Commonsense Reasoning and Large Network Analysis: A Computational Study of ConceptNet 4

In this report a computational study of ConceptNet 4 is performed using tools from the field of network analysis. Part I describes the process of extracting the data from the SQL database that is available online, as well as how the closure of the input among the assertions in the English language is computed. This part also performs a validation of the input as well as checks for the consistency of the entire database. Part II investigates the structural properties of ConceptNet 4. Different graphs are induced from the knowledge base by fixing different parameters. The degrees and the degree distributions are examined, the number and sizes of connected components, the transitivity and clustering coefficient, the cores, information related to shortest paths in the graphs, and cliques. Part III investigates non-overlapping, as well as overlapping communities that are found in ConceptNet 4. Finally, Part IV describes an investigation on rules.

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of $sOB(N^{14})$ for the purely projection-based method, and $sOB(N^{12})$ for two subresultant-based methods: this notation ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was $sOB(N^{14})$. Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in $sOB(N^{12})$, whereas the previous bound was $sOB(N^{14})$. All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE software