Reflective Cardinals

We introduce and axiomatize the notion of a reflective cardinal, use it to give semantics to higher order set theory, and explore connections between the notion of reflective cardinals and large cardinal axioms.Comment: 31 pages, added reflective sequences, original MathJax/html is in ancillary file

Quantum algorithms for formula evaluation

We survey the recent sequence of algorithms for evaluating Boolean formulas consisting of NAND gates.Comment: 11 pages, survey for NATO ARW "Quantum Cryptography and Computing", Gdansk, September 200

Open determinacy for class games

The principle of open determinacy for class games---two-player games of perfect information with plays of length $\omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals---is not provable in Zermelo-Fraenkel set theory ZFC or G\"odel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth and indeed a transfinite tower of truth predicates for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC+$\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.Comment: 18 pages; commentary concerning this article can be made at http://jdh.hamkins.org/open-determinacy-for-class-game

Tactics, dialectics, representation theory

This article is devoted to the tactical game theoretical interpretation of dialectics. Dialectical games are considered as abstractly as well as models of the internal dialogue and reflection. The models related to the representation theory (representative dynamics) are specially investigated in detail, they correlate with the hypothesis on the dialectical features of human thinking in general and mathematical thought (the constructing of a solution of mathematical problem) in particular.Comment: AMSTEX, 15 p

The Role of Time in Making Risky Decisions and the Function of Choice

The prospects of Kahneman and Tversky, Mega Million and Powerball lotteries, St. Petersburg paradox, premature profits and growing losses criticized by Livermore are reviewed under an angle of view comparing mathematical expectations with awards received. Original prospects have been formulated as a one time opportunity. An award value depends on the number of times the game is played. The random sample mean is discussed as a universal award. The role of time in making a risky decision is important as long as the frequency of games and playing time affect their number. A function of choice mapping properties of two-point random variables to fractions of respondents choosing them is proposed.Comment: 52 pages, 17 figure

Algebro-geometric solutions of the Schlesinger systems and the Poncelet-type polygons in higher dimensions

A new method to construct algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve represented as a ramified double covering of CP^1, a meromorphic differential is constructed with the following property: the common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of a Painleve VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve VI equations. A generalization of this differential to hyperelliptic curves is also constructed. In this case, positions of zeros of the differential provide part of a solution of the multidimensional Garnier system. The corresponding solutions of the rank two Schlesinger systems associated with elliptic and hyperelliptic curves are constructed in terms of this differential. The initial data for construction of the meromorphic differential include a point in the Jacobian of the curve, under the assumption that this point has nonvariable coordinates with respect to the lattice of the Jacobian while the branch points vary. It appears that the cases where the coordinates of the point are rational correspond to periodic trajectories of the billiard ordered games associated with g confocal quadrics in (g+1)-dimensional space. This is a generalization of a situation studied by Hitchin, who related algebraic solutions of a Painleve VI equation with the Poncelet polygons

Parrondo games with spatial dependence, II

Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N (3 or more) and p_0, p_1, p_2, p_3 in [0,1], and let game A be the special case p_0=p_1=p_2=p_3=1/2. In previous work we investigated mu_B and mu_(1/2,1/2), the mean profits per turn to the ensemble of N players always playing game B and always playing the randomly mixed game (1/2)(A+B). These means were computable for N=3,4,5,...,19, at least, and appeared to converge as N approaches infinity, suggesting that the Parrondo region (i.e., the region in which mu_B is nonpositive and mu_(1/2,1/2) is positive) has nonzero volume in the limit. The convergence was established under certain conditions, and the limits were expressed in terms of a parameterized spin system on the one-dimensional integer lattice. In this paper we replace the random mixture with the nonrandom periodic pattern A^r B^s, where r and s are positive integers. We show that mu_[r,s], the mean profit per turn to the ensemble of N players repeatedly playing the pattern A^r B^s, is computable for N=3,4,5,...,18 and r+s=2,3,4, at least, and appears to converge as N approaches infinity, albeit more slowly than in the random-mixture case. Again this suggests that the Parrondo region (mu_B is nonpositive and mu_[r,s] is positive) has nonzero volume in the limit. Moreover, we can prove this convergence under certain conditions and identify the limits.Comment: 18 pages, 2 figure

Rock bottom, the world, the sky: Catrobat, an extremely large-scale and long-term visual coding project relying purely on smartphones

Most of the 700 million teenagers everywhere in the world already have their own smartphones, but comparatively few of them have access to PCs, laptops, OLPCs, Chromebooks, or tablets. The free open source non-profit project Catrobat allows users to create and publish their own apps using only their smartphones. Initiated in 2010, with first public versions of our free apps since 2014 and 47 releases of the main coding app as of July 2018, Catrobat currently has more than 700,000 users from 180 countries, is available in 50+ languages, and has been developed so far by almost 1,000 volunteers from around the world ("the world"). Catrobat is strongly inspired by Scratch and indeed allows to import most Scratch projects, thus giving access to more than 30 million projects on our users' phones as of July 2018. Our apps are very intuitive ("rock bottom"), have many accessibility settings, e.g., for kids with visual or cognitive impairments, and there are tons of constructionist tutorials and courses in many languages. We also have created a plethora of extensions, e.g., for various educational robots, including Lego Mindstorms and flying Parrot quadcopters ("the sky"), as well as for controlling arbitrary external devices through Arduino or Raspberry Pi boards, going up to the stratosphere and even beyond to interplanetary space ("the sky"). A TurtleStitch extension allowing to code one's own embroidery patterns for clothes is currently being developed. Catrobat among others intensely focuses on including female teenagers. While a dedicated version for schools is being developed, our apps are meant to be primarily used outside of class rooms, anywhere and in particular outdoors ("rock bottom", "the world"). Catrobat is discovered by our users through various app stores such as Google Play and via social media channels such as YouTube as well as via our presence on Code.org.Comment: Constructionism 201

Location-Based Reasoning about Complex Multi-Agent Behavior

Recent research has shown that surprisingly rich models of human activity can be learned from GPS (positional) data. However, most effort to date has concentrated on modeling single individuals or statistical properties of groups of people. Moreover, prior work focused solely on modeling actual successful executions (and not failed or attempted executions) of the activities of interest. We, in contrast, take on the task of understanding human interactions, attempted interactions, and intentions from noisy sensor data in a fully relational multi-agent setting. We use a real-world game of capture the flag to illustrate our approach in a well-defined domain that involves many distinct cooperative and competitive joint activities. We model the domain using Markov logic, a statistical-relational language, and learn a theory that jointly denoises the data and infers occurrences of high-level activities, such as a player capturing an enemy. Our unified model combines constraints imposed by the geometry of the game area, the motion model of the players, and by the rules and dynamics of the game in a probabilistically and logically sound fashion. We show that while it may be impossible to directly detect a multi-agent activity due to sensor noise or malfunction, the occurrence of the activity can still be inferred by considering both its impact on the future behaviors of the people involved as well as the events that could have preceded it. Further, we show that given a model of successfully performed multi-agent activities, along with a set of examples of failed attempts at the same activities, our system automatically learns an augmented model that is capable of recognizing success and failure, as well as goals of peoples actions with high accuracy. We compare our approach with other alternatives and show that our unified model, which takes into account not only relationships among individual players, but also relationships among activities over the entire length of a game, although more computationally costly, is significantly more accurate. Finally, we demonstrate that explicitly modeling unsuccessful attempts boosts performance on other important recognition tasks

Locally Finite Knowledge Structures

In a game of incomplete information, an infinite state space can create problems. When the space is uncountably large, the strategy spaces of the players may be unwieldly, resulting in a lack of measurable equilibria. When the knowledge of a player allows for an infinite number of possibilities, without conditions on the behavior of the other players, that player may be unable to evaluate and compare the payoff consequences of her actions. We argue that local finiteness is an important and desirable property, namely that at every point in the state space every player knows that only a finite number of points are possible. Local finiteness implies a kind of common knowledge of a countable number of points. Unfortunately its relationship to other forms of common knowledge is complex. In the context of the multi-agent propositional calculus, if the set of formulas held in common knowledge is generated by a finite set of formulas but a finite structure is not determined then there are uncountably many locally finite structures sharing this same set of formulas in common knowledge and likewise uncountably many with uncountable size. This differs radically from the infinite generation of formulas in common knowledge, and we show some examples of this. One corollary is that if there are infinitely many distinct points but a uniform bound on the number of points any player knows is possible then the set of formulas in common knowledge cannot be finitely generated