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Arithmetic theories for computational complexity problems

1986
*
Information and Control
*

This paper considers a number

doi:10.1016/s0019-9958(86)80041-9
fatcat:mp344yvwefeitja4jhkenj3n7q
*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*. ...##
###
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

2007
*
Journal of Logic and Computation
*

We then present a number

doi:10.1093/logcom/exm040
fatcat:jatxxmbvnbbjxes42ytkrujmme
*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*...##
###
Characterizing the polynomial hierarchy by alternating auxiliary pushdown automata

1989
*
RAIRO - Theoretical Informatics and Applications
*

complement [1],

doi:10.1051/ita/1989230100871
fatcat:6uwn5bwe6vay5gyqihy7ldikr4
*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*. ...##
###
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]

2013
*
Lecture Notes in Computer Science
*

Parameterized Complexity is a new

doi:10.1007/978-3-642-41524-1_2
fatcat:ee2md36scbckvne73x7qfkw5ra
*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*. ...##
###
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

2013
*
2013 IEEE 25th International Conference on Tools with Artificial Intelligence
*

reasoning, both located at

doi:10.1109/ictai.2013.136
dblp:conf/ictai/Szeider13
fatcat:da5r7643mbafrkvaw74evlwbra
*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. ...##
###
The difference and truth-table hierarchies for NP

1987
*
RAIRO - Theoretical Informatics and Applications
*

Further, we define complete sets

doi:10.1051/ita/1987210404191
fatcat:u6ragzrbbndktbjjnoveibir74
*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*). ...##
###
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]

2005
*
Lecture Notes in Computer Science
*

In this paper, we formally investigate

doi:10.1007/11533719_98
fatcat:ewojc2i2z5gbllf6ynfulngu2a
*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 ...##
###
Preface to the Special Issue on Computer Science in Russia 2016

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*...

##
###
A View of Structural Complexity Theory
[chapter]

1993
*
Current Trends in Theoretical Computer Science
*

Acknowledgments Preparation

doi:10.1142/9789812794499_0034
dblp:series/wsscs/Book093
fatcat:jcrj2u7caveejeojt6w6ouzqvq
*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 ...##
###
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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