Lower Bounds for Achieving Synchronous Early Stopping Consensus with Orderly Crash Failures

In this paper, we discuss the consensus problem for synchronous distributed systems with orderly crash failures. For a synchronous distributed system of n processes with up to t crash failures and f failures actually occur, first, we present a bivalency argument proof to solve the open problem of proving the lower bound, min (t + 1, f + 2) rounds, for early-stopping synchronous consensus with orderly crash failures, where t < n - 1. Then, we extend the system model with orderly crash failures to a new model in which a process is allowed to send multiple messages to the same destination process in a round and the failing processes still respect the order specified by the protocol in sending messages. For this new model, we present a uniform consensus protocol, in which all non-faulty processes always decide and stop immediately by the end of f + 1 rounds. We prove that the lower bound of early stopping protocols for both consensus and uniform consensus are f + 1 rounds under the new model, and our proposed protocol is optimal.

Mean Field Theory for Sigmoid Belief Networks

We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition -- the classification of handwritten digits.

Computational Structure of Human Language

The central thesis of this report is that human language is NP-complete. That is, the process of comprehending and producing utterances is bounded above by the class NP, and below by NP-hardness. This constructive complexity thesis has two empirical consequences. The first is to predict that a linguistic theory outside NP is unnaturally powerful. The second is to predict that a linguistic theory easier than NP-hard is descriptively inadequate. To prove the lower bound, I show that the following three subproblems of language comprehension are all NP-hard: decide whether a given sound is possible sound of a given language; disambiguate a sequence of words; and compute the antecedents of pronouns. The proofs are based directly on the empirical facts of the language user's knowledge, under an appropriate idealization. Therefore, they are invariant across linguistic theories. (For this reason, no knowledge of linguistic theory is needed to understand the proofs, only knowledge of English.) To illustrate the usefulness of the upper bound, I show that two widely-accepted analyses of the language user's knowledge (of syntactic ellipsis and phonological dependencies) lead to complexity outside of NP (PSPACE-hard and Undecidable, respectively). Next, guided by the complexity proofs, I construct alternate linguisitic analyses that are strictly superior on descriptive grounds, as well as being less complex computationally (in NP). The report also presents a new framework for linguistic theorizing, that resolves important puzzles in generative linguistics, and guides the mathematical investigation of human language.