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

In this paper we show that programming languages can be translated into recurrent (analog, rational weighted) neural nets. Implementation of programming languages in neural nets turns to be not only theoretical exciting, but has also some practical implications in the recent efforts to merge symbolic and sub symbolic computation. To be of some use, it should be carried in a context of bounded resources. Herein, we show how to use resource bounds to speed up computations over neural nets, through suitable data type coding like in the usual programming languages. We introduce data types and show how to code and keep them inside the information flow of neural nets. Data types and control structures are part of a suitable programming language called NETDEF. Each NETDEF program has a specific neural net that computes it. These nets have a strong modular structure and a synchronization mechanism allowing sequential or parallel execution of subnets, despite the massive parallel feature of neural nets. Each instruction denotes an independent neural net. There are constructors for assignment, conditional and loop instructions. Besides the language core, many other features are possible using the same method.

/pdf/symbolic-processing-in-neural-networks-1tquxitqtp.pdf

Symbolic processing in neural networks

The class of Guaranteed Scoring Games (GS) are two-player combinatorial games with the property that Normal-play games (Conway et. al.) are ordered embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS form a partially ordered monoid, via defined Right- and Left-stops over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by 'switching the players'. The order relation between two games is defined by comparing their stops in any  disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013).

/pdf/guaranteed-scoring-games-49kflywmv5.pdf

Guaranteed Scoring Games

We show how to use recursive function theory to prove Turing universality of finite analog recurrent neural nets, with a piecewise linear sigmoid function as activation function We emphasize the modular construction of nets within nets, a relevant issue from the software engineering point of view

/pdf/turing-universality-of-neural-nets-revisited-1hyhlddrya.pdf

Turing Universality of Neural Nets (Revisited)

Duchenne muscular dystrophy (DMD) is a genetic disorder that results in progressive muscular degeneration. Although medical advances increased their life expectancy, DMD individuals are still highly dependent on caregivers. Hand/wrist function is central for providing independence, and robotic exoskeletons are good candidates for effectively compensating for deteriorating functionality. Robotic hand exoskeletons require the accurate decoding of motor intention typically via surface electromyography (sEMG). Traditional low-density sEMG was used in the past to explore the muscular activations of individuals with DMD; however, it cannot provide high spatial resolution. This study characterized, for the first time, the forearm high-density (HD) electromyograms of three individuals with DMD while performing seven hand/wrist-related tasks and compared them to eight healthy individuals (all data available online). We looked into the spatial distribution of HD-sEMG patterns by using principal component analysis (PCA) and also assessed the repeatability and the amplitude distributions of muscle activity. Additionally, we used a machine learning approach to assess DMD individuals' potentials for myocontrol. Our analysis showed that although participants with DMD were able to repeat similar HD-sEMG patterns across gestures (similarly to healthy participants), a fewer number of electrodes was activated during their gestures compared to the healthy participants. Additionally, participants with DMD activated their muscles close to maximal contraction level (0.63 ± 0.23), whereas healthy participants had lower normalized activations (0.26 ± 0.2). Lastly, participants with DMD showed on average fewer PCs (3), explaining 90% of the complete gesture space than the healthy (5). However, the ability of the DMD participants to produce repeatable HD-sEMG patterns was unexpectedly comparable to that of healthy participants, and the same holds true for their offline myocontrol performance, disproving our hypothesis and suggesting a clear potential for the myocontrol of wearable exoskeletons. Our findings present evidence for the first time on how DMD leads to progressive alterations in hand/wrist motor control in DMD individuals compared to healthy. The better understanding of these alterations can lead to further developments for the intuitive and robust myoelectric control of active hand exoskeletons for individuals with DMD.

Characterization of Forearm Muscle Activation in Duchenne Muscular Dystrophy via High-Density Electromyography: A Case Study on the Implications for Myoelectric Control.

The class of Guaranteed Scoring Games (GS) are two-player combinatorial games with the property that Normal-play games (Conway et. al.) are ordered embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS form a partially ordered monoid, via defined Right- and Left-stops over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by `switching the players'. The order relation between two games is defined by comparing their stops in \textit{any} disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013).

João Pedro Neto

Papers

Symbolic processing in neural networks

Guaranteed Scoring Games

Turing Universality of Neural Nets (Revisited)

Characterization of Forearm Muscle Activation in Duchenne Muscular Dystrophy via High-Density Electromyography: A Case Study on the Implications for Myoelectric Control.

Guaranteed Scoring Games