Abstract :
A fundamental question of complexity theory is the direct product question. A famous example is Yao´s (1982) XOR-lemma, in which one assumes that some function f is hard on average for small circuits, (meaning that every circuit of some fixed size s which attempts to compute f is wrong on a non-negligible fraction of the inputs) and concludes that every circuit of size s´ has a small advantage over guessing randomly when computing f⊕k (x1, …, xk)=f(x1)⊕…⊕f(xk ) on independently chosen x1, …, xk. All known proofs of this lemma have the feature that s´<s. In words, the circuit which attempts to compute f⊕k is smaller than the circuit which attempts to compute f on a single input. This paper addresses the issue of proving strong direct product assertions, that is ones in which s´≈ks and is in particular larger than s. Since we are unable to “handle” boolean circuits we follow a direction suggested by previous works (Nisan et al., 1994; Parnafes et al., 1997) and study this question in weaker computational models such as decision trees and communication complexity games
Keywords :
computational complexity; decision trees; theorem proving; XOR-lemma; communication complexity games; complexity theory; computational models; decision trees; strong direct product theorem proving; Boolean functions; Circuits; Complexity theory; Computational modeling; Decision trees; Polynomials; Protocols;
