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 a recent paper [Neto et al 97] we showed that programming languages can be translated on recurrent (analog, rational weighted) neural nets The goal was not efficiency but simplicity Indeed we used a number-theoretic approach to machine programming, where (integer) numbers were coded in a unary fashion, introducing a exponential slow down in the computations, with respect to a two-symbol tape Turing machine 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 subsymbolic computation To be of some use, it should be carried in a context of bounded resources Herein, we show how to use resource boundedness 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 synchronisation 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 There is also a NETDEF compiler, available at wwwdifculpt/~jpn/netdef/netdefhtm

/pdf/building-neural-net-software-5g3027g5se.pdf

Building Neural Net Software

Given a set of combinatorial games, the children are all those games that can be generated using as options the games of the original set. It is known that the partial order of the children of all games whose birthday is less than a fixed ordinal is a distributive lattice and also that the children of any set of games form a complete lattice. We are interested in the converse. In a previous paper, we showed that for any finite lattice there exists a finite set of games such that the partial order of the children, minus the top and bottom elements, is isomorphic to the original lattice. Here, the main part of the paper is to extend the result to infinite complete lattices. An original motivating question was to characterize those sets whose children generate distributive lattices. While we do not solve it, we show that if the process of taking children is iterated, eventually the corresponding lattice is distributive.

On lattices from combinatorial game theory: infinite case

Neural Networks are mainly seen as algorithmic solutions for optimization and learning tasks where the ability to spread the acquired knowledge into several neurons, i.e., the use of sub-symbolic computation, is the key. We have shown in previous works that neural networks can perform other types of computation, namely symbolic and chaotic computations. Here in, we show how these nets can be decomposed into tuples which can be efficient calculated by software or hardware simpler than previous neural solutions.

A virtual machine for neural computers

In a recent short paper and a report (see [Neto et al. 98] and [Neto and Costa 99]) it was shown that programming languages can be translated efficiently on recurrent (analog, rational weighted) neural nets, using bounded resources. This fact was achieved by creating a neural programming language called NETDEF, such that each program corresponds to a modular neural net that computes it. This framework has some practical implications in recent efforts to merge symbolic and subsymbolic computation. Adding neuron-synapse connections (high-order neurons) to the neural network model allows us to integrate learning into the NETDEF computing paradigm. Some possible enhancements are presented using this framework, namely structure selfmodification, and integration of sub-symbolic learning into the NETDEF neuron architecture. The Hebb learning rule is used to provide an illustrative example.

/pdf/merging-sub-symbolic-and-symbolic-computation-1bah67ei2w.pdf

Merging Sub-symbolic and Symbolic Computation

Abstract For games of complete information with no chance component, like Chess, Go, Hex, and Konane, some parameters have been identified that help us understand what makes a game pleasant to play. One of these goes by the name of drama. Briefly, drama is linked to the possibility of recovering from a seemingly weaker position, if the player is strong enough. This is an important requirement to prevent initial advantages to be amplified into unavoidable and thus uninteresting victories. Drama is a feature that arguably good board games should have, since it is relevant in the perception of the play experience as pleasant. Despite its intrinsic qualitative nature, we suggest the adaptation of the concept of drama to games of pure chance and propose a set of objective criteria to measure it. Some parameters are here used to compare Goose-like games, which we compute via computer simulation for some well-know games. A statistical analysis is performed based on the play of millions of matches done by computer simulation. The article discusses correlations and patterns found among the collected data. The methodology presented herein is general and can be used to compare other types of board games.

/pdf/measuring-drama-in-goose-like-games-4l6gongewr.pdf

João Pedro Neto

Papers

Building Neural Net Software

On lattices from combinatorial game theory: infinite case

A virtual machine for neural computers

Merging Sub-symbolic and Symbolic Computation

Measuring Drama in Goose-like Games