The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus
In a functional calculus, the so called \Omega-rule states that if two terms P and Q applied to any closed term <i>N</i> return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the \lambda\beta-calculus the \Omega-rule does not hold, even whe...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V.
2009-04-01
|
| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/1147/pdf |
| _version_ | 1827322963270041600 |
|---|---|
| author | Benedetto Intrigila Richard Statman |
| author_facet | Benedetto Intrigila Richard Statman |
| author_sort | Benedetto Intrigila |
| collection | DOAJ |
| description | In a functional calculus, the so called \Omega-rule states that if two terms
P and Q applied to any closed term <i>N</i> return the same value (i.e. PN =
QN), then they are equal (i.e. P = Q holds). As it is well known, in the
\lambda\beta-calculus the \Omega-rule does not hold, even when the \eta-rule
(weak extensionality) is added to the calculus. A long-standing problem of H.
Barendregt (1975) concerns the determination of the logical power of the
\Omega-rule when added to the \lambda\beta-calculus. In this paper we solve the
problem, by showing that the resulting theory is \Pi\_{1}^{1}-complete. |
| first_indexed | 2024-04-25T01:38:17Z |
| format | Article |
| id | doaj.art-e7fb7b426d444696aaacd334e5eb1d55 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:38:17Z |
| publishDate | 2009-04-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-e7fb7b426d444696aaacd334e5eb1d552024-03-08T09:00:32ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742009-04-01Volume 5, Issue 210.2168/LMCS-5(2:6)20091147The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-CalculusBenedetto IntrigilaRichard StatmanIn a functional calculus, the so called \Omega-rule states that if two terms P and Q applied to any closed term <i>N</i> return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the \lambda\beta-calculus the \Omega-rule does not hold, even when the \eta-rule (weak extensionality) is added to the calculus. A long-standing problem of H. Barendregt (1975) concerns the determination of the logical power of the \Omega-rule when added to the \lambda\beta-calculus. In this paper we solve the problem, by showing that the resulting theory is \Pi\_{1}^{1}-complete.https://lmcs.episciences.org/1147/pdfcomputer science - logic in computer sciencef.4.1 |
| spellingShingle | Benedetto Intrigila Richard Statman The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus Logical Methods in Computer Science computer science - logic in computer science f.4.1 |
| title | The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus |
| title_full | The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus |
| title_fullStr | The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus |
| title_full_unstemmed | The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus |
| title_short | The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus |
| title_sort | omega rule is mathbf pi 1 1 complete in the lambda beta calculus |
| topic | computer science - logic in computer science f.4.1 |
| url | https://lmcs.episciences.org/1147/pdf |
| work_keys_str_mv | AT benedettointrigila theomegaruleismathbfpi11completeinthelambdabetacalculus AT richardstatman theomegaruleismathbfpi11completeinthelambdabetacalculus AT benedettointrigila omegaruleismathbfpi11completeinthelambdabetacalculus AT richardstatman omegaruleismathbfpi11completeinthelambdabetacalculus |