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On the Impossibility of Probabilistic Proofs in Relativized Worlds

2020
*
Innovations in Theoretical Computer Science
*

We initiate

doi:10.4230/lipics.itcs.2020.57
dblp:conf/innovations/ChiesaL20
fatcat:nzwmckdggvbzjn2ni7djdnu52y
*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. ...##
###
Relativization: a Revisionistic Retrospective
[chapter]

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. ...

##
###
Simultaneous strong separations of probabilistic and unambiguous complexity classes

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. ...

##
###
Dialogue Concerning The Two Chief World Views
[article]

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. ...

##
###
Probabilistic computation and linear time

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. ...

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

1986
*
Lecture Notes in Computer Science
*

We suggest

doi:10.1007/3-540-16761-7_62
fatcat:erxbde3l5ngyhm5lwyyzvhcopq
*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. ...##
###
Complexity classes without machines: On complete languages for UP

1988
*
Theoretical Computer Science
*

We suggest

doi:10.1016/0304-3975(88)90022-9
fatcat:my7hpzvnk5eevmpsxfhs4rwbqa
*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. ...##
###
Notions of Black-Box Reductions, Revisited
[chapter]

2013
*
Lecture Notes in Computer Science
*

Third, we distinguish explicitly between efficient and inefficient primitives and adversaries, allowing us to determine how

doi:10.1007/978-3-642-42033-7_16
fatcat:ut2xagqdt5fsxd2clmqziucdle
*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. ...##
###
Oracles versus proof techniques that do not relativize
[chapter]

1990
*
Lecture Notes in Computer Science
*

This paper reviews

doi:10.1007/3-540-52921-7_54
fatcat:qsdgtlzrbnbfjgzjkutmddmsfi
*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 ...##
###
New Collapse Consequences of NP Having Small Circuits

1998
*
SIAM journal on computing (Print)
*

This improves

doi:10.1137/s0097539795296206
fatcat:fgjq46fytfh2rfxp6tphe463wm
*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. ...##
###
New collapse consequences of NP having small circuits
[chapter]

1995
*
Lecture Notes in Computer Science
*

This improves

doi:10.1007/3-540-60084-1_74
fatcat:biuud4ur65cmtfoa33pwh2xjmm
*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. ...##
###
One complexity theorist's view of quantum computing

2003
*
Theoretical Computer Science
*

Quantum computing often scares away many potential researchers

doi:10.1016/s0304-3975(01)00377-2
fatcat:5yviiw6j3reljpo6jjyhq7ri4i
*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. ...##
###
One Complexity Theorist's View of Quantum Computing

2000
*
Electronical Notes in Theoretical Computer Science
*

Quantum computing often scares away many potential researchers

doi:10.1016/s1571-0661(05)80330-5
fatcat:iz6kadfebzajjia5r34k6kcrmi
*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. ...##
###
One Complexity Theorist's View of Quantum Computing
[article]

2000
*
arXiv
*
pre-print

Quantum computing often scares away many potential researchers from computer science because

arXiv:quant-ph/0003035v1
fatcat:5f2kmepgsbdetcqiipkmuhif54
*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. ...##
###
Limits on the provable consequences of one-way permutations

1989
*
Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89
*

Since both sides

doi:10.1145/73007.73012
dblp:conf/stoc/ImpagliazzoR89
fatcat:luinwsh245d57bjzjgvgpbxez4
*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*. ...
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