f-models.) And we restrict the syntax to "game terms", with a
graphical representation. |
| first_indexed | 2024-04-25T01:37:03Z |
| format | Article |
| id | doaj.art-d6dc7389e9a04337964b130088ed1691 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:37:03Z |
| publishDate | 2012-10-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-d6dc7389e9a04337964b130088ed16912024-03-08T09:28:03ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742012-10-01Volume 8, Issue 410.2168/LMCS-8(4:7)2012925On Berry's conjectures about the stable order in PCFFritz MüllerPCF is a sequential simply typed lambda calculus language. There is a unique order-extensional fully abstract cpo model of PCF, built up from equivalence classes of terms. In 1979, G\'erard Berry defined the stable order in this model and proved that the extensional and the stable order together form a bicpo. He made the following two conjectures: 1) "Extensional and stable order form not only a bicpo, but a bidomain." We refute this conjecture by showing that the stable order is not bounded complete, already for finitary PCF of second-order types. 2) "The stable order of the model has the syntactic order as its image: If a is less than b in the stable order of the model, for finite a and b, then there are normal form terms A and B with the semantics a, resp. b, such that A is less than B in the syntactic order." We give counter-examples to this conjecture, again in finitary PCF of second-order types, and also refute an improved conjecture: There seems to be no simple syntactic characterization of the stable order. But we show that Berry's conjecture is true for unary PCF. For the preliminaries, we explain the basic fully abstract semantics of PCF in the general setting of (not-necessarily complete) partial order models (f-models.) And we restrict the syntax to "game terms", with a graphical representation.https://lmcs.episciences.org/925/pdfcomputer science - logic in computer sciencef.3.2f.4.1 |
| spellingShingle | Fritz Müller On Berry's conjectures about the stable order in PCF Logical Methods in Computer Science computer science - logic in computer science f.3.2 f.4.1 |
| title | On Berry's conjectures about the stable order in PCF |
| title_full | On Berry's conjectures about the stable order in PCF |
| title_fullStr | On Berry's conjectures about the stable order in PCF |
| title_full_unstemmed | On Berry's conjectures about the stable order in PCF |
| title_short | On Berry's conjectures about the stable order in PCF |
| title_sort | on berry s conjectures about the stable order in pcf |
| topic | computer science - logic in computer science f.3.2 f.4.1 |
| url | https://lmcs.episciences.org/925/pdf |
| work_keys_str_mv | AT fritzmuller onberrysconjecturesaboutthestableorderinpcf |