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Classical Descriptive Set Theory

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

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

On computable numbers

Game characterizations of classes of functions in descriptive set theory have their origins in the seminal work of Wadge, with further developments by several others. In this thesis we study such characterizations from several perspectives. We define modifications of Semmes's game characterization of the Borel functions, obtaining game characterizations of the Baire class $\alpha$ functions for each fixed $\alpha < \omega_1$. We also define a construction of games which transforms a game characterizing a class $\Lambda$ of functions into a game characterizing the class of functions which are piecewise $\Lambda$ on a countable partition by $\Pi^0_\alpha$ sets, for each $0 < \alpha < \omega_1$. We then define a parametrized Wadge game by using computable analysis, and show how the parameters affect the class of functions that is characterized by the game. As an application, we recast our games characterizing the Baire classes into this framework. Furthermore, we generalize our game characterizations of function classes to generalized Baire spaces, show how the notion of computability on Baire space can be transferred to generalized Baire spaces, and show that this is appropriate for computable analysis by defining a representation of Galeotti's generalized real line and analyzing the Weihrauch degree of the intermediate value theorem for that space. Finally, we show how the game characterizations of function classes discussed lead in a natural way to a stratification of each class into a hierarchy, intuitively measuring the complexity of functions in that class. This idea and the results presented open new paths for further research.

/pdf/games-for-functions-baire-classes-weihrauch-degrees-2btk3ypliw.pdf

Games for Functions: Baire Classes, Weihrauch Degrees, Transfinite Computations, and Ranks

In this paper we use infinitary Turing machines with tapes of length $\kappa $ and which run for time $ \kappa $ as presented, e.g., by Koepke & Seyfferth, to generalise the notion of type two computability to ${2}^{\kappa }$, where $\kappa $ is an uncountable cardinal with $\kappa ^{<\kappa }=\kappa $. Then we start the study of the computational properties of ${{\mathrm{\mathbb {R}}}}_\kappa $, a real closed field extension of ${{\mathrm{\mathbb {R}}}}$ of cardinality ${2}^{\kappa }$, defined by the first author using surreal numbers and proposed as the candidate for generalising real analysis. In particular we introduce representations of ${{\mathrm{\mathbb {R}}}}_\kappa $ under which the field operations are computable. Finally we show that this framework is suitable for generalising the classical Weihrauch hierarchy. In particular we start the study of the computational strength of the generalised version of the Intermediate Value Theorem.

Towards Computable Analysis on the Generalised Real Line

We introduce a parametrized version of the Wadge game for functions and show
that each lower cone in the Weihrauch degrees is characterized by such a game.
These parametrized Wadge games subsume the original Wadge game, the eraser and
backtrack games as well as Semmes's tree games. In particular, we propose that
the lower cones in the Weihrauch degrees are the answer to Andretta's question
on which classes of functions admit game characterizations.
 We then discuss some applications of such parametrized Wadge games. Using
machinery from Weihrauch reducibility theory, we introduce games characterizing
every (transfinite) level of the Baire hierarchy via an iteration of a pruning
derivative on countably branching trees.

/pdf/game-characterizations-and-lower-cones-in-the-weihrauch-f0t7v0hfnl.pdf

Game characterizations and lower cones in the Weihrauch degrees

/pdf/a-partial-characterization-by-forbidden-subgraphs-of-edge-2l8q5f4gff.pdf

A partial characterization by forbidden subgraphs of edge path graphs.

We introduce a parametrized version of the Wadge game for functions and show that each lower cone in the Weihrauch degrees is characterized by such a game. These parametrized Wadge games subsume the original Wadge game, the eraser and backtrack games as well as Semmes's tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta's question on which classes of functions admit game characterizations. 
We then discuss some applications of such parametrized Wadge games. Using machinery from Weihrauch reducibility theory, we introduce games characterizing every (transfinite) level of the Baire hierarchy via an iteration of a pruning derivative on countably branching trees.

Hugo Nobrega

Papers

Games for Functions: Baire Classes, Weihrauch Degrees, Transfinite Computations, and Ranks

Towards Computable Analysis on the Generalised Real Line

Game characterizations and lower cones in the Weihrauch degrees

A partial characterization by forbidden subgraphs of edge path graphs.

Game characterizations and lower cones in the Weihrauch degrees