Annals of Mathematics and Artificial Intelligence最新文献

Inspired by Fisher’s geometric approach to study beneficial mutations, we analyse probabilities of beneficial mutation and crossover recombination of strings in a general Hamming space with arbitrary finite alphabet. Mutations and recombinations that reduce the distance to an optimum are considered as beneficial. Geometric and combinatorial analysis is used to derive closed-form expressions for transition probabilities between spheres around an optimum giving a complete description of Markov evolution of distances from an optimum over multiple generations. This paves the way for optimization of parameters of mutation and recombination operators. Here we derive optimality conditions for mutation and recombination radii maximizing the probabilities of mutation and crossover into the optimum. The analysis highlights important differences between these evolutionary operators. While mutation can potentially reach any part of the search space, the probability of beneficial mutation decreases with distance to an optimum, and the optimal mutation radius or rate should also decrease resulting in a slow-down of evolution near the optimum. Crossover recombination, on the other hand, acts in a subspace of the search space defined by the current population of strings. However, probabilities of beneficial and deleterious crossover are balanced, and their characteristics, such as variance, are translation invariant in a Hamming space, suggesting that recombination may complement mutation and boost the rate of evolution near the optimum.

Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops from the ground up an abstract algebraic and qualitative notion of similarity based on the observation that sets of generalizations encode important properties of elements. We show that similarity defined in this way has appealing mathematical properties. As we construct our notion of similarity from first principles using only elementary concepts of universal algebra, to convince the reader of its plausibility, we show that it can model fundamental relations occurring in mathematics and can be naturally embedded into first-order logic via model-theoretic types. Finally, we sketch some potential applications to theoretical computer science and artificial intelligence.

We study the set of optimal solutions of the dual linear programming formulation of the linear assignment problem (LAP) to propose a method for computing a solution from the relative interior of this set. Assuming that an arbitrary dual-optimal solution and an optimal assignment are available (for which many efficient algorithms already exist), our method computes a relative-interior solution in linear time. Since the LAP occurs as a subproblem in the linear programming (LP) relaxation of the quadratic assignment problem (QAP), we employ our method as a new component in the family of dual-ascent algorithms that provide bounds on the optimal value of the QAP. To make our results applicable to the incomplete QAP, which is of interest in practical use-cases, we also provide a linear-time reduction from the incomplete LAP to the complete LAP along with a mapping that preserves optimality and membership in the relative interior. Our experiments on publicly available benchmarks indicate that our approach with relative-interior solution can frequently provide bounds near the optimum of the LP relaxation and its runtime is much lower when compared to a commercial LP solver.

We elaborate, as a benchmark, a list of geometry theorems that are regularly used in Hungarian high schools, in specialized tracks for mathematics. Then, these statements are implemented as GeoGebra files, and algebraically proven (with certification) in GeoGebra Discovery. Next, these files are imported in Java Geometry Expert (JGEX), and geometrically proven with the associated Geometry Deductive Database method. In both programs, through quite different approaches in each case, we define and implement an algorithmic measure of the difficulty of a geometric theorem. Then, we compute the corresponding measures on the selected list of statements, and we analyze the obtained collection of pairs of difficulty grades, finding a moderate positive correlation (0.51) between GeoGebra Discovery’s algebraic, and JGEX’s deductive difficulty formulations. A correlation that seems to support the adequacy of the chosen approximation in our on-going work to algorithmically establish some ad-hoc measure of the difficulty of geometric statements that could be reasonably coincidental with the appreciation of human expert users (e.g. teachers).

System W is an inference method for conditional belief bases with some notable properties like capturing system Z while, in contrast to system Z, avoiding the drowning problem. This paper further investigates the properties of system W. We show how system W behaves with respect to postulates put forward for inference relations. We develop postulates ensuring compliance with syntax splitting for inference operators based on a full strict partial order on worlds. By observing that system W satisfies these axioms, it is proven that system W satisfies syntax splitting. We also explore how syntax splitting affects the strict partial order underlying system W and exploit this for answering certain types of queries without having to determine this strict partial order completely. However, the original definition of system W and the results above only consider inference from belief bases satisfying a strong notion of consistency. In the second part of this paper, we lift this limitation and extend system W to also cover inference from belief bases that only satisfy a weaker notion of consistency. We investigate the properties of the such extended system W. Especially, it is shown that extended system W complies with syntax splitting and retains the desireable properties of system W. Furthermore, we give an overview of the relations of extended system W to other inductive inference operators.

In this study, we have chosen the computer network with the shape of a king’s graph. The king’s graph G is defined as a set of edges, that is (E={((i,j),(p,q))|i,p in [0,M], j,q in [0,N], M,N in textbf{Z},((i,j),(p,q))~{is, an, edge},iff i = p quad {and} quad j = qpm 1 quad {or} quad i = ppm 1 quad {and} quad j = q quad {or} quad i = ppm 1 quad {and} quad j = qpm 1}). We also set a delivery rule, in which the shortest paths in the graph are used for the message deliveries, to restrict the source consumption. Then, the paths are encoded in a way that we discover using binary arrays based on other well-known encoding methods. We prove that the path-coding method we present prevents errors denoted by false positives from the graph. Data transfer issues from computer science served as the motivation for this study.

An agent-based modelling approach is a powerful means of understanding social phenomena by modelling individual behaviours and interactions. However, the advancements in modelling pose challenges in the model analysis process for understanding the complex effects of input factors, especially when it comes to offering concrete policies for improving system outcomes. In this work, we propose a revised micro-dynamic analysis method that adopts pattern mining and causal discovery methods to enhance the model interpretation and to facilitate group-specific policy-making. It strengthens the explanation power of the conventional micro-dynamic analysis by eliminating ambiguity in the result interpretation and enabling a causal interpretation of a target phenomenon across subgroups. We applied our method to understand an agent-based model that evaluates the effects of a long-term care scheme on access to care. Our findings showed that the method can suggest policies for improving the equity of access more efficiently than the conventional scenario analysis.

In this paper, we present an Isabelle/HOL formalization of noncommutative and nonassociative algebraic structures known as gyrogroups and gyrovector spaces. These concepts were introduced by Abraham A. Ungar and have deep connections to hyperbolic geometry and special relativity. Gyrovector spaces can be used to define models of hyperbolic geometry. Unlike other models, gyrovector spaces offer the advantage that all definitions exhibit remarkable syntactical similarities to standard Euclidean and Cartesian geometry (e.g., points on the line between a and b satisfy the parametric equation ( a oplus totimes (ominus a oplus b)), for (t in mathbb {R}), while the hyperbolic Pythagorean theorem is expressed as (a^2oplus b^2 = c^2), where (otimes ), (oplus ), and (ominus ) represent gyro operations). We begin by formally defining gyrogroups and gyrovector spaces and proving their numerous properties. Next, we formalize Möbius and Einstein models of these abstract structures (formulated in the two-dimensional, complex plane), and then demonstrate that these are equivalent to the Poincaré and Klein-Beltrami models, satisfying Tarski’s geometry axioms for hyperbolic geometry.