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### A Simple Proof of Toda's Theorem

Lance Fortnow
2009 Theory of Computing
We give 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

Scott Aaronson, Dieter van Melkebeek
2011 Theory of Computing
Our 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]

ERIC W. ALLENDER, KLAUS W. WAGNER
1993 Current Trends in Theoretical Computer Science
Along the way, we'll review the important notion 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]

Cristopher Moore, Alexander Russell
2008 arXiv   pre-print
The 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

Saugata Basu
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

HERIBERT VOLLMER, KLAUS W. WAGNER
1993 International Journal of Foundations of Computer Science
Our results help in clarifying the status 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]

Haynes Miller
1990 Lecture notes in mathematics
For ttirosi Toda on his sixtieth birthday In this note we record 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]

Saugata Basu
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]

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.  ...  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

Raphael S. Zahler
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 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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