39 research outputs found
Quantifier Elimination over Finite Fields Using Gr\"obner Bases
We give an algebraic quantifier elimination algorithm for the first-order
theory over any given finite field using Gr\"obner basis methods. The algorithm
relies on the strong Nullstellensatz and properties of elimination ideals over
finite fields. We analyze the theoretical complexity of the algorithm and show
its application in the formal analysis of a biological controller model.Comment: A shorter version is to appear in International Conference on
Algebraic Informatics 201
A CDCL-style calculus for solving non-linear constraints
In this paper we propose a novel approach for checking satisfiability of
non-linear constraints over the reals, called ksmt. The procedure is based on
conflict resolution in CDCL style calculus, using a composition of symbolical
and numerical methods. To deal with the non-linear components in case of
conflicts we use numerically constructed restricted linearisations. This
approach covers a large number of computable non-linear real functions such as
polynomials, rational or trigonometrical functions and beyond. A prototypical
implementation has been evaluated on several non-linear SMT-LIB examples and
the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at
<http://informatik.uni-trier.de/~brausse/ksmt/
Learning a mixture of two multinomial logits
The classical Multinomial Logit (MNL) is a behavioral model for user choice. In this model, a user is offered a slate of choices (a subset of a finite universe of n items), and selects exactly one item from the slate, each with probability proportional to its (positive) weight. Given a set of observed slates and choices, the likelihood-maximizing item weights are easy to learn at scale, and easy to interpret. However, the model fails to represent common real-world behavior. As a result, researchers in user choice often turn to mixtures of MNLs, which are known to approximate a large class of models of rational user behavior. Unfortunately, the only known algorithms for this problem have been heuristic in nature. In this paper we give the first polynomial-time algorithms for exact learning of uniform mixtures of two MNLs. Interestingly, the parameters of the model can be learned for any n by sampling the behavior of random users only on slates of sizes 2 and 3; in contrast, we show that slates of size 2 are insufficient by themselves
A Global Approach for Solving Edge-Matching Puzzles
We consider apictorial edge-matching puzzles, in which the goal is to arrange
a collection of puzzle pieces with colored edges so that the colors match along
the edges of adjacent pieces. We devise an algebraic representation for this
problem and provide conditions under which it exactly characterizes a puzzle.
Using the new representation, we recast the combinatorial, discrete problem of
solving puzzles as a global, polynomial system of equations with continuous
variables. We further propose new algorithms for generating approximate
solutions to the continuous problem by solving a sequence of convex
relaxations