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On the theory of the PTIME degrees of the recursive sets

Juichi Shinoda, Theodore A. Slaman
1990 Journal of computer and system sciences (Print)  
There is an interpretation of first-order arithmetic in the theory of the PTIME degrees of the recursive sets.  ...  There is an interpretation of second-order arithmetic in the first-order theory of the PTIME degrees. These results characterize the Turing degrees of the first order theories of these structures.  ...  What is the Turing degree of the theory of the PTIME many-one degrees of the recursive sets ? A, B, C, K,,, K,, L, P,, Q,, P,, Q, and U of recursive sets.  ... 
doi:10.1016/0022-0000(90)90024-f fatcat:x6rmsefnxreepkfhrvyveejcnq

On the theory of the PTIME degrees of the recursive sets

J. Shinoda, T.A. Slaman
1988 [1988] Proceedings. Structure in Complexity Theory Third Annual Conference  
There is an interpretation of first order arithmetic in the theory of the PTIME degrees of the recursive sets.  ...  There is an interpretation of second order arithmetic in the first order theory of the PTIME degrees. These results characterize the Turing degrees of the first order theories of these structures.  ...  What is the Turing degree of the theory of the PTIME many-one degrees of the recursive sets? 55 Y $=\emptyset$ , then $q(A^{(e)})(x)=i$ .  ... 
doi:10.1109/sct.1988.5285 dblp:conf/coco/ShinodaS88 fatcat:n5hbeedjmbfnzgnfoxrw7wdq64

Page 1271 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews  
recursion on natural numbers, in the more general framework of primitive recursion on term algebras.  ...  It is a very important and difficult problem in recursion theory to characterize the embeddability of finite lattices in the recursively enumerable (r.e.) degrees.  ... 

Page 6027 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
Bisimulation-invariant PTIME, or the modal fragment of PTIME, thus proves to be one of the very rare cases in which a logical characterization is known in a setting of unordered structures.  ...  They show that the BSS model forces exactly one jump of unsolv- ability for the decidable sets, in contrast to the situation for type 2 recursion.  ... 

Automorphisms in the PTIME-Turing degrees of recursive sets

Christine Ann Haught, Theodore A. Slaman
1997 Annals of Pure and Applied Logic  
an automorphism of the ideal of PTIME-degrees below A.  ...  We consider questions related to the rigidity of the structure W, the PTZME-Turing degrees of recursive sets of strings together with PTME-Turing reducibility, <pr , and related structures; do these structures  ...  If X and Y are each recursive in the other, then X and Y have the same Turing degree. The recursive sets are those which are, in theory, computable.  ... 
doi:10.1016/s0168-0072(95)00065-8 fatcat:63si4h43srcwrnqtp56dju5hum

Page 3692 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews  
A Turing degree of enumerable sets is contiguous if all enumerable sets of that degree are also of the same weak-truth-table (wtt) degree.  ...  The expository paper under review, a part of a volume on recent advances in algebraic model theory, presents a very readable, self-contained proof of Khovanskii’s the- orem and some of its applications  ... 

Differences between Resource Bounded Degree Structures

Theodore A. Slaman, Michael~E. Mytilinaios
2003 Notre Dame Journal of Formal Logic  
We exhibit a structural difference between the truth-table degrees of the sets which are truth-table above 0 and the PTIME-Turing degrees of all sets.  ...  Though the structures do not have the same isomorphism type, demonstrating this fact relies on developing their common theory.  ...  If A and B are recursive in each other, we say that they have the same Turing degree. The Turing degree of a set is a measurement of the information which is contained in the diagram of that set.  ... 
doi:10.1305/ndjfl/1082637612 fatcat:cnjcerxjtvabrbctkkdbqnx4ee

Page 6154 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews  
second-order arithmetic is interpretable in the theory of PTIME degrees of all sets of binary strings.  ...  In both conception and proof this extends the earlier results of the authors that true first-order arithmetic is interpretable in the theory of PTIME degrees of recursive sets of binary strings and true  ... 

Page 8387 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
This survey explores the structural properties of polynomial-time bounded reducibilities defined on the recursive sets and of the corresponding degrees.  ...  Let w be the set of standard numbers and K be {(x,i): i¢€ w and x € @")}. We prove that the degree of w is definable in the ideal of degrees below the degree of K.  ... 

Collapsing degrees

Stuart A. Kurtz, Stephen R. Mahaney, James S. Royer
1988 Journal of computer and system sciences (Print)  
An m-degree is a collection of sets equivalent under polynomial-time many-one (Karp) reductions; for example, the complete sets for NP or PSPACE are m-degrees.  ...  Hartmanis showed that all the then known NP-complete sets are isomorphic, and conjectured that the m-degree of the NP-complete sets collapses, in essence claiming that there is only one NP-complete set  ...  Since the properties of sets studied in recursive function theory are invariant under recursive isomorphisms, the significance of this corollary is that the recursive m-complete r.e. sets are essentially  ... 
doi:10.1016/0022-0000(88)90007-4 fatcat:o7kwseb6rfcmjhjhzexqzv7gn4

Hyper-polynomial hierarchies and the polynomial jump

Stephen Fenner, Steven Homer, Randall Pruim, Marcus Schaefer
2001 Theoretical Computer Science  
Assuming that the polynomial hierarchy (PH) does not collapse, we show the existence of ascending sequences of ptime Turing degrees of length !  ...  The lack of uniform least upper bounds for ascending sequences of ptime degrees causes the limit levels of our hyper-polynomial hierarchy to be inherently non-canonical.  ...  In this generalization of classical computability theory, admissible recursion theory, the hyperarithmetic sets play the role of the computable sets and 1 1 corresponds to the computably enumerable sets  ... 
doi:10.1016/s0304-3975(00)00193-6 fatcat:52bsonjwfbgnld5ruoeou3yp6m

Page 16 of Mathematical Reviews Vol. , Issue 82a [page]

1982 Mathematical Reviews  
By information content he means essentially degree of nonrecursiveness in one of the usual degree notions. Most of these characterizations are in terms of the weak jump A‘" of an re. set A. (AS?  ...  The compression theorem says roughly that for any element g of a so-called measured set of functions one can compute n and a recursive A such that g is essentially inadequate for n, but A is strongly adequate  ... 

Tackling Polytype Queries in Inconsistent Databases: Theory and Algorithm

Dong Xie, Xinbo Chen, Yan Zhu
2012 Journal of Software  
To expand query types under a set of integrity constraints for obtaining consistent answers over inconsistent databases, a computational theory is proposed based on first-order logic.  ...  For a rewritable initial query, a consistent identification statement is constructed based on the join graph by recursive computation; and the statement combines with the initial query to construct a new  ...  Since our method is based on the join graph, the third algorithm considers recursive computation for subnodes.  ... 
doi:10.4304/jsw.7.8.1861-1866 fatcat:oleyvdabkvb7xfz2e2avof2are

Page 5878 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews  
Closing PTime under bounded primitive recursion on numbers yields PSpace.  ...  Known characterizations start with the PTime functions of Cob- ham, who employed bounded primitive recursion on notation and the initial function (among other simpler ones) x'’' (concatenate as many copies  ... 

On the Complexity of the Universality and Inclusion Problems for Unambiguous Context-Free Grammars

Lorenzo Clemente
2020 Electronic Proceedings in Theoretical Computer Science  
However, we show that computing the coin-flip measure of an unambiguous context-free language, a quantitative generalisation of universality, is hard for the long-standing open problem SQRTSUM.  ...  The latter problem has long been known to be decidable and we propose a PSPACE algorithm that works by reduction to the zeroness problem of recurrence equations with convolution.  ...  I also thank an anonymous reviewer for his helpful comments on a preliminary version of this draft.  ... 
doi:10.4204/eptcs.320.2 fatcat:mkv6x5s2mvhqjg423zjuydhkvm
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