There has been much work on the following question: given n how large can a subset of 1,...,n be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch applications of large 3-free sets, review the literature of how to construct large 3-free sets, and present empirical studies on how large such sets actually are. The two main questions considered are (1) How large can a 3-free set be when n is small, and (2) How do the methods in the literature compare to each other? In particular, when do the ones that are asymptotically better actually yield larger sets? (This paper overlaps with our previous paper with the title Finding Large 3-Free Sets I: the Small n Case.)

Cribbage is a popular multi-player imperfect-information card game. But despite recent advances in computational methods for imperfect information games, there has been little work on intelligent agents for cribbage. We present an agent for the two-player game based on Monte Carlo Counterfactual Regret Minimization and we describe the simplifications we chose in order to deal with the large state space; the end result achieves results superior to the best agents described in the literature to date.

The squeeziness of a sequence of program statements captures the loss of information (loss of entropy) caused by its execution. This information loss leads to problems such as failed error propagation. Intuitively, longer more complex statement sequences (more formally, longer paths of dependencies) bring greater squeeze. Using the cost of search-based test data generation as a measure of lost information, we investigate this intuition. Unexpectedly, we find virtually no correlation between dependence path length and information loss. Thus our study represents an (unexpected) negative result.Moreover, looking through the literature, this finding is in agreement with recent work of Masri and Podgurski. As such, our work replicates a negative result. More precisely, it provides a conceptual, generalization and extension replication. The replication falls into the category of a conceptual replication in that different methods are used to address a common problem, and into the category of generalization and extension in that we sample a different population of subjects and more rigorously consider the resulting data. Specifically, while Masri and Podgurski only informally observed the lack of a connection, we rigorously assess it using a range of statistical models.

Yahtzee is a dice game with elements of skill and chance. There are many numeric parameters that govern the game's scoring rules, and varying those parameters will affect many aspects of game play, including strategic depth. We take advantage of the ease of computing the optimal policy for solitaire Yahtzee to use supervised learning to develop near-optimal agents based on neural networks. With the aim of automatically selecting scoring parameters that increase strategic depth, we measure the agent's skill as we vary the resources available to the neural networks and use metrics derived from the resulting learning curve as an indicator of strategic depth.

“Bots” playing games is not a new concept, likely going back to the first video games. However, there has been a new wave recently using machine learning to learn to play games at a near optimal level essentially using neural networks to “solve” games. Depending on the game, this can be relatively straight forward using supervised learning. However, this requires having data for optimal play, which is often not possible due to the sheer complexity of many games. For example, solitaire Yahtzee has this data available, but two player Yahtzee does not due to the massive state space. A recent trend in response to this started with Google Deep Mind in 2013, who used Deep Reinforcement Learning to play various Atari games [4]. This project will apply Deep Reinforcement Learning (specifically Deep Q-Learning) and measure how an agent learns to play Yahtzee in the form of a strategy ladder. A strategy ladder is a way of looking at how the performance of an AI varies with the computational resources it uses. Different sets of rules changes how the the AI learns which varies the strategy ladder itself. This project will vary the upper bonus threshold and then attempt to measure how “good” the various strategy ladders are in essence attempting to find the set of rules which creates the “best” version of Yahtzee. We assume/expect that there is some correlation between strategy ladders for AI and strategy ladders for human, meaning that a game with a “good” strategy ladder for an AI indicates that game is interesting and challenging for humans.

This paper investigates the problem of scheduling a fixed number of evaluations for genetic algorithms in noisy environments. With a fixed number of evaluations there is a tradeoff between the time the population is allowed to evolve (that is, the number of generations), the size of the population, and the number of samples scheduled per individual in an effort to reduce the effects of noise. This paper focuses mostly on the balance between allocating evaluations to the evolutionary phase versus allocating evaluations to selecting the individual with the highest fitness from the final population (the “champion selection” phase). Several different algorithms for scheduling evaluations during the champion selection phase are compared using a common test function to see how often they find the optimal value. The best algorithm is enhanced to improve its running time. We find the optimal split between the evolutionary and champion selection phases for the selected test function and we examine the effect of varying other parameters such as number of generations (and hence population) and evaluations per individual.

Can't Stop is a jeopardy stochastic game played on an octagonal game board with four six-sided dice. Previous work generalized a well-known heuristic strategy for the solitaire game and attempted to optimize the parameters of the generalized strategy using a genetic algorithm (GA). There were two challenges in that optimization process: first, the stochastic nature of the game results in a very noisy fitness function; second, the fitness function is computationally expensive. In this work we continue the optimization process for the heuristic strategy by optimizing the GA: for a fixed number of fitness function evaluations, we investigate the effects of varying the GA parameters (in particular the population size and number of generations), which in turn affect the number of samples per individual and thus noise as well. We also examine different sampling schedules; our schedules are unique in that selecting the final champion is considered a schedulable phase. The GA parameters are first optimized on an easy-to-compute test function. The resulting GA parameters are effective on the original problem and as a result we obtain an improved heuristic strategy for Can't Stop.

There has been much work on the following question: given n, how large can a subset of {1, . . . , n} be that has no arithmetic progressions of length 3. We call such sets 3-free. In this paper we review the literature of how to construct large 3-free sets. We also show how such sets can be used to obtain lower bounds on W (3, c) and the L-problem.

Can’t Stop is a jeopardy stochastic game played on an octagonal game board with four six-sided dice. Optimal strategies have been computed for some simplified versions of Can’t Stop by employing retrograde analysis and value iteration combined with Newton’s method. These computations result in databases that map game positions to optimal moves. Solving the original game, however, is infeasible with current techniques and technology. This paper describes the creation of heuristic strategies for solitaire Can’t Stop by generalizing an existing heuristic and using genetic algorithms to optimize the generalized parameters. The resulting heuristics are easy to use and outperform the original heuristic by 19%. Results of the genetic algorithm are compared to the known optimal results for smaller versions of Can’t Stop, and data is presented showing the relative insensitivity of the particular genetic algorithm used to the balance between reduced noise and increased population diversity.

We study algorithms for solving oneand two-player stochas tic games with perfect information, by modeling a game as a finite, possibly cyclic graph. We analyze a family of jeopardy stochastic games, prove the existence and uniqueness of optimal solutions, an d give approximation algorithms to solve them by incorporating Newton’s method into retrograde anal ysis. Examples of jeopardy stochastic games include Can’t Stop, Pig, and some variants. Results of experiments on small versions of the game Can’t Stop are presented.

There has been much work on the following question: given n, how large can a subset of {1,…,n} be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch applications of large 3-free sets, present techniques to find large 3-free sets of {1,…,n} for n⩽250, and give empirical results obtained by coding up those techniques. In the sequel we survey the known techniques for finding large 3-free sets of {1,…,n} for large n, discuss variants of them, and give empirical results obtained by coding up those techniques and variants.

Hierarchical bit vectors represent sets of integers using a collection of bit vectors. At the top level, a single bit is set iff the set is non-empty. The bits of this next level summarize ranges of the elements. In the case of a binary hierarchical b it vector the two bits of the next level summarize two ranges: the lower half and the upper half of the possible elements. At the lowest level each bit records the membership of a particu lar integer. Hierarchical bit vectors find application in information re tri val, bioinformatics, nonaveraging sets, and the conversion of NFAs to DFAs. Competing data structures for such applications include simple bit vectors and tree-based str uctu es such as skip-lists. A comparison of hierarchical bit vectors with two other represen tations (simple bit vectors and binary search trees) is presented. The comparison includes both analytical and empirical analysis of the hierarchical bit vectors. The analytical results show that as the size of the set used in creases, hierarchical bit vectors enjoy an advantage over tree-based structures and that as the sets used become sparser, hierarchical bit vectors perform better than standard bit v ectors. The empirical results confirm that hierarchical bit vectors sit in between the other tw o. The empirical investigation also highlights the impact that the processor cache has on th e break-even points.

Solitaire Yahtzee has been solved completely. However, the optimal strategy is not one a human could practically use, and for computer play it requires either a very large database or significant CPU time. We present some refinements to the techniques used to solve solitaire Yahtzee and give a method for analyzing other solitaire strategies and give some examples of this analysis for some non-optimal strategies, including some produced by evolutionary algorithms

A one-player, finite, probabilistic game with perfect information can be presented as a bipartite graph. For one-player Can't Stop, the graph is cyclic and the challenge is to determine the game-theoretical values of the positions in the cycles. In this contribution we prove the existence and uniqueness of the solution to one-player Can't Stop, and give an efficient approximation algorithm to solve it by incorporating Newton's method with retrograde analysis. We give results of applying this method to small versions of one-player Can't Stop.

Let x i ,...,x k be n-bit numbers and T ∈ ℕ. Assume that P 1,...,P k are players such that P i knows all of the numbers exceptx i . They want to determine if $\sum^{k}_{j=1}{\it x}_{j}$ = T by broadcasting as few bits as possible. In [7] an upper bound of $O(\sqrt n )$ bits was obtained for the k=3 case, and a lower bound of ω(1) for k ≥3 when T=Θ(2 n ). We obtain (1) for k ≥3 an upper bound of $k+O((n+\log k)^{1/(\lfloor{\rm lg(2k-2)}\rfloor)})$ , (2) for k=3, T=Θ(2 n ), a lower bound of Ω(loglogn), (3) a generalization of the protocol to abelian groups, (4) lower bounds on the multiparty communication complexity of some regular languages, and (5) empirical results for k = 3.

It has long been known that WSIS is decidable through the use of finite automata. However, since the worst-case running time has been proven to grow extremely quickly, few have explored the implementation of the algorithm. In this paper we describe some of the points of interest that have come up while coding and running the algorithm. These points include the data structures used as well as the special features of the automata, which we can exploit to perform minimization very quickly in certain cases. We also present some data that enable us to gain insight into how the algorithm performs in the average case, both on random inputs and on inputs that come from the use of Presburger Arithmetic (which can be converted to WSIS) in compiler optimization.

If Alice has x,y, Bob has x,z and Carol has y,z can they determine if x+y +z = N? They can if (say) Alice broadcasts x to Bob and Carol; can they do better? Chandra, Furst, and Lipton studied this problem and showed sublinear upper bounds. They also had matching (up to an additive constant) lower bounds. We give an exposition of their result with some attention to what happens for particular values of N.