The articles in this book had their origins in three mini-courses offered at the conference “Analyse & Logique” held August 25–29, 1997, at the University of Mons-Hainaut in Möns, Belgium. For a long time there have been rich connections between analysis and logic; these articles bear witness that this relationship is still very active, and continues to be important for both areas.

Here we briefly describe these three articles; each one has a more detailed Introduction as its first chapter.

Part One: Ultraproducts in Analysis by C. Ward Henson and Jose Iovino

Applications of model theory in functional analysis have been pursued since the mid 1960s, beginning with the introduction of Banach space ultraproducts by Bretagnolle, Dacunha-Castelle, and Krivine, and of nonstandard hulls by Luxemburg. These constructions have been widely and successfully used in many parts of analysis. This paper presents the basic aspects of a systematic model theoretic framework within which these tools are naturally situated.

The logic developed here has its origin in the question: "what does a normed space structure have in common with its Banach space ultrapowers?" To give a precise answer it is necessary to introduce a suitable formal language of positive bounded formulas together with a semantics of approximate satisfaction. The resulting theory is developed here for a very general class of structures based on normed spaces; our treatment has the most general context possible within functional analysis in which the Banach space ultraproduct and nonstandard hull constructions apply.

In this chapter we prove isomorphism theorems for ultrapowers and ultraproducts of normed space structures. These results show that there is a very tight connection between (a) properties that are preserved under the ultraproduct construction and (b) properties that are expressible using the logic for normed space structures that is described in this paper.

Let L be a signature and let ℳ and N be two normed space L-structures. If ℳ and N have isomorphic ultrapowers, by Corollary 9.4 they must be approximately elementarily equivalent. Theorem 10.7 below gives the converse (in a strong form). Together these results show that ultrapower equivalence of ℳ and N is the same as approximate elementary equivalence. (See the discussion of this issue in the Introduction.)

Theorem 10.8 below is a similar result for ultraproducts. Among other things, it shows that the ultrafilter guaranteed by Theorem 10.7 can be chosen in a highly uniform way and that the ultrapowers in question can be taken to be highly saturated. The uniformity will be exploited in Chapter 12 to prove the existence of ultrapowers that are highly homogeneous (in addition to being highly saturated).

The results in this chapter are analogous to the Keisler-Shelah Theorem in ordinary model theory. (See [She71] and Chapter 6 in [CK90].) Moreover, our proof follows a similar line of argument, with adjustments appropriate to the handling of positive bounded formulas and their approximations.

The ideas and methods of model theory are being applied today in nearly all parts of mathematics. Here we concentrate on a framework for applications in functional analysis. Model theory has already provided several tools for the research analyst, of which the most important are: (a) the Banach space ultraproduct and nonstandard hull constructions in functional analysis; (b) spaces of (model-theoretic) types as used in the geometry of Banach spaces; and (c) Loeb measure spaces in stochastic analysis and its applications. In this paper we explain a systematic model theoretic framework within which these tools (especially (a) and (b)) are naturally situated. Our main intended audience consists of analysts who are familiar with the ultraproduct construction, and this perspective has strongly influenced our presentation of the material. We expect that many model theorists will also find something of interest in this subject. In particular, we indicate the initial steps of a program for introducing the key ideas and methods of model theory into functional analysis in a systematic and comprehensive way.

Applications of model theory in functional analysis have been pursued since the mid 1960s, beginning with the introduction of Banach space ultraproducts by Bretagnolle, Dacunha-Castelle, and Krivine [BDCK66] [Kri67] [DCK72] and nonstandard hulls by Luxemburg [Lux69b]. These two constructions Avere first used at about the same time and they are essentially the same; however, these initial steps led to largely independent lines of research that have still not been fully integrated.

1. Introduction

The game of Amazons was invented by Walter Zamkauskas. Two players with four playing pieces each compete on a 10 x 10 board. Figure 1 shows the initial position of the game. The pieces, called queens or amazons, move like chess queens. After each move an amazon shoots an arrow, which travels in the same way as a chess queen moves. The point where an arrow lands is burned off the playing board, reducing the effective playing area. Neither queens nor arrows can travel across a burned off square or another queen. The first player who cannot move with any queen loses.

Amazons endgames share many characteristics with Go endgames, but avoid the extra complexity of Go such as ko fights or the problem of determining the safety of stones and territories. Just like Go, Amazons endgames are being studied by combinatorial games researchers. Berlekamp and Snatzke have investigated play on sums of long narrow n x 2 strips containing one amazon of each player [1; 15]. Even though n x 2 areas have a simple structure, sum game play is surprisingly subtle, and full combinatorial game values become very complex.

1. Introduction

Cellular Automata Games have traditionally been 0-player games such as Conway's Life, or solitaire games played on a grid or digraph G = (V, E). (This includes undirected graphs, since every undirected edge {u, v} can be interpreted as the pair of directed edges (u,v) and (v,u).) Each cell or vertex of the graph can assume a finite number of possible states. The set of all states is the alphabet. We restrict attention to the binary alphabet {0,1}. A position is an assignment of states to all the vertices. There is a local transition rule from one position to another: pick a vertex u and fire it, i.e., complement it together with its neighborhood. The aim is to move from a given position (such as all 0s) to a target position (such as all 0s). In many of these games any order of the moves produces the same result, so the outcome depends on the set of moves, not on the sequence of moves. Two commercial manifestations are Lights Out manufactured by Tiger Electronics, and Merlin Magic Square by Parker Brothers (but Arthur-Merlin games are something else again). Quite a bit is known about such solitaire games.

1. Global Wins and Global Threats

Combinatorial game theory (CGT) applies the divide and conquer paradigm to game analysis and game tree search. We decompose a game into independent components (local games) and compute its value as the sum of all local games. The end of play in a sum of combinatorial games is determined by the normal termination rule: A player unable to move loses. Thus, in a sum of games, no single move or game can be decisive by itself. We investigate a class of games where a move in a local game may lead to an overall win in the sum of all local games. We call such a move globally winning.

Examples of globally winning moves are moves that capture a vital opponent piece such as checkmate, moves that promote a piece to a much more powerful one like promoting a king in checkers, or moves that “escape” in games where one side has to try to catch the other side's pieces like in the game Fox and Geese (Winning Ways [2], chapter 20). Figure 1 shows a Fox and Geese position where the fox escapes with his last move from e5 to d4 and obtains “an infinite number of free moves”. If this game were a component of a sum game S, the fox side would never lose in S.

1. Introduction

Hypercube tic-tac-toe is a two-person game played on an nk “board” (i.e. a /c-dimensional hypercube of side n). (The familiar 3 x 3 game has k = 2 and n = 3. Several editions of the 43 game, k = 3 and n = 4, are commercially available.) In all these games the two players take turns. Each player claims a single one of the nk cells with his/her symbol (traditionally O's and X's, or “noughts and crosses”, as the game is known in the UK), and the first player to complete a “path” of length n (in any straight line, including any type of diagonal) is the winner. If all nk cells are filled (with the two kinds of symbols) but no solid-symbol path has been completed, the game is declared a draw.

Since the first move cannot be a disadvantage, with best play the first player should never lose. Hence, in the ideal world, the first player seeks a win, while the second player tries to draw. For each given value of k, we believe there is a critical value nd of n below which the first player can force a win, while at or above this critical value, the second player can obtain a draw. This exact value of n is exceedingly difficult to determine as a function of k. (The larger the value of n for a given k, the easier it becomes for the second player to draw.)

This volume arose from the second Combinatorial Games Theory Workshop and Conference, held at MSRI from July 24 to 28, 2000. The first such conference at MSRI, which took place in 1994, gave a boost to the relatively new field of Combinatorial Game Theory (CGT); its excitement is captured in Games of No Chance (Cambridge University Press, 1996), which includes an introduction to CGT and a brief history of the subject. In this volume we pick up where Games of No Chance left off.

Although Game Theory overlaps many disciplines, the majority of the researchers are in mathematics and computer science. This was the first time that the practioners from both camps were brought together deliberately, and the results are impressive. This bringing together seems to have formed a critical mass. There has already been a follow-up workshop at Dagstuhl (February 2002) and more are planned.

This conference greatly expanded upon the accomplishments of and questions posed at the first conference. What is missing from this volume are the reports of games that were played and analyzed at the conference; of Grossman's Dotsand- Boxes program beating everyone in sight, except for the top four humans who had it beat by the fifth move.

This volume is divided into five parts. The first deals with new theoretical developments. Calistrate, Paulhus, and Wolfe correct a mistake about the ordering of the set of game values, a mistake that has been around for three decades or more. Not only do they show that the ordering is much richer than previously thought, they open up a whole new avenue of investigations. Conway echoes this theme of fantastic and weird structures in CGT (2 being the cube root of), and he introduces the smallest infinite games.

The classical games are well represented. Elkies continues his investigations in Chess. There are many new results and tantalizing hints about the deep structure of Go. Moore and Eppstein turn one-dimensional solitaire into a twoplayer game, and conjecture that the S—values are unbounded. (In attempting to solve this, Albert, Grossman and Nowakowski defined Clobber, a big hit at the Dagstuhl conference.)

1. Introduction

Clickomania is a one-player game (puzzle) with the following rules. The board is a rectangular grid. Initially the board is full of square blocks each colored one of k colors. A group is a maximal connected monochromatic polyomino; algorithmically, start with each block as its own group, then repeatedly combine groups of the same color that are adjacent along an edge. At any step, the player can select (click) any group of size at least two. This causes those blocks to disappear, and any blocks stacked above them fall straight down as far as they can (the settling process). Thus, in particular, there is never an internal hole. There is an additional twist on the rules: if an entire column becomes empty of blocks, then this column is “removed,” bringing the two sides closer to each other (the column shifting process).

The basic goal of the game is to remove all of the blocks, or to remove as many blocks as possible. Formally, the basic decision question is whether a given puzzle is solvable: can all blocks of the puzzle be removed? More generally, the algorithmic problem is to find the maximum number of blocks that can be removed from a given puzzle. We call these problems the decision and optimization versions of Clickomania.

There are several parameters that influence the complexity of Clickomania. One obvious parameter is the number of colors. For example, the problem is trivial if there is only one color, or every block is a different color.

1. Introduction

“The Game of the Amazons” — or simply “Amazons” — is a relatively new star on the sky of abstract strategic two-person games, invented in 1988 by Argentinian Walter Zamkauskas. Compared with other non-classical games it seems to be well done — easy rules, many interesting choices for each move, a big range of different tactics and strategies to employ, challenging even after many games played. It has yet to prove if it can stand the comparison to classics like chess and go.

Amazons has already built a solid base of players and followers. It has especially produced interest in programmers, leading to several Amazons programming competitions including a yearly world-championship and a tournament at the Mind Sports Olympiad 2000, see for example [Iida and Müller 2000] and [Hensgens and Uiterwijk 2000]. Already there are about a dozen different Amazons programs competing at these tournaments. Some of these programs play strong enough to beat average human players easily.

In contrast to most games of this type combinatorial game theory can be applied directly to Amazons without any changes to the rules. Amazons also often decomposes into at least two independent subgames when the endgame is played out, making the application of combinatorial game theory worthwhile.

The first analysis of Amazons with the means of combinatorial game theory was done by E. Berlekamp, who looked at positions with one amazon per player on boards of size 2 by n [Berlekamp 2000]. Berlekamp calculated only the thermographs for these positions, but even these proved to be quite difficult to analyze.

1. Introduction

J. H. Conway's Philosophers Football, otherwise known as Phutball [1], is played by two players, Left and Right, who move alternately. The game is usually played on a 19 by 19 board, starts with a ball on a square, one side is designated the Left goal line and the opposite side the Right goal line. A move consists of either placing a stone on an unoccupied square or jumping the ball over a (horizontal, vertical or diagonal) line of stones one end of which is adjacent to the square containing the ball. The stones are removed immediately after being jumped. Jumps can be chained and a player does not have to jump when one is available. A player wins by getting the ball on or over the opponent's goal line. Demaine, Demaine and Eppstein [2] have shown that deciding whether or not a player has a winning jump is NP complete.

In this paper we consider the 1-dimensional version of the game. For example:

is a position on a finite linear strip of squares. Initially, there is a black stone, •, which represents the ball. A player on his turn can either place a white stone, o, on an empty square, or jump the ball over a string of contiguous stones one end of which is adjacent to the ball; the ball ends on the next empty square. The stones are removed immediately upon jumping. A jump can be continued if there is another group adjacent to the ball's new position. Left wins by jumping the ball onto or over the rightmost square —Right's goalline; Right wins by jumping on over the leftmost square — Left's goalline

It would seem clear that

• • Your position can only improve by having a stone placed between the ball and the opponent's goalline; and

• • Your position can only improve if an empty square between the ball and the opponent's goalline is deleted.

My PhD thesis was on transfinite numbers and ordered types and when I got my first job in Cambridge it was as a resident mathematical logician. This was very fortunate for me since I am also interested in wasting time, professionally, and I have invented some very powerful ways of wasting time. One of these is combinatorial game theory. Most combinatorial game theorists automatically have a finite mind set when they look at games — a game is a finite set of positions. However, as a logician, developing surreal numbers, this was irrelevant. I just took whatever was needed to make the theory work. For additive games the notion of sum worked very well. One does not need finiteness just, essentially, the idea that you cannot make an infinite sequence of legal moves.

One of the main results, the existence of strategies for games with no infinite chain of moves, has two proofs one of which works well for finite games the other for all games. The first involves drawing the game tree and, starting from the leaves, marking a position with a P, for Previous, if the next player to play from this position cannot win and otherwise mark it with an N, for a Next player win. The other proof is a reductio ad absurdum. Suppose we have a game with no strategies. Prom this position look at the options: some will have strategies and some maybe not. If one option is marked with a P then we can move there and win so the original position is marked with an N. If all options are marked N then we can mark it as P. If there is no strategy then one of the options has no strategy and we iterate the argument and we get an infinite chain of moves which we supposing there wasn't. However, one of the interesting things about this argument is that a game can be perfectly well defined and computable but its winning strategy not be computable. The mathematical logician Michael Rabin studied this situation and found some interesting results.