Conspiracy Number Research Articles

Sibling Conspiracy Number Search

For some two-player games (e.g. Go), no accurate and inexpensive heuristic is known for evaluating leaves of a search tree. For other games (e.g. chess), a heuristic is known (sum of piece values). For other games (e.g. Hex), only a local heuristic — one that compares children reliably, but non-siblings poorly — is known (cell voltage drop in the Shannon/Anshelevich electric circuit model). In this paper we introduce a search algorithm for a two-player perfect information game with a reasonable local heuristic. Sibling Conspiracy Number Search (SCNS) is an anytime best-first version of Conspiracy Number Search based not on evaluation of leaf states of the search tree, but — for each node — on relative evaluation scores of all children of that node. SCNS refines CNS search value intervals, converging to Proof Number Search. SCNS is a good framework for a game player. We tested SCNS in the domain of Hex, with promising results. We implemented an 11-by-11 SCNS Hex bot, DeepHex. We competed DeepHex against current Hex bot champion MoHex, a Monte-Carlo Tree Search player, and previous Hex bot champion Wolve, an Alpha-Beta Search player. DeepHex widely outperforms Wolve at all time levels, and narrowly outperforms MoHex once time reaches 4min/move.

Computing Games: Bridging the Gap Between Search and Entertainment

Over history, games have served multiple purposes. It serves as a fun activity for players who need the entertainment to become test-beds for artificial intelligence. Solving games is beneficial in providing a better understanding of how information is progressing throughout the game. Uncertainty in games affects the way a game is solved and the way the game is experienced. Previous works have interpreted uncertainty in the game progress through various means, but there have been no clear links among those interpretations. In this paper, the probability-based proof number search (PPNS) and single conspiracy number (SCN) were used as the domain-independent indicators to analyze how uncertainty affects various game elements. PPNS exploits information from certain and uncertain information to reach convergence in solving games. Meanwhile, SCN evaluates the game states’ difficulty and describes game-playing patterns to understand play positions better. The study’s objective focuses on finding the optimal difficulty ordering of a game solver, defining the indicator for entertainment, and linking game-tree search and entertainment in different game environments. Experiments results demonstrate the link between the search indicators and the measure of entertainment where uncertainty plays a vital role in both contexts, verified from both two-person and single-agent games. Such a situation is also crucial for both computation and entertainment measures since it impacts both the quality of information and the expected game-playing experience.

Using single conspiracy number for long term position evaluation

Single Conspiracy Number (SCN) is a variant concept of conspiracy number and proof number which indicates the difficulty of a root node changing its MIN/MAX value to a certain score. It makes up the drawbacks of conspiracy number on computing complexity, and can be easily applied into different sea rch frameworks. This paper explores the potential usage of SCN as a long-term position evaluation to understand in-depth game progress patterns. Chinese Chess is chosen as a testbed for this study, whereas a strong AI engine ‘Xiangqi Wizard’ is used. It is implemented with alpha-beta search and modified to produce SCNs during the search process. Experiments are conducted on different types of positions including tactical positions, drawn positions and opening positions. The experimental results show that SCN is more consistent and accurate for long-term position evaluation than the conventional way using evaluation function values only, and using SCN together with evaluation function values enables us to better understand game progress patterns.

Conspiracy number search with relative sibling scores

For some two-player games (e.g. Go), no accurate and inexpensive heuristic is known for evaluating leaves of a search tree. For other games (e.g. chess), a heuristic is known (sum of piece values). For other games (e.g. Hex), only a local heuristic — one that compares children reliably, but non-siblings poorly — is known (cell voltage drop in the Shannon/Anshelevich electric circuit model). In this paper we introduce a search algorithm for a two-player perfect information game with a reasonable local heuristic.Sibling Conspiracy Number Search (SCNS) is an anytime best-first version of Conspiracy Number Search based not on evaluation of leaf states of the search tree, but — for each node — on relative evaluation scores of all children of that node. SCNS refines CNS search value intervals, converging to Proof Number Search. SCNS is a good framework for a game player.We tested SCNS in the domain of Hex, with promising results. We implemented an 11-by-11 SCNS Hex bot, DeepHex. We competed DeepHex against current Hex bot champion MoHex, a Monte Carlo Tree Search player, and previous Hex bot champion Wolve, an Alpha–Beta Search player. DeepHex widely outperforms Wolve at all time levels, and narrowly outperforms MoHex once time reaches 4 min/move.We tested the strength of SCNS features: most critical is to initialize leaves via a multi-step process. Also, we show a simple parallel version of SCNS: it scales well for 2 threads but less efficiently for 4 or 8 threads.

An analysis of the conspiracy numbers algorithm

McAllester's conspiracy numbers algorithm is a minimax search procedure that builds game trees to variable depths without application-dependent knowledge. The algorithm gathers information to determine how likely it is that the search of a sub-tree will produce a useful result. “Likeliness” is measured by the conspiracy numbers, the minimum number of leaf nodes that must change their value (by being searched deeper) to cause the minimax value of a sub-tree to change. The search is controlled by the conspiracy threshold ( CT), the minimum number of conspirators beyond which it is considered unlikely that a sub-tree's value can be changed. This paper analyzes the best case performance for the algorithm. McAllester's original algorithm is shown to build a tree of size O( w ( CT)/( w − 1) ), where w is the branching factor of the tree. A new improvement to the algorithm is shown to reduce this complexity to O( CT 2). Hence, for a given fixed w, the algorithm's best case can be improved from exponential to quadratic growth. Although the minimal tree case is not necessarily representative of the search trees built in practice, experimental results are presented that demonstrate the improvement appears to be significant in the expected case as well.

Conspiracy numbers

McAllester's conspiracy numbers algorithm is an exciting, new minimax search procedure that builds trees to variable depths in an application-independent manner. The algorithm selectively expands nodes in the tree trying to narrow the range of plausible minimax values at the root of the search tree. It does so by computing the conspiracy number for each possible root value; the minimum number of leaf nodes that must change their score to cause the root to change to that value. This paper describes the algorithm and presents an implementation in a computer chess program. Experimental results give rise to optimism about the algorithm's future, although there are still some serious problems to be overcome.

Empirical results with conspiracy numbers

McAllester's conspiracy numbers algorithm is an exciting new approach to minimax search that builds trees to variable depth without application‐dependent knowledge. The likelihood of the root taking on a value is expressed by its conspiracy number: the minimum number of leaf nodes that must change their value to cause the root to change to that value. This paper describes experiences with the algorithm, using both random and application‐generated trees. Experiments suggest that more emphasis on breadth, rather than depth, can lead to significant performance improvements. New enhancements to the algorithm are capable of solving 41% more problems than McAllester's original proposal.

Conspiracy numbers for min-max search

A new procedure is presented for growing min-max game trees. In large games, such as chess, decisions must be based on incomplete search trees. The new tree-growth procedure is based on “conspiracy numbers” as a measure of the accuracy of the root minimax value of an incomplete tree. Conspiracy numbers measure the number of leaf nodes whose value must change in order to change the minimax root value by a given amount. Trees are grown in a way that maximizes the conspiracy required to change the root value. The trees grown by this procedure are often deep and narrow. However, if all static values in a game are the same, this new procedure reduces to d-ply search with α-β pruning. Unlike B∗ search, nonuniform growth is achieved without any modification of the static-board evaluator.

Advances in computer chess

A fully distributed chess program fundamental concepts in search selective trees and majority systems - two experiments with commercial chess computers sundry computer chess topics alpha-beta conspiracy number search new ideas in the field of problem-solving and composing programs can a computer compose chess problems? chunking for experience to make dumb endgame databases speak ALEXS - genetic-learning optimization for KNNKP(h) a taxonomy of concepts for evaluating chess strength - examples from two difficult categories.