Final algebras, cosemicomputable algebras, and degrees of unsolvability:
Abstract: "This paper studies some computability notions for abstract data types, and in particular compares cosemicomputable many- sorted algebras with a notion of finality to model minimal-state realizations of abstract (software) machines. Given a finite many-sorted signature [sigma] and a s...
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Stanford, Calif.
1991
|
| Schriftenreihe: | Computer Science Laboratory <Menlo Park, Calif.>: SRI-CSL
91,6 |
| Schlagworte: | |
| Zusammenfassung: | Abstract: "This paper studies some computability notions for abstract data types, and in particular compares cosemicomputable many- sorted algebras with a notion of finality to model minimal-state realizations of abstract (software) machines. Given a finite many-sorted signature [sigma] and a set V of visible sorts, for every [sigma]-algebra A with co-r.e. behavior and nontrivial, computable V-behavior, there is a finite signature extension [sigma] of [sigma] (without new sorts) and a finite set E of [sigma]-equations such that A is isomorphic to a reduct of the final ([sigma], E)-algebra relative to V This uses a theorem due to Bergstra and Tucker [3]. If A is computable, then A is also isomorphic to the reduct of the initial ([sigma], E)-algebra. We also prove some results on congruences of finitely generated free algebras. We show that for every finite signature [sigma], there are either countably many [sigma]-congruences on the free [sigma]-algebra or else there is a continuum of such congruences. There are several necessary and sufficient conditions that separate these two cases We introduce the notion of the Turing degree of minimal algebra. Using the results above, we prove that there is a fixed one-sorted signature such that for every r.e. degree d, there is a finite set E of [sigma]-equations such the initial ([sigma], E)-algebra has degree d. There is a two-sorted signature [sigma] b0 s and a single visible sort such that for every r.e. degree d there is a finite set E of [sigma]-equations such that the initial ([sigma], E, V)-algebra is computable and the final ([sigma], E, V)-algebra is cosemicomputable and has degree d. |
| Beschreibung: | 37 S. |
Internformat
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| 100 | 1 | |a Moss, Lawrence S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Final algebras, cosemicomputable algebras, and degrees of unsolvability |c Lawrence S. Moss, José Meseguer, and Joseph A. Goguen |
| 264 | 1 | |a Stanford, Calif. |c 1991 | |
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| 490 | 1 | |a Computer Science Laboratory <Menlo Park, Calif.>: SRI-CSL |v 91,6 | |
| 520 | 3 | |a Abstract: "This paper studies some computability notions for abstract data types, and in particular compares cosemicomputable many- sorted algebras with a notion of finality to model minimal-state realizations of abstract (software) machines. Given a finite many-sorted signature [sigma] and a set V of visible sorts, for every [sigma]-algebra A with co-r.e. behavior and nontrivial, computable V-behavior, there is a finite signature extension [sigma] of [sigma] (without new sorts) and a finite set E of [sigma]-equations such that A is isomorphic to a reduct of the final ([sigma], E)-algebra relative to V | |
| 520 | 3 | |a This uses a theorem due to Bergstra and Tucker [3]. If A is computable, then A is also isomorphic to the reduct of the initial ([sigma], E)-algebra. We also prove some results on congruences of finitely generated free algebras. We show that for every finite signature [sigma], there are either countably many [sigma]-congruences on the free [sigma]-algebra or else there is a continuum of such congruences. There are several necessary and sufficient conditions that separate these two cases | |
| 520 | 3 | |a We introduce the notion of the Turing degree of minimal algebra. Using the results above, we prove that there is a fixed one-sorted signature such that for every r.e. degree d, there is a finite set E of [sigma]-equations such the initial ([sigma], E)-algebra has degree d. There is a two-sorted signature [sigma] b0 s and a single visible sort such that for every r.e. degree d there is a finite set E of [sigma]-equations such that the initial ([sigma], E, V)-algebra is computable and the final ([sigma], E, V)-algebra is cosemicomputable and has degree d. | |
| 650 | 4 | |a Algebra, Abstract |x Computer programs | |
| 650 | 4 | |a Logic, Symbolic and mathematical | |
| 700 | 1 | |a Meseguer, José |e Verfasser |4 aut | |
| 700 | 1 | |a Goguen, Joseph |d 1941-2006 |e Verfasser |0 (DE-588)172100275 |4 aut | |
| 830 | 0 | |a Computer Science Laboratory <Menlo Park, Calif.>: SRI-CSL |v 91,6 |w (DE-604)BV008930658 |9 91,6 | |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-006133980 | ||
Datensatz im Suchindex
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| author | Moss, Lawrence S. Meseguer, José Goguen, Joseph 1941-2006 |
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| author_facet | Moss, Lawrence S. Meseguer, José Goguen, Joseph 1941-2006 |
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Given a finite many-sorted signature [sigma] and a set V of visible sorts, for every [sigma]-algebra A with co-r.e. behavior and nontrivial, computable V-behavior, there is a finite signature extension [sigma] of [sigma] (without new sorts) and a finite set E of [sigma]-equations such that A is isomorphic to a reduct of the final ([sigma], E)-algebra relative to V</subfield></datafield><datafield tag="520" ind1="3" ind2=" "><subfield code="a">This uses a theorem due to Bergstra and Tucker [3]. If A is computable, then A is also isomorphic to the reduct of the initial ([sigma], E)-algebra. We also prove some results on congruences of finitely generated free algebras. We show that for every finite signature [sigma], there are either countably many [sigma]-congruences on the free [sigma]-algebra or else there is a continuum of such congruences. 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| id | DE-604.BV009224825 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:33:25Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006133980 |
| oclc_num | 24764523 |
| open_access_boolean | |
| owner | DE-29T |
| owner_facet | DE-29T |
| physical | 37 S. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| record_format | marc |
| series | Computer Science Laboratory <Menlo Park, Calif.>: SRI-CSL |
| series2 | Computer Science Laboratory <Menlo Park, Calif.>: SRI-CSL |
| spelling | Moss, Lawrence S. Verfasser aut Final algebras, cosemicomputable algebras, and degrees of unsolvability Lawrence S. Moss, José Meseguer, and Joseph A. Goguen Stanford, Calif. 1991 37 S. txt rdacontent n rdamedia nc rdacarrier Computer Science Laboratory <Menlo Park, Calif.>: SRI-CSL 91,6 Abstract: "This paper studies some computability notions for abstract data types, and in particular compares cosemicomputable many- sorted algebras with a notion of finality to model minimal-state realizations of abstract (software) machines. Given a finite many-sorted signature [sigma] and a set V of visible sorts, for every [sigma]-algebra A with co-r.e. behavior and nontrivial, computable V-behavior, there is a finite signature extension [sigma] of [sigma] (without new sorts) and a finite set E of [sigma]-equations such that A is isomorphic to a reduct of the final ([sigma], E)-algebra relative to V This uses a theorem due to Bergstra and Tucker [3]. If A is computable, then A is also isomorphic to the reduct of the initial ([sigma], E)-algebra. We also prove some results on congruences of finitely generated free algebras. We show that for every finite signature [sigma], there are either countably many [sigma]-congruences on the free [sigma]-algebra or else there is a continuum of such congruences. There are several necessary and sufficient conditions that separate these two cases We introduce the notion of the Turing degree of minimal algebra. Using the results above, we prove that there is a fixed one-sorted signature such that for every r.e. degree d, there is a finite set E of [sigma]-equations such the initial ([sigma], E)-algebra has degree d. There is a two-sorted signature [sigma] b0 s and a single visible sort such that for every r.e. degree d there is a finite set E of [sigma]-equations such that the initial ([sigma], E, V)-algebra is computable and the final ([sigma], E, V)-algebra is cosemicomputable and has degree d. Algebra, Abstract Computer programs Logic, Symbolic and mathematical Meseguer, José Verfasser aut Goguen, Joseph 1941-2006 Verfasser (DE-588)172100275 aut Computer Science Laboratory <Menlo Park, Calif.>: SRI-CSL 91,6 (DE-604)BV008930658 91,6 |
| spellingShingle | Moss, Lawrence S. Meseguer, José Goguen, Joseph 1941-2006 Final algebras, cosemicomputable algebras, and degrees of unsolvability Computer Science Laboratory <Menlo Park, Calif.>: SRI-CSL Algebra, Abstract Computer programs Logic, Symbolic and mathematical |
| title | Final algebras, cosemicomputable algebras, and degrees of unsolvability |
| title_auth | Final algebras, cosemicomputable algebras, and degrees of unsolvability |
| title_exact_search | Final algebras, cosemicomputable algebras, and degrees of unsolvability |
| title_full | Final algebras, cosemicomputable algebras, and degrees of unsolvability Lawrence S. Moss, José Meseguer, and Joseph A. Goguen |
| title_fullStr | Final algebras, cosemicomputable algebras, and degrees of unsolvability Lawrence S. Moss, José Meseguer, and Joseph A. Goguen |
| title_full_unstemmed | Final algebras, cosemicomputable algebras, and degrees of unsolvability Lawrence S. Moss, José Meseguer, and Joseph A. Goguen |
| title_short | Final algebras, cosemicomputable algebras, and degrees of unsolvability |
| title_sort | final algebras cosemicomputable algebras and degrees of unsolvability |
| topic | Algebra, Abstract Computer programs Logic, Symbolic and mathematical |
| topic_facet | Algebra, Abstract Computer programs Logic, Symbolic and mathematical |
| volume_link | (DE-604)BV008930658 |
| work_keys_str_mv | AT mosslawrences finalalgebrascosemicomputablealgebrasanddegreesofunsolvability AT meseguerjose finalalgebrascosemicomputablealgebrasanddegreesofunsolvability AT goguenjoseph finalalgebrascosemicomputablealgebrasanddegreesofunsolvability |