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In this paper we develop novel algorithmic ideas for building a natural language parser grounded upon the hypothesis of incrementality. Although widely accepted and experimentally supported under a cognitive perspective as a model of the human parser, the incrementality assumption has never been exploited for building automatic parsers of unconstrained real texts. The essentials of the hypothesis are that words are processed in a left-to-right fashion, and the syntactic structure is kept totally connected at each step.Our proposal relies on a machine learning technique for predicting the correctness of partial syntactic structures that are built during the parsing process. A recursive neural network architecture is employed for computing predictions after a training phase on examples drawn from a corpus of parsed sentences, the Penn Treebank. Our results indicate the viability of the approach and lay out the premises for a novel generation of algorithms for natural language processing which more closely model human parsing. These algorithms may prove very useful in the development of efficient parsers.