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Arithmetic theories for computational complexity problems

Steve Homer, John Reif
1986 Information and Control  
This paper considers a number of arithmetic theories and shows how the strength of these theories relates to certain open problems in complexity theory concerning the polynomial-time hierarchy.  ...  These results are proved quite generally and hold for a wide class of subrecursive hierarchies. They can be used to characterize certain properties of functions provable in these theories.  ...  We first consider applications to the polynomial-time hierarchy as this hierarchy was the original motivation for looking at these theories.  ... 
doi:10.1016/s0019-9958(86)80041-9 fatcat:mp344yvwefeitja4jhkenj3n7q

Page 5450 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
The results yield problems that are complete not only for NUOGSPACE and PTIME but also for second and third levels of the restricted logspace hierarchy.  ...  “For these reasons it appears worthwhile at this time to study the properties of the class AN.  ... 

Page 467 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
There it is shown that under a suitable oracle, the collapse cannot go down to the next lower level of the polynomial-time hierarchy, AD = P'?.  ...  Also, this paper shows that the complete sets for the levels of the Boolean hierarchy above the second level cannot have AND or OR unless the polynomial hierarchy collapses.  ... 

Third-Order Computation and Bounded Arithmetic

A. Skelley
2007 Journal of Logic and Computation  
We then present a number of third-order theories of bounded arithmetic whose definable functions are the classes of the EXP-time hierarchy in the third-order setting.  ...  We describe a natural generalization of ordinary computation to a third-order setting and give a function calculus with nice properties and recursion-theoretic characterizations of several large complexity  ...  In the hands of an NEXP machine, however, an unbounded (ordinary) oracle from some level of the quasi-polynomial-time hierarchy is no more powerful than one from the same level of the polynomial-time hierarchy  ... 
doi:10.1093/logcom/exm040 fatcat:jatxxmbvnbbjxes42ytkrujmme

Characterizing the polynomial hierarchy by alternating auxiliary pushdown automata

Birgit Jenner, Bernd Kirsig
1989 RAIRO - Theoretical Informatics and Applications  
complement [1], it is shown that, surprisingly, the further levels ofthis alternating auxiliary pushdown hierarchy coincide level by level with the Polynomial Hierarchy.  ...  -An alternating auxiliary pushdown hierarchy is defined by extending the machine model of the Logarithmic Alternation Hierarchy by a pushdown store while keeping a polynomial time bound.  ...  In particular, we show: which means that the fe-th level of the Polynomial Hierarchy is just the /c-hlst level of the A Y? PLU pr Hierarchy.  ... 
doi:10.1051/ita/1989230100871 fatcat:6uwn5bwe6vay5gyqihy7ldikr4

Page 2054 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
We prove that the levels of the polynomial hierarchy are order-oblivious: pri = pur.  ...  In the present paper, we study the power of query order when accessing levels of the polynomial hierarchy, and we show that here order does not matter.  ... 

The Parameterized Complexity of Constraint Satisfaction and Reasoning [chapter]

Stefan Szeider
2013 Lecture Notes in Computer Science  
Parameterized Complexity is a new and increasingly popular theoretical framework for the rigorous analysis of NP-hard problems and the development of algorithms for their solution.  ...  The framework provides adequate concepts for taking structural aspects of problem instances into account.  ...  on higher levels of the Polynomial Hierarchy.  ... 
doi:10.1007/978-3-642-41524-1_2 fatcat:ee2md36scbckvne73x7qfkw5ra

Page 6682 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  
Allender and Hemachandra, and Long and Sheu, introduced refinements of the extended low hierarchy based on the A- and @-levels, respectively, of the polynomial-time hierarchy: for k > 2, ELE“ = {A| AP(  ...  Summary: “Balcazar, Book, and Schéning introduced the extended low hierarchy based on the £-levels of the polynomial-time hier- archy as follows: for k > 1, level k of the extended low hierar- 68 COMPUTER  ... 

Capturing structure in hard combinatorial problems

Stefan Szeider
2013 2013 IEEE 25th International Conference on Tools with Artificial Intelligence  
reasoning, both located at the second level of the Polynomial Hierarchy.  ...  SAT is well-suited as a target problem since by means of FPT-reductions to SAT we can make today's extremely powerful SAT solvers applicable to problems at higher levels of the Polynomial Hierarchy.  ...  Acknowledgment: The author gratefully acknowledges the support by the European Research Council (ERC), project COMPLEX REASON 239962.  ... 
doi:10.1109/ictai.2013.136 dblp:conf/ictai/Szeider13 fatcat:da5r7643mbafrkvaw74evlwbra

The difference and truth-table hierarchies for NP

Johannes Köbler, Uwe Schöning, Klaus W. Wagner
1987 RAIRO - Theoretical Informatics and Applications  
Further, we define complete sets for each level of this hierarchy and study conditions under which the hierarchy "collapses", i. e. it consists only of finitely many levels.  ...  In [16] several natural problems are shown to be complete for every level of the Boolean NP-hierarchy (and thus also for every level of the différence hierarchy).  ... 
doi:10.1051/ita/1987210404191 fatcat:u6ragzrbbndktbjjnoveibir74

Page 5724 of Mathematical Reviews Vol. , Issue 92j [page]

1992 Mathematical Reviews  
consequence for the polynomial-time hierarchy.  ...  In particular, the new model satisfies the Immerman-Szelepcsenyi theorem and Savitch’s theorem in the rel- ativized worlds for oracle sets in each level of the polynomial-time hierarchy.” 92j:68030 68Q15  ... 

W-Hardness Under Linear FPT-Reductions: Structural Properties and Further Applications [chapter]

Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, Ge Xia
2005 Lecture Notes in Computer Science  
In this paper, we formally investigate the notions of W [t]-hardness and W [t]-completeness under the linear fpt-reduction, and study structural properties of the corresponding complexity classes.  ...  The notion of linear fpt-reductions has been recently used to derive strong computational lower bounds for well-known NP-hard problems.  ...  For example, if for any integer t > 0, the t-th level of the polynomial time hierarchy collapses to the (t − 1)-st level, Σ p t = Σ p t−1 , then the entire polynomial time hierarchy collapses to the (t  ... 
doi:10.1007/11533719_98 fatcat:ewojc2i2z5gbllf6ynfulngu2a

Preface to the Special Issue on Computer Science in Russia 2016

Alexander S. Kulikov, Gerhard J. Woeginger
2018 Theory of Computing Systems  
The paper Level two of the quantifier alternation hierarchy over infinite words by Manfred Kufleitner and Tobias Walter addresses the membership problem for a certain fragment of first-order logic over  ...  CSR represents Theoretical Computer Science in all its aspects, a vibrant field that is turning into an abundant source of ideas and concepts for a variety of other research disciplines and applications  ...  The paper The word problem for omega-terms over the Trotter-Weil Hierarchy by Manfred Kufleitner and Jan Philipp Wächter proves that the word problem for ω-terms over each level of the Trotter-Weil Hierarchy  ... 
doi:10.1007/s00224-018-9854-5 fatcat:sbiokecipfaqldekzhdviifzx4

A View of Structural Complexity Theory [chapter]

Ronald V. BOOK, Osamu WATANABE
1993 Current Trends in Theoretical Computer Science  
Acknowledgments Preparation of this paper was supported in part by the National Science Foundation under grant CCR86-11980 and CCR89-13584.  ...  If S is a low set in the polynomial-time hierarchy and NP ⊆ P(S), then the polynomial-time hierarchy extends to only finitely many levels -the level depends on the "degree of lowness" of set S.  ...  The polynomial-time hierarchy extends to infinitely many levels if and only if for every sparse set S, the polynomial-time hierarchy relative to S extends to infinitely many levels if and only if there  ... 
doi:10.1142/9789812794499_0034 dblp:series/wsscs/Book093 fatcat:jcrj2u7caveejeojt6w6ouzqvq

Page 4184 of Mathematical Reviews Vol. , Issue 2001F [page]

2001 Mathematical Reviews  
This hierarchy is a refinement of the polynomial hierarchy. It is shown that the plus hierarchy is proper if the polynomial hierarchy is proper. The proof uses ideas from a variant of the proof of J.  ...  in the length of the in- stance’ and ‘easy’ is formalized as ‘doable in polynomial time’.  ... 
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