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    Real root finding for equivariant semi-algebraic systems

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    Let RR be a real closed field. We consider basic semi-algebraic sets defined by nn-variate equations/inequalities of ss symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2d<n2d < n. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most 2d12d-1 distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by ss polynomials of degree dd in time (sn)O(d)(sn)^{O(d)}. This improves the state-of-the-art which is exponential in nn. When the variables x1,,xnx_1, \ldots, x_n are quantified and the coefficients of the input system depend on parameters y1,,yty_1, \ldots, y_t, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time (sn)O(dt)(sn)^{O(dt)}

    Dimensions of Neural-symbolic Integration - A Structured Survey

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    Research on integrated neural-symbolic systems has made significant progress in the recent past. In particular the understanding of ways to deal with symbolic knowledge within connectionist systems (also called artificial neural networks) has reached a critical mass which enables the community to strive for applicable implementations and use cases. Recent work has covered a great variety of logics used in artificial intelligence and provides a multitude of techniques for dealing with them within the context of artificial neural networks. We present a comprehensive survey of the field of neural-symbolic integration, including a new classification of system according to their architectures and abilities.Comment: 28 page
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