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Closure properties of GapP and #P

R. Beigel
Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems  
We classify the univariate functions that are relativizable closure properties of GapP, solving a problem posed by Hertrampf, Vollmer, and Wagner (Structures '95).  ...  We also give a simple proof of their classi cation of univariate functions that are relativizable closure properties of #P.  ...  Acknowledgments We thank Bill Gasarch for proofreading and Uli Hertrampf, Heribert Vollmer, and Klaus Wagner for an advance copy of their paper.  ... 
doi:10.1109/istcs.1997.595166 dblp:conf/istcs/Beigel97 fatcat:axcer644obhgplkvu4h73lcpzi

Gap-definable counting classes

Stephen A. Fenner, Lance J. Fortnow, Stuart A. Kurtz
1994 Journal of computer and system sciences (Print)  
GapP is the closure of #P under subtraction and has all the other useful closure properties of ga P as well.  ...  The function class gap lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P.  ...  We also thank Krzysztof Lorys for pointing out an error in an earlier version of the paper.  ... 
doi:10.1016/s0022-0000(05)80024-8 fatcat:iamrtup2ofdgrino3oq63msqhq

Page 7654 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
The authors investigate the closure properties of GapP and GapP , under some simple arithmetic operations.  ...  GapP is the closure of #P under subtraction and GapP , is the subset of GapP consisting of nonnegative functions.  ... 

Gap-definability as a closure property [chapter]

Stephen Fermer, Lance Fortnow, Lide Li
1993 Lecture Notes in Computer Science  
This class is exactly the closure of *P under subtraction. GapP also has all the other nice closure properties of *P, such as addition, multiplication, and binomial coefficients.  ...  We use the results of the previous sections to describe some properties of gap-closure and gapdefinability.  ... 
doi:10.1007/3-540-56503-5_48 fatcat:m5u2ur2x6zglbhprv6m67gl4uq

Gap-Definability as a Closure Property

Stephen Fenner, Lance Fortnow, Lide Li
1996 Information and Computation  
This class is exactly the closure of *P under subtraction. GapP also has all the other nice closure properties of *P, such as addition, multiplication, and binomial coefficients.  ...  We use the results of the previous sections to describe some properties of gap-closure and gapdefinability.  ... 
doi:10.1006/inco.1996.0080 fatcat:3fr7xlh4ezfahl3iat6yymxduy

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

Edith Hemaspaandra, Lane A. Hemaspaandra, Holger Spakowski, Osamu Watanabe, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
polynomial-sized lists; yet on the other hand, we show that for the #P-based analog of LWPP the behavior much differs in that, in some relativized worlds, even two target values already yield a richer  ...  The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction) is in LWPP (and thus low for PP, i.e., PP Legitimate Deck = PP) if the weakened version of the  ...  Closure Property 3.2 ([8] ). GapP • FP = GapP and FP ⊆ GapP. Closure Property 3.3 ([8] ). If g ∈ GapP then −g ∈ GapP. Closure Property 3.4 ([8] ).  ... 
doi:10.4230/lipics.mfcs.2018.51 dblp:conf/mfcs/HemaspaandraHS018 fatcat:g7jtwaapwjaltkgf6dfctwnck4

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction [article]

Edith Hemaspaandra, Lane A. Hemaspaandra, Holger Spakowski, Osamu Watanabe
2018 arXiv   pre-print
Despite that nonrobustness result for a #P-based class, we show that the #P-based "exact counting" class C_=P remains unchanged even if one allows a polynomial number of target values for the number of  ...  The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction) is in LWPP (and thus low for PP, i.e., PP^Legitimate Deck = PP) if the weakened version of the  ...  Closure Property 3.2 ( [FFK94] ). GapP • FP = GapP and FP ⊆ GapP. Closure Property 3.3 ( [FFK94] ). If g ∈ GapP then −g ∈ GapP. Closure Property 3.4 ( [FFK94] ).  ... 
arXiv:1711.01250v3 fatcat:ijariarmwjbejcosyfo6zyjske

What is in #P and what is not? [article]

Christian Ikenmeyer, Igor Pak
2022 arXiv   pre-print
We initiate the study of the polynomial closure properties of #P on affine varieties, i.e., if all problem instances satisfy algebraic constraints.  ...  We investigate #TFNP and obtain oracle separations that prove the strict inclusion of #P in all standard syntactic subclasses of #TFNP-1.  ...  We thank Markus Bläser and Paul Goldberg for helpful comments on a draft version of this paper.  ... 
arXiv:2204.13149v1 fatcat:mbx6hsldbnf5nku2d7o5gws3we

Complexity limitations on quantum computation [article]

Lance Fortnow, John D. Rogers
1998 arXiv   pre-print
There exists a relativized world where P=BQP but P is not equal to UP intersect coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.  ...  We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation.  ...  Acknowledgments We would like to thank André Berthiaume, Harry Buhrman, Richard Cleve, Ronald de Wolf, Wim van Dam and John Watrous for a number of illuminating conversations on quantum computation.  ... 
arXiv:cs/9811023v1 fatcat:mxiruxhcyneehe3qa6pey6wcoq

Complexity Limitations on Quantum Computation

Lance Fortnow, John Rogers
1999 Journal of computer and system sciences (Print)  
world, where P=BQP, but P{UP & coUP and one-way functions exist.  ...  We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation.  ...  ACKNOWLEDGMENTS We thank Andre Berthiaume, Harry Buhrman, Richard Cleve, Ronald de Wolf, Wim van Dam, and John Watrous for a number of illuminating conversations on quantum computation.  ... 
doi:10.1006/jcss.1999.1651 fatcat:2pkcasoih5enzg3lvd7e2b7ayi

Page 5542 of Mathematical Reviews Vol. , Issue 95i [page]

1995 Mathematical Reviews  
System Sci. 48 (1994), no. 1, 116-148; MR 95d:68047] considered the class GapP, the closure of #P under subtraction, and showed many interesting properties of this class.  ...  “We introduce the class GapSpanP, the closure of SpanP under subtraction. We show that this class of functions coincides with the class GapP’?.”  ... 

Revisiting a limit on efficient quantum computation

Tarsem S. Purewal
2006 Proceedings of the 44th annual southeast regional conference on - ACM-SE 44  
Our proof follows the one given by Fortnow and Rogers that relates quantum computing to counting complexity classes by way of GapP functions.  ...  The contribution of this paper is an exposition of an important result that assumes a minimal background in computational complexity theory and no knowledge of quantum mechanics.  ...  Also, thanks to Shelby Funk and Bob Robinson for reading earlier drafts of this paper and offering suggestions.  ... 
doi:10.1145/1185448.1185502 dblp:conf/ACMse/Purewal06 fatcat:xw4tip375ve5bpjbue3jz2jcze

My favorite ten complexity theorems of the past decade [chapter]

Lance Fortnow
1994 Lecture Notes in Computer Science  
We use each of the theorems as a springboard to discuss work done in various areas of complexity theory.  ...  Lokam,Dieter Van Melkebeek and Sophie Laplante for their comments and help on this paper.  ...  Fenner, Fortnow and Kurtz show that many of the #P closure properties also hold for GapP and that looking at GapP functions simpli ed many counting complexity arguments.  ... 
doi:10.1007/3-540-58715-2_130 fatcat:oqj4nco6ozaxpclwtipoor2egi

Revealed Price Preference: Theory and Stochastic Testing

Rahul Deb, Yuichi Kitamura, John Kim-Ho Quah, JJrg Stoye
2017 Social Science Research Network  
If at observations t and t , we find that p t · x t < p t · x t , then 4 DEB, KITAMURA, QUAH, AND STOYE  ...  In formal terms, this requires that the consumer has a preference over grocery bundles that is weakly separable from her consumption of all other goods. 2 But, presented with the same data set D = {(p  ...  We denote the transitive closure of p by * p , that is, for p t and p t in P, we have p t * p p t if there are t 1 , t 2 ,...  ... 
doi:10.2139/ssrn.2970512 fatcat:zcqehiyhkfflfhq5chpraai4pa

Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy

S. Fenner, F. Green, S. Homer, R. Pruim
1999 Proceedings of the Royal Society A  
This result is achieved by showing that the complexity class NQP of Adleman, Demarrais, and Huang 1], a quantum analog of NP, is equal to the counting class coC = P.  ...  It is shown that determining whether a quantum computation has a non-zero probability of accepting is at least as hard as the polynomial time hierarchy.  ...  W atrous and the referees for helpful comments. An earlier version of this paper appeared in the Sixth Italian Conference on Theoretical Computer Science, October, 1998 11].  ... 
doi:10.1098/rspa.1999.0485 fatcat:jhzu3u4kyvf2dptulz3p4gvyn4
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