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

This chapter presents a method to merge symbolic, sub-symbolic and chaotic computation into a single neural network architecture using higher order neural networks. First, we introduce a high-level programming language to hard-wire the neural net model. In this language we can handle symbolic computation (meaning that the information flow of a given synapse has a fixed semantic content, like a value of a certain data type or a synchronization signal). Using higher order neurons – corresponding to special constructs called neuron-synapse connections – we show how to include learning processes and sub-symbolic computation (where content is spread in several data units). The end result is a neural net partially hard-wired and partially soft-wired by a suitable learning algorithm with modules performing programming tasks and modules supporting sub-symbolic tasks. The last main section shows uses of chaotic properties inscribed in these complex dynamical systems, namely, a mechanism of blind search and a random number generator. ABSTRACT

Higher Order Neural Networks for Symbolic, Sub-symbolic and Chaotic Computations

Abstract. Wythoff Queens is a classical combinatorial game related to very interesting mathematical results. An amazing one is the fact that the -positions are given by and where . In this paper, we analyze a different version where one player (Left) plays with a chess bishop and the other (Right) plays with a chess knight. The new game (call it Chessfights) lacks a Beatty sequence structure in the -positions as in Wythoff Queens. However, it is possible to formulate and prove some general results of a general recursive law which is a particular case of a Partizan Subtraction game.

/pdf/a-recursive-process-related-to-a-partizan-variation-of-1ew7t8rrpi.pdf

A Recursive Process Related to a Partizan Variation of Wythoff

The increasing complexity associated to both structures and materials have been posing a continuous demand for numerical tools that allow predicting their behavior and simultaneously consider the influence of the uncertainty associated to multiple parameters. This work presents a Bayesian approach to characterize the influence of each material phase’s properties and the volume fraction parameters in the behavior of biphasic functionally graded plates.

/pdf/a-bayesian-approach-to-predict-the-structural-responses-of-2rv1pusya4.pdf

A Bayesian approach to predict the structural responses of FGM plates with uncertain parameters

Neural Networks are more than just mathematical tools to achieve optimization and learning via subsymbolic computations. Neural networks can perform several other types of computation, namely symbolic and chaotic computations. The discrete time neural model presented here can perform those three types of computations in a modular way. This paper focuses on how neural networks within this model can be used to automatically parallelize computational processes.

Automatic Parallelization in Neural Computers.

We present a model of a computer aided assessment system -- CATS (Computer Assessable Task System) -- for mathematics, which is both modular and has rich feedback. Modular in the sense that elaborated tasks need not be built from scratch but, instead, from already existing tasks. Modules are made general enough to be reusable in many different contexts. As a consequence, the construction and management of multiple-step problems with complex system-user interaction, is facilitated for non-expert users. By rich feedback we mean that the system automatically generates messages explaining errors, whenever students make algebraic manipulation and logical mistakes. This contrasts with current assessment software where feedback must be hard coded by the teacher, therefore constraining feedback richness by the teacher programming skills.

João Pedro Neto

Papers

Higher Order Neural Networks for Symbolic, Sub-symbolic and Chaotic Computations

A Recursive Process Related to a Partizan Variation of Wythoff

A Bayesian approach to predict the structural responses of FGM plates with uncertain parameters

Automatic Parallelization in Neural Computers.

Enhancing Modularity and Feedback in Computer Aided Assessment