On the Structure of Valiant's Complexity Classes
In Valiant developed an algebraic analogue of the theory of NP-completeness for computations of polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay, Ladner, and Schöning.\par We show th...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
1999-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/260/pdf |
| _version_ | 1797270190187085824 |
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| author | Peter Bürgisser |
| author_facet | Peter Bürgisser |
| author_sort | Peter Bürgisser |
| collection | DOAJ |
| description | In Valiant developed an algebraic analogue of the theory of NP-completeness for computations of polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay, Ladner, and Schöning.\par We show that if Valiant's hypothesis is true, then there is a p-definable family, which is neither p-computable nor \textitVNP-complete. More generally, we define the posets of p-degrees and c-degrees of p-definable families and prove that any countable poset can be embedded in either of them, provided Valiant's hypothesis is true. Moreover, we establish the existence of minimal pairs for \textitVP in \textitVNP.\par Over finite fields, we give a \emphspecific example of a family of polynomials which is neither \textitVNP-complete nor p-computable, provided the polynomial hierarchy does not collapse.\par We define relativized complexity classes VP^h and VNP^h and construct complete families in these classes. Moreover, we prove that there is a p-family h satisfying VP^h = VNP^h. |
| first_indexed | 2024-04-25T02:00:20Z |
| format | Article |
| id | doaj.art-1c421578a12744dd8bd6ab40c6b815ee |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:00:20Z |
| publishDate | 1999-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-1c421578a12744dd8bd6ab40c6b815ee2024-03-07T15:00:15ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80501999-01-01Vol. 3 no. 310.46298/dmtcs.260260On the Structure of Valiant's Complexity ClassesPeter Bürgisser0Institut für Mathematik [Zürich]In Valiant developed an algebraic analogue of the theory of NP-completeness for computations of polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay, Ladner, and Schöning.\par We show that if Valiant's hypothesis is true, then there is a p-definable family, which is neither p-computable nor \textitVNP-complete. More generally, we define the posets of p-degrees and c-degrees of p-definable families and prove that any countable poset can be embedded in either of them, provided Valiant's hypothesis is true. Moreover, we establish the existence of minimal pairs for \textitVP in \textitVNP.\par Over finite fields, we give a \emphspecific example of a family of polynomials which is neither \textitVNP-complete nor p-computable, provided the polynomial hierarchy does not collapse.\par We define relativized complexity classes VP^h and VNP^h and construct complete families in these classes. Moreover, we prove that there is a p-family h satisfying VP^h = VNP^h.https://dmtcs.episciences.org/260/pdfposet of degreesstructural complexityalgebraic theories of np-completeness diagonalizationposet of degrees.[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Peter Bürgisser On the Structure of Valiant's Complexity Classes Discrete Mathematics & Theoretical Computer Science poset of degrees structural complexity algebraic theories of np-completeness diagonalization poset of degrees. [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | On the Structure of Valiant's Complexity Classes |
| title_full | On the Structure of Valiant's Complexity Classes |
| title_fullStr | On the Structure of Valiant's Complexity Classes |
| title_full_unstemmed | On the Structure of Valiant's Complexity Classes |
| title_short | On the Structure of Valiant's Complexity Classes |
| title_sort | on the structure of valiant s complexity classes |
| topic | poset of degrees structural complexity algebraic theories of np-completeness diagonalization poset of degrees. [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/260/pdf |
| work_keys_str_mv | AT peterburgisser onthestructureofvaliantscomplexityclasses |