Abstract
We investigate the polynomial time isomorphic type structure of (the class of tally, polynomial time computable sets). We partition P T into six parts: D −, D^ −, C, S, F, F^, and study their p-isomorphic properties separately. The structures of , , and are obvious, where F, F^, and C are the class of tally finite sets, the class of tally co-finite sets, and the class of tally bi-dense sets respectively. The following results for the structures of and will be proved, where D^ is the class of tally, co-dense, polynomial time computable sets and S is the class of tally, scatted (i.e., neither dense nor co-dense), polynomial time computable sets.
1. is a countable distributive lattice with the greatest element.
2. Infinitely many intervals in can be distinguished by first order formulas.
3. There exist infinitely many nontrivial automorphisms for .
4. is not distributive, but any interval in is a countable distributive lattice. RID=""ID="" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID=""ID="" <E5>Key words or phrases:</E5> Computational complexity – Polynomial time – Degree structure – Lattice – Isomorphism RID=""ID="" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies.
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Received: 23 July 1999 / Published online: 27 March 2002
RID=""
ID="" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20
RID=""
ID="" <E5>Key words or phrases:</E5> Computational complexity – Polynomial time – Degree structure – Lattice – Isomorphism
RID=""
ID="" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies.
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Wang, Y. The algebraic structure of the isomorphic types of tally, polynomial time computable sets . Arch. Math. Logic 41 , 215 –244 (2002). https://doi.org/10.1007/s001530100117
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DOI: https://doi.org/10.1007/s001530100117