Annals of Mathematics and Artificial Intelligence最新文献

Families of new, multi-level integer 2D arrays are introduced here as an extension of the well-known binary Legendre sequences that are derived from quadratic residues. We present a construction, based on Fourier and Finite Radon Transforms, for families of periodic perfect arrays, each of size (ptimes p) for many prime values p. Previously delta functions were used as the discrete projections which, when back-projected, build 2D perfect arrays. Here we employ perfect sequences as the discrete projected views. The base family size is (p+1). All members of these multi-level array families have perfect autocorrelation and constant, minimal cross-correlation. Proofs are given for four useful and general properties of these new arrays. 1) They are comprised of odd integers, with values between at most (-p) and (+p), with a zero value at just one location. 2) They have the property of ‘conjugate’ spatial symmetry, where the value at location (i, j) is always the negative of the value at location ((p-i, p-j)). 3) Any change in the value assigned to the array’s origin leaves all of its off-peak autocorrelation values unchanged. 4) A family of (p+1), (ptimes p) arrays can be compressed to size ((p+1)^2) and each family member can be exactly and rapidly unpacked in a single (ptimes p) decompression pass.

We introduce a hedonic game form, Hedonic Expertise Games (HEGs), that naturally models a variety of settings where agents with complementary qualities would like to form groups. Students forming groups for class projects, and hackathons in which software developers, graphic designers, project managers, and other domain experts collaborate on software projects, are typical scenarios modeled by HEGs. This game form possesses the common ranking property, and additionally, the coalitional utility function is monotone. We present comprehensive results for the existence/nonexistence of stable and efficient partitions of HEGs with respect to the most common stability and optimality concepts used in the literature. Specifically, we show that an HEG instance may not have a strict core stable partition, and yet every HEG instance has a strong Nash stable and Pareto optimal partition. Furthermore, it may be the case that none of the socially-optimal partitions of an HEG instance is Nash stable or core stable. However, it is guaranteed that every socially-optimal partition is contractually Nash stable. We show that all these existence/nonexistence results also hold for the monotone hedonic games with common ranking property (monotone HGCRP). We also present several results for HEGs from the computational complexity perspective, some of which are as follows: A contractually Nash stable partition (and a Nash stable partition in a restricted setting) can be found in polynomial time. A strong Nash stable partition can be approximated within a factor of (1-1/e), and this bound is tight even for approximating core stable partitions. We present a natural game dynamics for monotone HGCRP that converges to a Nash stable partition in a relatively low number of moves.

In this paper we consider a new class of RBF (Radial Basis Function) neural networks, in which smoothing factors are replaced with shifts. We prove under certain conditions on the activation function that these networks are capable of approximating any continuous multivariate function on any compact subset of the d-dimensional Euclidean space. For RBF networks with finitely many fixed centroids we describe conditions guaranteeing approximation with arbitrary precision.

In this note, I present my personal view on the interaction of the three areas Automated Programming, Symbolic Computation, and Machine Learning. Programming is the activity of finding a (hopefully) correct program (algorithm) for a given problem. Programming is central to automation in all areas and is considered one of the most creative human activities. However, already very early in the history of programming, people started to “jump to the meta-level” of programming, i.e., started to develop procedures that automate, or semi-automate, (various aspects or parts of) the process of programming. This area has various names like “Automated Programming”, “Automated Algorithm Synthesis”, etc. Developing compilers can be considered an early example of a problem in automated programming. Automated reasoners for proving the correctness of programs with respect to a specification is an advanced example of a topic in automated programming. ChatGPT producing (amazingly good) programs from problem specifications in natural language is a recent example of automated programming. Programming tends to become the most important activity as the level of technological sophistication increases. Therefore, automating programming is maybe the most exciting and relevant technological endeavor today. It also will have enormous impact on the global job market in the software industry. Roughly, I see two main approaches to automated programming:

• symbolic computation

• and machine learning.

In this note, I explain how the two approaches work and that they are fundamentally different because they address two completely different ways of how problems are specified. Together, the two approaches constitute (part of) what some people like to call “artificial intelligence”. In my analysis, both approaches are just part of (algorithmic) mathematics. The approaches, like all non-trivial mathematical methods, need quite some intelligence on the side of the human inventors of the methods. However, applying the methods is just “machine execution” of algorithms. It is misleading to call the application “machine intelligence” or “artificial intelligence”. The analysis of the two approaches to automated programming also suggests that the two approaches, in the future, should be combined to achieve even higher levels of sophistication. At the end of this note, I propose some research questions for this new direction.

Agent Programming Languages have been studied for over 20 years for programming complex decision-making for autonomous systems. The GOAL agent programming language is particularly interesting since it depends on automated planning based on beliefs and goals to determine behavior rather than preprogrammed planning by developers. Model checking is a powerful verification technique to guarantee the safety of an autonomous system. Despite studies of model checking in other agent programming languages, GOAL lacks support for model checking of GOAL programs. The fundamental challenge is to make GOAL programs feasible for model checking. In this paper, we tackle this fundamental issue. First, we formalize the syntax and semantics of the logic underpinning stratified single-agent GOAL programs. Second, we devise an algorithm for transforming a stratified single-agent GOAL program to a transition system that is equivalent in terms of operational semantics, enabling model checking. Third, we develop an automated translator for a stratified single-agent GOAL program. The translator consists of (1) the automated transformation of a GOAL program into its operational semantically equivalent transition system, and (2) the interface generation of the generated transition system into a Prism model, an input for two probabilistic symbolic model checkers: Storm and Prism. Moreover, we point out that we will extend the applicability of the transformation algorithm and its implementation to all stratified GOAL programs.

The continuous Hopfield network (CHN) has provided a powerful approach to optimization problems and has shown good performance in different domains. However, two primary challenges still remain for this network: defining appropriate parameters and hyperparameters. In this study, our objective is to address these challenges and achieve optimal solutions for combinatorial optimization problems, thereby improving the overall performance of the continuous Hopfield network. To accomplish this, we propose a new technique for tuning the parameters of the CHN by considering its stability. To evaluate our approach, three well-known combinatorial optimization problems, namely, weighted constraint satisfaction problems, task assignment problems, and the traveling salesman problem, were employed. The experiments demonstrate that the proposed approach offers several advantages for CHN parameter tuning and the selection of optimal hyperparameter combinations.

Finding a collision-free feasible path for mobile robots is very important because they are essential in many fields such as healthcare, military, and industry. In this paper, a novel Clustering Obstacles (CO)-based path planning algorithm for mobile robots is presented using a quintic trigonometric Bézier curve and its two shape parameters. The CO-based algorithm forms clusters of geometrically shaped obstacles and finds the cluster centers. Moreover, the proposed waypoint algorithm (WP) finds the waypoints of the predefined skeleton path in addition to the start and destination points in an environment. Based on all these points, the predefined quintic trigonometric Bézier path candidates, taking the skeleton path as their convex hull, are then generated using the shape parameters of this curve. Moreover, the performance of the proposed algorithm is compared with K-Means and agglomerative hierarchical algorithms to obtain the quintic trigonometric Bézier paths desired by the user. The experimental results show that the CO-based path planning algorithm achieves better cluster centers and consequently better collision-free predefined paths.

A digitized rigid motion is called digitally continuous if two neighbor pixels still stay neighbors after the motion. This concept plays important role when people or computers (artificial intelligence, machine vision) need to recognize the object shown in the image. In this paper, digital rotations of a pixel with its closest neighbors are of our interest. We compare the neighborhood motion map results among the three regular grids, when the center of rotation is the midpoint of a main pixel, a grid point (corner of a pixel) or an edge midpoint. The first measure about the quality of digital rotations is based on bijectivity, e.g., measuring how many of the cases produce bijective and how many produce not bijective neighborhood motion maps (Avkan et. al, 2022). Now, a second measure is investigated, the quality of bijective digital rotations is measured by the digital continuity of the resulted image: we measure how many of the cases are bijective and also digitally continuous. We show that rotations on the triangular grid prove to be digitally continuous at many more real angles and also as a special case, many more integer angles compared to the square grid or to the hexagonal grid with respect to the three different rotation centers.