ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.

On numbers and games, by J. H. Conway. Pp ix, 238. £6·50. 1976. SBN 0 12 186350 6 (Academic Press)

Some readers know to play the game of nim well, fewer play a perfect annihilation game, and nobody knows whether there exists an opening move in chess that will guarantee a win for white. These games and many more, belong to the family of combinatorial games, by which we mean the set of all two-player perfect-information games without chance moves and with outcomes lose or win (and sometimes: dynamic tie). The motivation for ONAG may have been, and perhaps was-and I would like to think that it was-the attempt to bridge the theory gap between nim-like and chess-like games. Why is there a gap? Every combinatorial game can be described as a directed graph called game-graph, whose vertices are the game positions, and (u, v) is a directed edge if and only if there is a move from position u to position v. Denote by N the set of all positions from which the Next (first) player can force a win; by P the set of all positions from which the Previous (second) player can force a win; and by T the set of all (dynamic) Tie positions, which are positions from which no player can force a win and therefore both can avoid losing. In an acyclic game-graph there cannot be any tie positions. The N, P, T classification of any game graph R = (V, E) can be determined in 0(\V\ + \E\) steps [8]. For both nim and chess, a finite game-graph can be constructed and the N, P, T classification can be determined. So both games are solvable in principle. If we play nim with n piles, each pile containing at most k tokens, then the game-graph contains (k + \) vertices. Suppose that in (generalized) chess played on an « X « board there are k different pieces. If k is about n/2, then the game-graph of chess contains O (2") vertices. So both game-graphs have exponentially many vertices, and thus both games appear intractable in the usual sense of computational complexity [1, Chapter 10], [14, Chapter 9], namely a computation appears to be required which is asymptotically exponential. From a computational efficiency standpoint, the essential difference between nim and chess is that nim can be viewed as a disjunctive compound (sum) of independent games, namely the individual piles. A disjunctive

http://ieeexplore.ieee.org/iel5/5/31266/01455274.pdf

On numbers and games

Every programmer has a characteristic style, ranging from preferences about identifier naming to preferences about object relationships and design patterns. Coding conventions define a consistent syntactic style, fostering readability and hence maintainability. When collaborating, programmers strive to obey a project's coding conventions. However, one third of reviews of changes contain feedback about coding conventions, indicating that programmers do not always follow them and that project members care deeply about adherence. Unfortunately, programmers are often unaware of coding conventions because inferring them requires a global view, one that aggregates the many local decisions programmers make and identifies emergent consensus on style. We present NATURALIZE, a framework that learns the style of a codebase, and suggests revisions to improve stylistic consistency. NATURALIZE builds on recent work in applying statistical natural language processing to source code. We apply NATURALIZE to suggest natural identifier names and formatting conventions. We present four tools focused on ensuring natural code during development and release management, including code review. NATURALIZE achieves 94% accuracy in its top suggestions for identifier names and can even transfer knowledge about conventions across projects, leveraging a corpus of 10,968 open source projects. We used NATURALIZE to generate 18 patches for 5 open source projects: 14 were accepted.

/pdf/learning-natural-coding-conventions-f25hqo8tpr.pdf

Learning Natural Coding Conventions

This paper studies theoretically the capacity limits of graph neural networks (GNN) falling within the message-passing framework. Two main results are presented. First, GNN are shown to be Turing universal under sufficient conditions on their depth, width, node identification, and layer expressiveness. Second, it is discovered that GNN can lose a significant portion of their power when their depth and width is restricted. The proposed impossibility statements stem from a new technique that enables the repurposing of seminal results from theoretical computer science and leads to lower bounds for an array of decision, optimization, and estimation problems involving graphs. Strikingly, several of these problems are deemed impossible unless the product of a GNN's depth and width exceeds (a function of) the graph size; this dependence remains significant even for tasks that appear simple or when considering approximation.

/pdf/what-graph-neural-networks-cannot-learn-depth-vs-width-7far4uu2ol.pdf

What graph neural networks cannot learn: depth vs width

A thorough revision of a popular text in combinatorial game theory, this second edition reorganizes presentation to make it more widely accessible. The beginning focuses less on technical and more on conceptual material and applications. Still written in a textbook style with supporting evidence and proofs, the authors add many more exercises and examples and implement a two-step approach for some aspects of the material involving an initial introduction, examples, and basic results to be followed later by more technical and abstract results.

Lessons in Play: An Introduction to Combinatorial Game Theory, Second Edition

Ordinal sums of impartial games

On lattices from combinatorial game theory modularity and a representation theorem: Finite case

Digital Archaeoludology (DAL) is a new field of study involving the analysis and reconstruction of ancient games from incomplete descriptions and archaeological evidence using modern computational techniques. The aim is to provide digital tools and methods to help game historians and other researchers better understand traditional games, their development throughout recorded human history, and their relationship to the development of human culture and mathematical knowledge. This work is being explored in the ERC-funded Digital Ludeme Project. 

The aim of this inaugural international research meeting on DAL is to gather together leading experts in relevant disciplines – computer science, artificial intelligence, machine learning, computational phylogenetics, mathematics, history, archaeology, anthropology, etc. – to discuss the key themes and establish the foundations for this new field of research, so that it may continue beyond the lifetime of its initiating project.

Foundations of Digital Archæoludology

The evolution from individual to collective learning opens a new dimension of solutions to address problems that appeal for gradual adaptation in dynamic and unpredictable environments. A team of individuals has the potential to outperform any sum of isolated efforts, and that potential is materialized when a good system of interaction is considered. In this paper, we describe two forms of cooperation that allow multi-agent learning: the sharing of partial results obtained during the learning activity, and the social adaptation to the stages of collective learning. We consider different ways of sharing information and different options for social reconfiguration, and apply them to the same learning problem. The results show the effects of cooperation and help to put in perspective important properties of the collective learning activity.

Multi-agent learning: how to interact to improve collective results

This paper deals with computation over dynamical analog systems, exploring applications of chaos. The dynamical system used is an artificial neural network model that can code any symbolic algorithm. Herein, we propose the integration of chaotic dynamics onto this model, to implement computational tasks, namely, a blind search algorithm and a pseudo-random number generator.

João Pedro Neto

Papers

Ordinal sums of impartial games

On lattices from combinatorial game theory modularity and a representation theorem: Finite case

Foundations of Digital Archæoludology

Multi-agent learning: how to interact to improve collective results

On computation over chaos using neural networks: application to blind search and random number generation