Stephen R. Mahaney

Inexact agreement: accuracy, precision, and graceful degradation

An Inexact Agreement protocol alows processors that each have a value approximating

The isomorphism conjecture fails relative to a random oracle

\hat{\nu}

One-way log-tape reductions

to compute new values that are closer to each other and close to

Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis

\hat{\nu}

The ismorphism conjecture fails relative to a random oracle

. Two fault-tolerant protocols for Inexact Agreement are described. As long as fewer than 1/3 of the processors are faulty, the protocols give the required convergence; they also permit iteration and thus convergence to any desired precision. When between 1/3 and 2/3 of the processors are faulty, the protocols may not converge. However, then processors either detect that too many faults have occurred or the new values computed by processors remain close to each other and to

The Structure of Complete Degrees

\hat{\nu}

Collapsing degrees

. In this case, the divergence is bounded. Use of the protocols for clock synchronization in a distributed system is explained.

Reductions among polynomial isomorphism types

Berman and Hartmanis [1977] conjectured that there is a polynomial-time computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [1985] gave a structural definition of a class of NP-complete sets-the k-creative sets-and defined a class of sets (the K f k s) that are necessarily k-creative. They went on to conjecture that certain of these K f t s are not isomorphic to the standard NP-complete sets. Clearly, the Berman-Hartmanis and Joseph-Young conjectures cannot both be correct. We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K f k is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scrambling functions, we show that much more powerful one-way functions-the annihilating functions-exist relative to a random oracle. Random oracles are the first examples of oracles relative to which the isomorphism conjecture fails with respect to higher classes such as EXP and NEXP.

Languages Simultaneously Complete for One-Way and Two-Way Log-Tape Automata

One-way log-tape (1-L) reductions are mappings defined by log-tape Turing machines whose read head on the input can only move to the right. The 1-L reductions provide a more refined tool for studying the feasible complexity classes than the P-time [2,7] or log-tape [4] reductions. Although the 1-L computations are provably weaker than the feasible classes L, NL, P and NP, the known complete sets for those classes are complete under 1-L reductions. However, using known techniques of counting arguments and recursion theory we show that certain log-tape reductions cannot be 1-L and we construct sets that are complete under log-tape reductions but not under 1-L reductions.

Relativizing relativized computations

A set S ⊂ {0,1}* is sparse if there is a polynomial p such that the number of strings in S of size at most n is at most p(n). All known NP-complete sets, such as SAT, are not sparse. The main result of this paper is that if there is a sparse NP-complete set under many-one reductions, then P = NP. We also show that if there is a sparse NP-complete set under Turing reductions, then the polynomial time hierarchy collapses to Δ2P.