Entropy-Driven Evolutionary Approaches to the Mastermind Problem C. Cotta J.J. Merelo A. Mora T.P. Runarsson University of Málaga University of Granada University of Iceland M M ASTERMIND ASTERMIND Find the secret combination of colors using the information provided by the codemaker, e.g., h( , ) = hidden combination played combination 1 correct peg 2 pegs of the right colot but out of place Mastermind is a dynamic constraint satisfaction problem. Consistency Consistency is the key. Assume these are the available colors: and let the hidden combination be ... 1 st guess 64 combinations are initially possible 12 combinations remain feasible after 1 st move 2 nd guess Only 2 possible combinations remained after 2 nd move 3 rd guess A combination c is feasible iff h(c,g i ) = h(g i ,c h ) for all i, where g i are previous guesses and c h is the secret combination. Let Φ={c 1 ,...c k } be a collection of feasible combinations, given the information gathered in previous moves. We compute the partition matrix Ξ Ξ Ξ ibw ibw = |{c = |{c Φ | h( Φ | h( c c , , c c i i ) = < ) = < b b , , w w > > }| }| We pick c i such that the entropy of Ξ i[··] is maximal, i.e., we maximize the information obtained when playing a combination. An EA tries to find feasible solutions (1st-level goal) maximizing entropy (2nd-level goal). Panmictic approach (EGA) Entropy is measured against solutions in the population. Cooperative approach (EGACO) Two populations coevolve, and entropy is measured against solutions in the other population Competitive approach (EGACM) Two populations coevolve, and one of them tries to minimize the entropy of the other one. Experiments Experiments The algorithms have been tested on instances with 4 pegs and 6 or 8 colors. A comparison is done with EvoRank, a state-of-the-art EA for this problem. EGAs perform comparably to EvoRank with a much lower computational effort. EGACM is the best EGA, statistically indistinguishable of EvoRank in number of guesses. Seeded initialization is essential for good performance.
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
- 1. Entropy-Driven Evolutionary Approaches to the Mastermind Problem C. Cotta J.J. Merelo A. Mora T.P. Runarsson University of Mlaga University of Granada University of Iceland M ASTERMIND Find the secret combination of colors using the information provided by the codemaker, e.g., h (,) =hidden combination played combination 1 correct peg 2 pegs of the right colot butout of place Mastermind is a dynamic constraint satisfaction problem.Consistencyis the key. Assume these are the available colors:and let the hidden combination be... 1 stguess 64 combinations are initially possible 12 combinations remain feasible after 1 stmove 2 ndguess Only 2 possible combinations remained after 2 ndmove 3 rdguess Acombinationcis feasibleiff h( c , g i ) = h( g i ,c h ) for alli , whereg iare previous guesses andc his the secret combination. Let={ c 1 ,... c k }be a collection of feasible combinations, given the information gathered in previous moves. We compute the partition matrix ibw= |{c | h( c , c i ) = < b , w > }| We pickc isuch that theentropy of i[]is maximal , i.e., we maximize the information obtained when playing a combination. An EA tries to findfeasible solutions(1st-level goal)maximizing entropy(2nd-level goal). Experiments The algorithms have been tested on instances with4 pegsand6 or 8 colors . A comparison is done withEvoRank , a state-of-the-art EA for this problem. E GAs perform comparably to EvoRank with amuch lower computational effort . E GAC Mis the bestE GA,statistically indistinguishableof EvoRank in number of guesses. Seeded initializationis essential for good performance.
Related Documents
