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A Simple Proof of Toda's Theorem

2009
*
Theory of Computing
*

We give

doi:10.4086/toc.2009.v005a007
dblp:journals/toc/Fortnow09
fatcat:b2uxjulvufd6jiot25t7cwudnm
*a*short and*simple**proof**of*the first half*of**Toda's**Theorem*that the polynomial-time hierarchy is contained in BPP ⊕P . ... Our*proof*uses easy consequences*of*relativizable*proofs**of*results that predate Toda. For completeness we also include*a**proof**of*the second half*of**Toda's**Theorem*. ... In this paper we give*a*short*proof**of*Lemma 1.2 using relativizable versions*of*results that predate*Toda's**Theorem*. For completeness we will give*a**proof**of*Lemma 1.3 as well. ...##
###
On Circuit Lower Bounds from Derandomization

2011
*
Theory of Computing
*

Our

doi:10.4086/toc.2011.v007a012
dblp:journals/toc/AaronsonM11
fatcat:qtjslytlj5f2nncr7jmk6rlyi4
*proof*is simpler, scales better, and yields*a*somewhat stronger result than the original argument. ... We present an alternate*proof**of*the result by Kabanets and Impagliazzo (2004) that derandomizing polynomial identity testing implies circuit lower bounds. ... Here we merely present*a*sketch*of*the modifications to our*proof**of**Theorem*2.1 and an explanation*of*how*Toda's**Theorem*helps. ...##
###
COUNTING HIERARCHIES: POLYNOMIAL TIME AND CONSTANT DEPTH CIRCUITS
[chapter]

1993
*
Current Trends in Theoretical Computer Science
*

Along the way, we'll review the important notion

doi:10.1142/9789812794499_0035
dblp:series/wsscs/AllenderW93
fatcat:l4d7d6qhzvbzjen54ekvvxujie
*of*an operator on*a*complexity class. ... In this Structural Complexity Column, we will brie y review*Toda's*result, and explore how it relates to other topics*of*interest in computer science. ... (The*proof**of*the uniform version still does not appeal to*Toda's**theorem*.) The*proof**of*the*theorem*for uniform circuits is found in AH-90]. ...##
###
Page 1058 of Mathematical Reviews Vol. 26, Issue 5
[page]

1963
*
Mathematical Reviews
*

.

*of*Math. (2) 72 (1960), 292-311; MR 22 #11403] he had given*a*roundabout*proof**of*this fact. ... ./, satisfies all the above conditions, including Condition C, and the author obtains*a*direct*proof*that the stable cohomology*of*the Thom space M(SO(n)) is*a**simple*»/,-module; in an earlier paper [Ann ...##
###
Page 990 of Mathematical Reviews Vol. 30, Issue 6
[page]

1965
*
Mathematical Reviews
*

The diffeomorphism V,,#=*"~-' ~ V, is proved by

*a**simple*geometric argument, using Smale’s h-cobordism*theorem*. M.*A*. Kervaire (New York) Hsiang, W. C.; Levine, J.; Szezarba, R. ... The*proofs*are fairly*simple*and the exposition is very careful. M.*A*. Kervaire (New York) Libermann, Paulette 5326 Surconnexions et singularités des applicati C. R. Acad. Sci. ...##
###
Page 4750 of Mathematical Reviews Vol. , Issue 94h
[page]

1994
*
Mathematical Reviews
*

Summary: “We present

*a**simple**proof**of*S.*Toda’s*result [in 30th Annual Symposium on Foundations*of*Computer Science (Research Triangle Park, NC, 1989), 514-519, IEEE Comput. Sci. ... (1-CMU-SC; Pittsburgh, PA); Venkateswaran, H. (1-GAIT; Atlanta, GA); Vinay, V. (6-IIS-CS; Bangalore); Yao, Andrew C. (1-PRIN-CS; Princeton, NJ)*A*circuit-based*proof**of**Toda’s**theorem*. ...##
###
Page 995 of American Journal of Mathematics Vol. 80, Issue 4
[page]

1958
*
American Journal of Mathematics
*

The following

*theorems*concern the homotopy groups 7;*of*Lie groups. It is*a*classical fact that 7.(K) =0 for semi-*simple*K; and it is known [6] that 73(K) = Z for*simple*K. ... The*proofs**of*the*theorems**of*no. 2 depend on the construction*of*certain iterated 2-sphere bundles that appear as K-cycles for suitable actions*of*K. ...##
###
A simple constant-probability RP reduction from NP to Parity P
[article]

2008
*
arXiv
*
pre-print

The

arXiv:0810.1018v1
fatcat:mgf3jdglazggjgqcu5w3tn4p7q
*proof**of**Toda's*celebrated*theorem*that the polynomial hierarchy is contained in ^# P relies on the fact that, under mild technical conditions on the complexity class C, we have ∃ C ⊂ BP ·⊕ C. ... Our reduction is very*simple*, and its analysis relies on well-known properties*of*the Legendre symbol in finite fields. ... Our reduction is quite*simple*, and may be*of*independent interest. ...##
###
Page 8278 of Mathematical Reviews Vol. , Issue 2001K
[page]

2001
*
Mathematical Reviews
*

*A*consequence

*of*the main

*theorem*and the

*proof*

*of*

*Toda’s*

*theorem*is an exponential-size lower bound for every depth-three circuit that computes Mod, and consists

*of*an “exact counting” gate over Mod, ... But this claim is false:

*a*

*simple*counterexample is given by the set

*of*all circuits which evaluate to 0 on the all-0 assignment. ...

##
###
A Complex Analogue of Toda's Theorem

2011
*
Foundations of Computational Mathematics
*

*A*key role is played by the complex join

*of*quasi-projective complex varieties. As

*a*consequence we obtain

*a*complex analogue

*of*

*Toda's*

*theorem*. ... Unlike

*Toda's*

*proof*in the discrete case, which relied on sophisticated combinatorial arguments, the

*proof*in [1] is topological in nature in which the properties

*of*the topological join is used in

*a*fundamental ... It is interesting to observe that in complete analogy with the

*proof*

*of*the classical

*Toda's*

*theorem*the

*proof*

*of*

*Theorem*2.1 also requires just one call to the oracle at the end. ...

##
###
THE COMPLEXITY OF FINDING MIDDLE ELEMENTS

1993
*
International Journal of Foundations of Computer Science
*

Our results help in clarifying the status

doi:10.1142/s0129054193000195
fatcat:lpgswhfhnrgazftwbybye46bii
*of**Toda's*very important class Mid P in showing that it is closely related to the class PP NP . ... Seinosuke Toda introduced the class Mid P*of*functions that yield the middle element in the set*of*output values over all paths*of*nondeterministic polynomial time Turing machines. ... Thanks to Johannes K obler (Ulm) for pointing out and correcting an error in*a*previous*proof**of*one*of*our*theorems*. ...##
###
On Jones's Kahn-Priddy theorem
[chapter]

1990
*
Lecture notes in mathematics
*

For ttirosi Toda on his sixtieth birthday In this note we record

doi:10.1007/bfb0083705
fatcat:jjrar53nbzd3jc27zqdon6gbmq
*a**simple**proof**of**a*beautiful result*of*J.D.S. ... We end with*a**proof**of*Mahowald's*theorem*that*a*i e R(pi), and the observation that Jones's*theorem*allows one to translate*a*20-year old result*of**Toda's*into the assertion that up to*a*unit, 81 e R( ... P-s-1 is nulliff PSr ~ Pqk-e q(*a*), s-t The latter is surely null if s < qk-e; and this is Jones's*theorem*(1.5) and its extension to odd primes,*of*(3.1) is quite*simple*. Let s = 2~+6. ...##
###
A complex analogue of Toda's Theorem
[article]

2011
*
arXiv
*
pre-print

*A*key role is played by the complex join

*of*quasi-projective complex varieties. As

*a*consequence we obtain

*a*complex analogue

*of*

*Toda's*

*theorem*. ... Unlike

*Toda's*

*proof*in the discrete case, which relied on sophisticated combinatorial arguments, the

*proof*in BZ09 is topological in nature in which the properties

*of*the topological join is used in

*a*... It is interesting to observe that in complete analogy with the

*proof*

*of*the classical

*Toda's*

*theorem*the

*proof*

*of*

*Theorem*2.1 also requires just one call to the oracle at the end. ...

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

1994
*
Lecture Notes in Computer Science
*

We use each

doi:10.1007/3-540-58715-2_130
fatcat:oqj4nco6ozaxpclwtipoor2egi
*of*the*theorems*as*a*springboard to discuss work done in various areas*of*complexity theory. ... We review the past ten years in computational complexity theory by focusing on ten*theorems*that the author enjoyed the most. ... We highlight the last*of*the truly great results in interactive*proof*systems,*a*paper by Arora, Lund, Motwani, Sudan and Szegedy that characterizes NP by*a**simple*veri cation procedure: Favorite*Theorem*...##
###
Fringe families in stable homotopy

1976
*
Transactions of the American Mathematical Society
*

The method involves constructing the stable maps by factoring through "F(zz)-spaces"spaces whose Brown-Peterson cohomology has

doi:10.1090/s0002-9947-1976-0431160-5
fatcat:pjrxlqc3fjh7fpih4vhs5jum54
*a**simple*structure. ... Recent work on the p-primary stable homotopy groups*of*spheres has shown that it is possible to give*a*unified interpretation*of*the three infinite families at, ßt, yt*of*indecomposable elements (p is ...*Proof**of**Theorem*10. Compute ExtA+x¿p2+p)q-x(BP*,BP*l(p, u^)) as*a*subquotient*of*Homjf +p+l\Cq+x, BP*/(p, ifx)), using the resolution C*of*Proposition 15. Ifp>5, l<r<p-l,the*Proof**of**Theorem*1(c). ...
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