Cs221: Computational Complexity Lecture 27: Average-case Complexity, Low-degree Extensions
نویسندگان
چکیده
Theorem 1 If there exists a probabilistic polynomial time (PPT) algorithm A such that PrM←μk,p [A(1 k, 1p,M) 6= Perm(M) mod p ] ≤ 1 3(k+1) , then there exists a PPT algorithm B such that for every M ∈ Zk×k p , Prcoins of B [B(1k, 1p,M) 6= Perm(M) mod p] ≤ 1/3 In particular, we’d have that ModPerm ∈ BPP, and thus P#P = BPP. (We proved ModPerm was #P-hard last class). Note that this theorem is usually read negatively—since we don’t believe these consequences hold, it is unlikely such an A exists. The proof of this theorem follows (as shown in the last class), from this more general one, which we will now prove:
منابع مشابه
Cs221: Computational Complexity Lecture 20: Circuit Complexity
2 Existential Lower Bounds Theorem 1 (Lupanov, Shannon) For each ε > 0 and for each sufficiently large n, ∃f : {0, 1}n → {0, 1} with circuit complexity (over the full basis B2) at least (1− ε)2n n . Proof: We will prove the required result using a counting argument. In particular, we will show that the number of functions mapping {0, 1}n to {0, 1} is much larger than the number of ’distinct’ ci...
متن کاملReduction of Computational Complexity in Finite State Automata Explosion of Networked System Diagnosis (RESEARCH NOTE)
This research puts forward rough finite state automata which have been represented by two variants of BDD called ROBDD and ZBDD. The proposed structures have been used in networked system diagnosis and can overcome cominatorial explosion. In implementation the CUDD - Colorado University Decision Diagrams package is used. A mathematical proof for claimed complexity are provided which shows ZBDD ...
متن کاملA MULTI-OBJECTIVE OPTIMIZATION MODEL FOR PROJECT PORTFOLIO SELECTION CONSIDERING AGGREGATE COMPLEXITY: A CASE STUDY
Existing project selection models do not consider the complexity of projects as a selection criterion, while their complexity may prolong the project duration and even result in its failure. In addition, existing models cannot formulate the aggregate complexity of the selected projects. The aggregated complexity is not always equal to summation of complexity of projects because of possible syne...
متن کاملNotes for Lecture 1
This course assumes CS170, or equivalent, as a prerequisite. We will assume that the reader is familiar with the notions of algorithm and running time, as well as with basic notions of discrete math and probability. We will occasionally refer to Turing machines, especially in this lecture. A main objective of theoretical computer science is to understand the amount of resources (time, memory, c...
متن کاملBiclique Cryptanalysis of Block Ciphers LBlock and TWINE-80 with Practical Data Complexity
In the biclique attack, a shorter biclique usually results in less data complexity, but at the expense of more computational complexity. The early abort technique can be used in partial matching part of the biclique attack in order to slightly reduce the computations. In this paper, we make use of this technique, but instead of slight improvement in the computational complexity, we keep the amo...
متن کامل