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On the Impossibility of Probabilistic Proofs in Relativized Worlds

Alessandro Chiesa, Siqi Liu, Michael Wagner
2020 Innovations in Theoretical Computer Science  
We initiate the systematic study of probabilistic proofs in relativized worlds, where the goal is to understand, for a given oracle, the possibility of "non-trivial" proof systems for deterministic or  ...  Informally, negative results about probabilistic proofs in relativized worlds provide evidence that this line of work is inherent and, conversely, positive results provide a way to bypass it.  ...  (which we then solve). 57:8 On the Impossibility of Probabilistic Proofs in Relativized Worlds 3.  ... 
doi:10.4230/lipics.itcs.2020.57 dblp:conf/innovations/ChiesaL20 fatcat:nzwmckdggvbzjn2ni7djdnu52y

Relativization: a Revisionistic Retrospective [chapter]

Juris Hartmanis, Richard Chang, Suresh Chari, Desh Ranjan, Pankaj Rohatgi
1993 Current Trends in Theoretical Computer Science  
In this column we examine the role of relativization in complexity theory in light of recent non-relativizing results involving interactive protocols.  ...  We begin with the twice-told tale of the relativization principle and ponder upon its possible demise. Then, we discuss whether usual assumptions are historically accurate.  ...  Intuitively, probabilistic proof checking is a generalization of the concept of NP.  ... 
doi:10.1142/9789812794499_0040 dblp:series/wsscs/HartmanisCCRR93 fatcat:xi32jv2ukjg4lbvrk62gorwjyu

Simultaneous strong separations of probabilistic and unambiguous complexity classes

David Eppstein, Lane A. Hemachandra, James Tisdall, B�lent Yener
1992 Mathematical Systems Theory  
In particular, we display a relativized world in which the complexity classes embodying these paradigms of computation are mutually immune.  ...  We study the relationship between probabilistic and unambiguous computation, and provide strong relativized evidence that they are incomparable.  ...  Informally, if statement S fails in some relativized world, we can never hope to prove S in the real world by any relativizable proof technique.  ... 
doi:10.1007/bf01368782 fatcat:hrdtdasxlzd7bpqabjyz7qq2vy

Dialogue Concerning The Two Chief World Views [article]

Craig Alan Feinstein
2016 arXiv   pre-print
In 1632, Galileo Galilei wrote a book called Dialogue Concerning the Two Chief World Systems which compared the new Copernican model of the universe with the old Ptolemaic model.  ...  His book took the form of a dialogue between three philosophers, Salviati, a proponent of the Copernican model, Simplicio, a proponent of the Ptolemaic model, and Sagredo, who was initially open-minded  ...  Wapner: And yes indeed, I am convinced that Feinstein's proof is valid and that P = NP.  ... 
arXiv:1605.08639v1 fatcat:j3kivq2bzzcwnpvn4t6ze2b6jm

Probabilistic computation and linear time

L. Fortnow, M. Sipser
1989 Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89  
In addition, we also give oracles where P 2 is contained in probabilistic linear time and where BPP has linear sized circuits, as well as oracles for the negation of these questions.  ...  In this paper, we give an oracle under which BPP is equal to probabilistic linear time, an unusual collapse of a complexity time hierarchy.  ...  Acknowledgments The authors would like to thank Eric Schwabe, Richard Beigel, Eric Allender and Rutger Verbeek for their useful comments on this paper.  ... 
doi:10.1145/73007.73021 dblp:conf/stoc/FortnowS89 fatcat:hxyksoak6fafpnabyotzr4xdhy

Complexity classes without machines: On complete languages for UP [chapter]

Juris Hartmanis, Lane Hemachandra
1986 Lecture Notes in Computer Science  
We suggest the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class BPP.  ...  S in P of uniquely satisfiable Boolean formulas such that no polynomial-time machine can compute the solutions for the formulas in S.  ...  US [2], a close cousin to UP, has complete languages in every relativized world. Nonetheless, our techniques yield noncompleteness results for probabilistic compu- tation models.  ... 
doi:10.1007/3-540-16761-7_62 fatcat:erxbde3l5ngyhm5lwyyzvhcopq

Complexity classes without machines: On complete languages for UP

Juris Hartmanis, Lane A. Hemachandra
1988 Theoretical Computer Science  
We suggest the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class BPP.  ...  S in P of uniquely satisfiable Boolean formulas such that no polynomial-time machine can compute the solutions for the formulas in S.  ...  US [2], a close cousin to UP, has complete languages in every relativized world. Nonetheless, our techniques yield noncompleteness results for probabilistic compu- tation models.  ... 
doi:10.1016/0304-3975(88)90022-9 fatcat:my7hpzvnk5eevmpsxfhs4rwbqa

Notions of Black-Box Reductions, Revisited [chapter]

Paul Baecher, Christina Brzuska, Marc Fischlin
2013 Lecture Notes in Computer Science  
Third, we distinguish explicitly between efficient and inefficient primitives and adversaries, allowing us to determine how relativizing reductions in the sense of Impagliazzo and Rudich (STOC, 1989) fit  ...  An impossibility result, which clearly specifies the type of reduction it rules out, enables us to identify the potential leverages to bypass the separation.  ...  Acknowledgments We thank the anonymous reviewers and Pooya Farshim for valuable comments on previous versions of this work.  ... 
doi:10.1007/978-3-642-42033-7_16 fatcat:ut2xagqdt5fsxd2clmqziucdle

Oracles versus proof techniques that do not relativize [chapter]

Eric Allender
1990 Lecture Notes in Computer Science  
This paper reviews the status of nonrelativizing proof techniques, and argues that many oracle constructions still yield valuable information about problems in complexity theory.  ...  However, the existence of nonrelativizing proof techniques seems to call this practice into question.  ...  Acknowledgments: I thank Ron Book, Jack Lutz, Pekka Orponen, and Seinosuke Toda for some interesting discussions that took place at the Workshop in Structural Complexity Theory at the University of California  ... 
doi:10.1007/3-540-52921-7_54 fatcat:qsdgtlzrbnbfjgzjkutmddmsfi

New Collapse Consequences of NP Having Small Circuits

Johannes Köbler, Osamu Watanabe
1998 SIAM journal on computing (Print)  
This improves on the well-known result of Karp, Lipton, and Sipser [KL80] stating a collapse of PH to its second level Σ P 2 under the same assumption.  ...  We show that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP(NP).  ...  Thierauf for permitting us to include their observations in the paper.  ... 
doi:10.1137/s0097539795296206 fatcat:fgjq46fytfh2rfxp6tphe463wm

New collapse consequences of NP having small circuits [chapter]

Johannes Köbler, Osamu Watanabe
1995 Lecture Notes in Computer Science  
This improves on the well-known result of Karp, Lipton, and Sipser [KL80] stating a collapse of PH to its second level Σ P 2 under the same assumption.  ...  We show that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP(NP).  ...  Thierauf for permitting us to include their observations in the paper.  ... 
doi:10.1007/3-540-60084-1_74 fatcat:biuud4ur65cmtfoa33pwh2xjmm

One complexity theorist's view of quantum computing

Lance Fortnow
2003 Theoretical Computer Science  
Quantum computing often scares away many potential researchers in computer science because of the apparent background need in quantum mechanics and the alien looking notation used in papers on the topic  ...  The tools of computational complexity theory should come to bear on these important questions.  ...  of the conference.  ... 
doi:10.1016/s0304-3975(01)00377-2 fatcat:5yviiw6j3reljpo6jjyhq7ri4i

One Complexity Theorist's View of Quantum Computing

Lance Fortnow
2000 Electronical Notes in Theoretical Computer Science  
Quantum computing often scares away many potential researchers in computer science because of the apparent background need in quantum mechanics and the alien looking notation used in papers on the topic  ...  The tools of computational complexity theory should come to bear on these important questions.  ...  of the conference.  ... 
doi:10.1016/s1571-0661(05)80330-5 fatcat:iz6kadfebzajjia5r34k6kcrmi

One Complexity Theorist's View of Quantum Computing [article]

Lance Fortnow
2000 arXiv   pre-print
Quantum computing often scares away many potential researchers from computer science because of the apparent background need in quantum mechanics and the alien looking notation used in papers on the topic  ...  The tools of computational complexity theory should come to bear on these important questions.  ...  The opinions expressed in this paper remain solely my own. Dieter van Melkebeek gave many useful comments on an earlier draft of this paper.  ... 
arXiv:quant-ph/0003035v1 fatcat:5f2kmepgsbdetcqiipkmuhif54

Limits on the provable consequences of one-way permutations

R. Impagliazzo, S. Rudich
1989 Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89  
Since both sides of this implication are widely believed true in real life, to show that the implication is false requires a new model.  ...  Thus, no technique which relativizes can prove that secret exchange can be based on any one-way permutation.  ...  We give Manuel and Umesh a second helping of thanks for their support and encouragement during the many times we found a fatal flaw in what we thought was a proof.  ... 
doi:10.1145/73007.73012 dblp:conf/stoc/ImpagliazzoR89 fatcat:luinwsh245d57bjzjgvgpbxez4
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