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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| Published in: | Theoretical computer science Vol. 412; no. 37; pp. 4893 - 4904 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
26.08.2011
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| Subjects: | |
| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. An expression such as ax ( P ( x ) a" phi ( P ) ) , where P occurs in phi ( P ) , does not always define P . When such an expression implicitly defines P , in the sense of Beth (1953) and Padoa (1900) , 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 , Libkin, 2004 ). 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 , Hodkinson, 1993 , Dawar, 1995 ). 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. |
| Author | Martins, Ana Teresa Ferreira, Francicleber Martins |
| Author_xml | – sequence: 1 givenname: Francicleber Martins surname: Ferreira fullname: Ferreira, Francicleber Martins email: fran@lia.ufc.br, francicleber.ferreira@gmail.com – sequence: 2 givenname: Ana Teresa surname: Martins fullname: Martins, Ana Teresa email: ana@lia.ufc.br |
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| Cites_doi | 10.1109/LICS.2002.1029848 10.1016/j.entcs.2009.07.046 10.1016/S1385-7258(53)50042-3 10.2140/pjm.1955.5.285 10.2307/2275675 10.2178/bsl/1182353853 |
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| Keywords | Beth’s definability theorem Recursive definitions Fixed-points |
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| References | Dawar, Gurevich (br000010) 2002; 8 Dawar, Hella, Kolaitis (br000015) 1995 S. Kreutzer, Pure and applied fixed-point logics, Ph.D. Thesis, von der Fakultät für Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen, 2002. Tarski (br000070) 1955; 5 Kolaitis (br000045) 1990 Gurevich, Shelah (br000035) 1996; 61 Libkin (br000060) 2004 Hodkinson (br000040) 1993; 51 Ebbinghaus, Flum, Thomas (br000025) 1994 S. Kreutzer, Expressive equivalence of least and inflationary fixed-point logic, in: Proceedings of 17th IEEE Symp. on Logic in Computer Science, LICS’02, 2002, pp. 403–410. Ebbinghaus, Flum (br000020) 1995 Beth (br000005) 1953; 15 Ferreira, Martins (br000030) 2009; 247 Padoa (br000065) 1900; vol. 3 Ebbinghaus (10.1016/j.tcs.2011.01.028_br000025) 1994 Libkin (10.1016/j.tcs.2011.01.028_br000060) 2004 10.1016/j.tcs.2011.01.028_br000055 Tarski (10.1016/j.tcs.2011.01.028_br000070) 1955; 5 Gurevich (10.1016/j.tcs.2011.01.028_br000035) 1996; 61 Hodkinson (10.1016/j.tcs.2011.01.028_br000040) 1993; 51 Dawar (10.1016/j.tcs.2011.01.028_br000010) 2002; 8 Ferreira (10.1016/j.tcs.2011.01.028_br000030) 2009; 247 Ebbinghaus (10.1016/j.tcs.2011.01.028_br000020) 1995 Dawar (10.1016/j.tcs.2011.01.028_br000015) 1995 Padoa (10.1016/j.tcs.2011.01.028_br000065) 1900; vol. 3 Beth (10.1016/j.tcs.2011.01.028_br000005) 1953; 15 Kolaitis (10.1016/j.tcs.2011.01.028_br000045) 1990 10.1016/j.tcs.2011.01.028_br000050 |
| References_xml | – volume: 15 start-page: 330 year: 1953 end-page: 339 ident: br000005 article-title: On Padoa’s method in the theory of definitions publication-title: Indagtiones Mathematicae – year: 1995 ident: br000020 article-title: Finite Model Theory – start-page: 624 year: 1995 end-page: 635 ident: br000015 article-title: Implicit definability and infinitary logic in finite model theory publication-title: Proceedings 22nd Int’l Coll. on Automata, Languages, and Programming, ICALP’95, Szeged, Hungary, 10–14 July 1995, Vol. 944 – volume: 8 start-page: 65 year: 2002 end-page: 88 ident: br000010 article-title: Fixed-point logics publication-title: Bulletin of Symbolic Logic – reference: S. Kreutzer, Pure and applied fixed-point logics, Ph.D. Thesis, von der Fakultät für Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen, 2002. – volume: 5 start-page: 285 year: 1955 end-page: 309 ident: br000070 article-title: A lattice-theoretical fixpoint theorem and its applications publication-title: Pacific Journal of Mathematics – year: 1994 ident: br000025 article-title: Mathematical Logic – reference: S. Kreutzer, Expressive equivalence of least and inflationary fixed-point logic, in: Proceedings of 17th IEEE Symp. on Logic in Computer Science, LICS’02, 2002, pp. 403–410. – volume: 61 start-page: 549 year: 1996 end-page: 562 ident: br000035 article-title: On finite rigid structures publication-title: Journal of Symbolic Logic – volume: 247 start-page: 19 year: 2009 end-page: 37 ident: br000030 article-title: Recursive definitions and fixed-points publication-title: Electronic Notes in Theoretical Computer Science – volume: 51 start-page: 111 year: 1993 end-page: 140 ident: br000040 article-title: Finite variable logics publication-title: Bulletin of the EATCS – start-page: 168 year: 1990 end-page: 180 ident: br000045 article-title: Implicit definability on finite structures and unambiguous computations (preliminary report) publication-title: Proceedings, Fifth Annual IEEE Symposium on Logic in Computer Science – year: 2004 ident: br000060 article-title: Elements of Finite Model Theory – volume: vol. 3 start-page: 118 year: 1900 end-page: 123 ident: br000065 article-title: Essai d’une théorie algébrique des nombres entiers, precedé d’une introduction logique a une théorie deductive quelconque publication-title: Bibliothèque Du Congrès International de Philosophie – ident: 10.1016/j.tcs.2011.01.028_br000050 doi: 10.1109/LICS.2002.1029848 – start-page: 168 year: 1990 ident: 10.1016/j.tcs.2011.01.028_br000045 article-title: Implicit definability on finite structures and unambiguous computations (preliminary report) – volume: 51 start-page: 111 year: 1993 ident: 10.1016/j.tcs.2011.01.028_br000040 article-title: Finite variable logics publication-title: Bulletin of the EATCS – year: 1995 ident: 10.1016/j.tcs.2011.01.028_br000020 – volume: 247 start-page: 19 year: 2009 ident: 10.1016/j.tcs.2011.01.028_br000030 article-title: Recursive definitions and fixed-points publication-title: Electronic Notes in Theoretical Computer Science doi: 10.1016/j.entcs.2009.07.046 – year: 2004 ident: 10.1016/j.tcs.2011.01.028_br000060 – volume: 15 start-page: 330 year: 1953 ident: 10.1016/j.tcs.2011.01.028_br000005 article-title: On Padoa’s method in the theory of definitions publication-title: Indagtiones Mathematicae doi: 10.1016/S1385-7258(53)50042-3 – start-page: 624 year: 1995 ident: 10.1016/j.tcs.2011.01.028_br000015 article-title: Implicit definability and infinitary logic in finite model theory – year: 1994 ident: 10.1016/j.tcs.2011.01.028_br000025 – volume: 5 start-page: 285 year: 1955 ident: 10.1016/j.tcs.2011.01.028_br000070 article-title: A lattice-theoretical fixpoint theorem and its applications publication-title: Pacific Journal of Mathematics doi: 10.2140/pjm.1955.5.285 – volume: 61 start-page: 549 year: 1996 ident: 10.1016/j.tcs.2011.01.028_br000035 article-title: On finite rigid structures publication-title: Journal of Symbolic Logic doi: 10.2307/2275675 – ident: 10.1016/j.tcs.2011.01.028_br000055 – volume: vol. 3 start-page: 118 year: 1900 ident: 10.1016/j.tcs.2011.01.028_br000065 article-title: Essai d’une théorie algébrique des nombres entiers, precedé d’une introduction logique a une théorie deductive quelconque – volume: 8 start-page: 65 year: 2002 ident: 10.1016/j.tcs.2011.01.028_br000010 article-title: Fixed-point logics publication-title: Bulletin of Symbolic Logic doi: 10.2178/bsl/1182353853 |
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| Snippet | 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... An expression such as ax ( P ( x ) a" phi ( P ) ) , where P occurs in phi ( P ) , does not always define P . When such an expression implicitly defines P , in... |
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| StartPage | 4893 |
| SubjectTerms | Beth’s definability theorem Computer simulation Constrictions Fixed points (mathematics) Fixed-points Fragments Logic Mathematical analysis Mathematical models Recursive Recursive definitions Theorems |
| Title | Recursive definitions and fixed-points on well-founded structures |
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