7-3. PCP

Cover vol.7, no.3

2005 ) Lecture 3 : Expander Graphs and PCP Theorem Proof Overview

Recall in last lecture that we defined a (n, d, λ)-expander to be a d-regular n-vertex undirected graph with second eigenvalue λ. We also defined the edge expansion of a graph G with vertex set V to be φ(G) = min S⊂V |S|≤n/2 |E(S, S)| |S| , where E(S, S) is the set of edges between a vertex set S and its complement. The following lemma shows that the eigenvalue formulation of an expander is ess...

Notes for Lecture 3 Some Consequences of the Pcp Theorem

Let now φ be an instance of 3SAT with n variables x1, . . . , xn and m clauses. For each variable xi, let occi be the number of occurrences of xi, that is, the number of clauses that involve the literal xi or the literal x̄i. We write xi ∈ Cj if the variable xi occurs in clause Cj . Notice that ∑n i=1 occi = 3m. For each i, construct a 1-expander graph Gi = (Vi, Ei) where Vi has occi vertices, o...

سنتز 1، 3، 5، 7-تترا استیل-1، 3، 5، 7-تترا آزاسیکلواکتان (TAT) از طریق استیل‌دار کردن 3، 7-دی‌استیل-1، 3، 5، 7-تترا آزا بی‌سیکلو-[3.3.1]-نونان (DAPT)

In this study, a novel method to synthesis of 1,3,5,7-tetraacetyl octahydro-1,3,5,7-tetraazocine or 1,3,5,7-tetraacetyl-1,3,5,7-tetraazacyclooctane (TAT) was used through acetolysis of 3,7-diacetyl-1,3,5,7-tetraazabicyclo-[3.3.1]nonane (DAPT) with acetic anhydride in the presence of sodium acetate as catalyst. Optimized reaction conditions were obtained by changing temperature, time and startin...

Pcp Verifiers

Definition 1 ((r, q)-verifier) An algorithm V is an (r, q)-verifier for a language L if on input strings x and y: step 1 V reads input x and draws r random bits ρ ∈ {0, 1}. step 2 V decides (after a polynomial-time computation in |x|) on q indexes: i1, . . . , iq and on a predicate φ : {0, 1} → {0, 1}. step 3 V outputs φ(y[i1], . . . , y[iq]). and the following holds, 1. (Completeness) If x ∈ L...