1,601 research outputs found
Equilibrium Points of an AND-OR Tree: under Constraints on Probability
We study a probability distribution d on the truth assignments to a uniform
binary AND-OR tree. Liu and Tanaka [2007, Inform. Process. Lett.] showed the
following: If d achieves the equilibrium among independent distributions (ID)
then d is an independent identical distribution (IID). We show a stronger form
of the above result. Given a real number r such that 0 < r < 1, we consider a
constraint that the probability of the root node having the value 0 is r. Our
main result is the following: When we restrict ourselves to IDs satisfying this
constraint, the above result of Liu and Tanaka still holds. The proof employs
clever tricks of induction. In particular, we show two fundamental
relationships between expected cost and probability in an IID on an OR-AND
tree: (1) The ratio of the cost to the probability (of the root having the
value 0) is a decreasing function of the probability x of the leaf. (2) The
ratio of derivative of the cost to the derivative of the probability is a
decreasing function of x, too.Comment: 13 pages, 3 figure
A new paradigm for minimax search
This paper introduces a new paradigm for minimax game-tree search algorithms. MT is a memory-enhanced version of Pearl's Test procedure. By changing the way MT is called, a number of best-first game-tree search algorithms can be simply and elegantly constructed (including SSS*).
Most of the assessments of minimax search algorithms have been based on simulations.
However, these simulations generally do not address two of the key ingredients of high
performance game-playing programs: iterative deepening and memory usage. This paper
presents experimental data from three game-playing programs (checkers, Othello and chess),
covering the range from low to high branching factor. The improved move ordering due to
iterative deepening and memory usage results in significantly different results from those
portrayed in the literature. Whereas some simulations show alpha-beta expanding almost
100% more leaf nodes than other algorithms [Marsland, Reinefeld & Schaeffer, 1987],
our results showed variations of less than 20%.
One new instance of our framework MTD(f) out-performs our best alpha-beta searcher
(aspiration NegaScout) on leaf nodes, total nodes and execution time. To our knowledge,
these are the first reported results that compare both depth-first and best-first algorithms given the same amount of memory
Complexity, Heuristic, and Search Analysis for the Games of Crossings and Epaminondas
Games provide fertile research domains for algorithmic research. Often, game research helps solve real-world problems through the testing and refinement of search algorithms in game domains. Other times, game research finds limits for certain algorithms. For example, the game of Go proved intractable for the Min-Max with Alpha-Beta pruning algorithm leading to the popularity of Monte-Carlo based search algorithms. Although effective in Go, and game domains once ruled by Alpha-Beta such as Lines of Action, Monte-Carlo methods appear to have limits too as they fall short in tactical domains such as Hex and Chess. In a continuation of this type of research, two new games, Crossings and Epaminondas, are presented, analyzed and used to test two Monte-Carlo based algorithms: Upper Confidence Bounds applied to Trees (UCT) and Heuristic Guided UCT (HUCT). Results indicate that heuristic knowledge can positively affect UCT\u27s performance in the lower complexity domain of Crossings. However, both agents perform worse in the higher complexity domain of Epaminondas. This identifies Epaminondas as another domain that poses difficulties for Monte Carlo agents
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