Recursive definitions and fixed-points on well-founded structures
An expression such as ∀ x ( P ( x ) ↔ ϕ ( P ) ) , where P occurs in ϕ ( P ) , does not always define P . When such an expression implicitly defines P , in the sense of Beth (1953) [1] and Padoa (1900) [13], we call it a recursive definition. In the Least Fixed-Point Logic (LFP), we have theories whe...
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| Vydáno v: | Theoretical computer science Ročník 412; číslo 37; s. 4893 - 4904 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier B.V
26.08.2011
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| Témata: | |
| ISSN: | 0304-3975, 1879-2294 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | An expression such as
∀
x
(
P
(
x
)
↔
ϕ
(
P
)
)
, where
P
occurs in
ϕ
(
P
)
, does not always define
P
. When such an expression
implicitly defines
P
, in the sense of Beth (1953)
[1] and Padoa (1900)
[13], we call it a
recursive definition. In the Least Fixed-Point Logic (LFP), we have theories where interesting relations can be recursively defined (Ebbinghaus, 1995
[4], Libkin, 2004
[12]). We will show that for some sorts of recursive definitions there are explicit definitions on sufficiently strong theories of LFP. It is known that LFP, restricted to finite models, does not have Beth’s Definability Theorem (Gurevich, 1996
[7], Hodkinson, 1993
[8], Dawar, 1995
[3]). Beth’s Definability Theorem states that, if a relation is implicitly defined, then there is an explicit definition for it. We will also give a proof that Beth’s Definability Theorem fails for LFP without this finite model restriction. We will investigate fragments of LFP for which Beth’s Definability Theorem holds, specifically theories whose models are well-founded structures. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 0304-3975 1879-2294 |
| DOI: | 10.1016/j.tcs.2011.01.028 |