Backtracking Search (Csps) ■Chapter 5 5.3 Is About Local Search Which Is a Very Useful Idea but We Won’T Cover It in Class
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Adversarial Search
Adversarial Search In which we examine the problems that arise when we try to plan ahead in a world where other agents are planning against us. Outline 1. Games 2. Optimal Decisions in Games 3. Alpha-Beta Pruning 4. Imperfect, Real-Time Decisions 5. Games that include an Element of Chance 6. State-of-the-Art Game Programs 7. Summary 2 Search Strategies for Games • Difference to general search problems deterministic random – Imperfect Information: opponent not deterministic perfect Checkers, Backgammon, – Time: approximate algorithms information Chess, Go Monopoly incomplete Bridge, Poker, ? information Scrabble • Early fundamental results – Algorithm for perfect game von Neumann (1944) • Our terminology: – Approximation through – deterministic, fully accessible evaluation information Zuse (1945), Shannon (1950) Games 3 Games as Search Problems • Justification: Games are • Games as playground for search problems with an serious research opponent • How can we determine the • Imperfection through actions best next step/action? of opponent: possible results... – Cutting branches („pruning“) • Games hard to solve; – Evaluation functions for exhaustive: approximation of utility – Average branching factor function chess: 35 – ≈ 50 steps per player ➞ 10154 nodes in search tree – But “Only” 1040 allowed positions Games 4 Search Problem • 2-player games • Search problem – Player MAX – Initial state – Player MIN • Board, positions, first player – MAX moves first; players – Successor function then take turns • Lists of (move,state)-pairs – Goal test -
Artificial Intelligence Spring 2019 Homework 2: Adversarial Search
Artificial Intelligence Spring 2019 Homework 2: Adversarial Search PROGRAMMING In this assignment, you will create an adversarial search agent to play the 2048-puzzle game. A demo of the game is available here: gabrielecirulli.github.io/2048. I. 2048 As A Two-Player Game II. Choosing a Search Algorithm: Expectiminimax III. Using The Skeleton Code IV. What You Need To Submit V. Important Information VI. Before You Submit I. 2048 As A Two-Player Game 2048 is played on a 4×4 grid with numbered tiles which can slide up, down, left, or right. This game can be modeled as a two player game, in which the computer AI generates a 2- or 4-tile placed randomly on the board, and the player then selects a direction to move the tiles. Note that the tiles move until they either (1) collide with another tile, or (2) collide with the edge of the grid. If two tiles of the same number collide in a move, they merge into a single tile valued at the sum of the two originals. The resulting tile cannot merge with another tile again in the same move. Usually, each role in a two-player games has a similar set of moves to choose from, and similar objectives (e.g. chess). In 2048 however, the player roles are inherently asymmetric, as the Computer AI places tiles and the Player moves them. Adversarial search can still be applied! Using your previous experience with objects, states, nodes, functions, and implicit or explicit search trees, along with our skeleton code, focus on optimizing your player algorithm to solve 2048 as efficiently and consistently as possible. -
Constraint Satisfaction with Preferences Ph.D
MASARYK UNIVERSITY BRNO FACULTY OF INFORMATICS } Û¡¢£¤¥¦§¨ª«¬Æ !"#°±²³´µ·¸¹º»¼½¾¿45<ÝA| Constraint Satisfaction with Preferences Ph.D. Thesis Brno, January 2001 Hana Rudová ii Acknowledgements I would like to thank my supervisor, Ludˇek Matyska, for his continuous support, guidance, and encouragement. I am very grateful for his help and advice, which have allowed me to develop both as a person and as the avid student I am today. I want to thank to my husband and my family for their support, patience, and love during my PhD study and especially during writing of this thesis. This research was supported by the Universities Development Fund of the Czech Repub- lic under contracts # 0748/1998 and # 0407/1999. Declaration I declare that this thesis was composed by myself, and all presented results are my own, unless otherwise stated. Hana Rudová iii iv Contents 1 Introduction 1 1.1 Thesis Outline ..................................... 1 2 Constraint Satisfaction 3 2.1 Constraint Satisfaction Problem ........................... 3 2.2 Optimization Problem ................................ 4 2.3 Solution Methods ................................... 5 2.4 Constraint Programming ............................... 6 2.4.1 Global Constraints .............................. 7 3 Frameworks 11 3.1 Weighted Constraint Satisfaction .......................... 11 3.2 Probabilistic Constraint Satisfaction ........................ 12 3.2.1 Problem Definition .............................. 13 3.2.2 Problems’ Lattice ............................... 13 3.2.3 Solution -
Best-First and Depth-First Minimax Search in Practice
Best-First and Depth-First Minimax Search in Practice Aske Plaat, Erasmus University, [email protected] Jonathan Schaeffer, University of Alberta, [email protected] Wim Pijls, Erasmus University, [email protected] Arie de Bruin, Erasmus University, [email protected] Erasmus University, University of Alberta, Department of Computer Science, Department of Computing Science, Room H4-31, P.O. Box 1738, 615 General Services Building, 3000 DR Rotterdam, Edmonton, Alberta, The Netherlands Canada T6G 2H1 Abstract Most practitioners use a variant of the Alpha-Beta algorithm, a simple depth-®rst pro- cedure, for searching minimax trees. SSS*, with its best-®rst search strategy, reportedly offers the potential for more ef®cient search. However, the complex formulation of the al- gorithm and its alleged excessive memory requirements preclude its use in practice. For two decades, the search ef®ciency of ªsmartº best-®rst SSS* has cast doubt on the effectiveness of ªdumbº depth-®rst Alpha-Beta. This paper presents a simple framework for calling Alpha-Beta that allows us to create a variety of algorithms, including SSS* and DUAL*. In effect, we formulate a best-®rst algorithm using depth-®rst search. Expressed in this framework SSS* is just a special case of Alpha-Beta, solving all of the perceived drawbacks of the algorithm. In practice, Alpha-Beta variants typically evaluate less nodes than SSS*. A new instance of this framework, MTD(ƒ), out-performs SSS* and NegaScout, the Alpha-Beta variant of choice by practitioners. 1 Introduction Game playing is one of the classic problems of arti®cial intelligence. -
General Branch and Bound, and Its Relation to A* and AO*
ARTIFICIAL INTELLIGENCE 29 General Branch and Bound, and Its Relation to A* and AO* Dana S. Nau, Vipin Kumar and Laveen Kanal* Laboratory for Pattern Analysis, Computer Science Department, University of Maryland College Park, MD 20742, U.S.A. Recommended by Erik Sandewall ABSTRACT Branch and Bound (B&B) is a problem-solving technique which is widely used for various problems encountered in operations research and combinatorial mathematics. Various heuristic search pro- cedures used in artificial intelligence (AI) are considered to be related to B&B procedures. However, in the absence of any generally accepted terminology for B&B procedures, there have been widely differing opinions regarding the relationships between these procedures and B &B. This paper presents a formulation of B&B general enough to include previous formulations as special cases, and shows how two well-known AI search procedures (A* and AO*) are special cases o,f this general formulation. 1. Introduction A wide class of problems arising in operations research, decision making and artificial intelligence can be (abstractly) stated in the following form: Given a (possibly infinite) discrete set X and a real-valued objective function F whose domain is X, find an optimal element x* E X such that F(x*) = min{F(x) I x ~ X}) Unless there is enough problem-specific knowledge available to obtain the optimum element of the set in some straightforward manner, the only course available may be to enumerate some or all of the elements of X until an optimal element is found. However, the sets X and {F(x) [ x E X} are usually tThis work was supported by NSF Grant ENG-7822159 to the Laboratory for Pattern Analysis at the University of Maryland. -
Trees, Binary Search Trees, Heaps & Applications Dr. Chris Bourke
Trees Trees, Binary Search Trees, Heaps & Applications Dr. Chris Bourke Department of Computer Science & Engineering University of Nebraska|Lincoln Lincoln, NE 68588, USA [email protected] http://cse.unl.edu/~cbourke 2015/01/31 21:05:31 Abstract These are lecture notes used in CSCE 156 (Computer Science II), CSCE 235 (Dis- crete Structures) and CSCE 310 (Data Structures & Algorithms) at the University of Nebraska|Lincoln. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License 1 Contents I Trees4 1 Introduction4 2 Definitions & Terminology5 3 Tree Traversal7 3.1 Preorder Traversal................................7 3.2 Inorder Traversal.................................7 3.3 Postorder Traversal................................7 3.4 Breadth-First Search Traversal..........................8 3.5 Implementations & Data Structures.......................8 3.5.1 Preorder Implementations........................8 3.5.2 Inorder Implementation.........................9 3.5.3 Postorder Implementation........................ 10 3.5.4 BFS Implementation........................... 12 3.5.5 Tree Walk Implementations....................... 12 3.6 Operations..................................... 12 4 Binary Search Trees 14 4.1 Basic Operations................................. 15 5 Balanced Binary Search Trees 17 5.1 2-3 Trees...................................... 17 5.2 AVL Trees..................................... 17 5.3 Red-Black Trees.................................. 19 6 Optimal Binary Search Trees 19 7 Heaps 19 -
Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space
applied sciences Article Parallel Technique for the Metaheuristic Algorithms Using Devoted Local Search and Manipulating the Solutions Space Dawid Połap 1,* ID , Karolina K˛esik 1, Marcin Wo´zniak 1 ID and Robertas Damaševiˇcius 2 ID 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland; [email protected] (K.K.); [email protected] (M.W.) 2 Department of Software Engineering, Kaunas University of Technology, Studentu 50, LT-51368, Kaunas, Lithuania; [email protected] * Correspondence: [email protected] Received: 16 December 2017; Accepted: 13 February 2018 ; Published: 16 February 2018 Abstract: The increasing exploration of alternative methods for solving optimization problems causes that parallelization and modification of the existing algorithms are necessary. Obtaining the right solution using the meta-heuristic algorithm may require long operating time or a large number of iterations or individuals in a population. The higher the number, the longer the operation time. In order to minimize not only the time, but also the value of the parameters we suggest three proposition to increase the efficiency of classical methods. The first one is to use the method of searching through the neighborhood in order to minimize the solution space exploration. Moreover, task distribution between threads and CPU cores can affect the speed of the algorithm and therefore make it work more efficiently. The second proposition involves manipulating the solutions space to minimize the number of calculations. In addition, the third proposition is the combination of the previous two. All propositions has been described, tested and analyzed due to the use of various test functions. -
Best-First Minimax Search Richard E
Artificial Intelligence ELSEVIER Artificial Intelligence 84 ( 1996) 299-337 Best-first minimax search Richard E. Korf *, David Maxwell Chickering Computer Science Department, University of California, Los Angeles, CA 90024, USA Received September 1994; revised May 1995 Abstract We describe a very simple selective search algorithm for two-player games, called best-first minimax. It always expands next the node at the end of the expected line of play, which determines the minimax value of the root. It uses the same information as alpha-beta minimax, and takes roughly the same time per node generation. We present an implementation of the algorithm that reduces its space complexity from exponential to linear in the search depth, but at significant time cost. Our actual implementation saves the subtree generated for one move that is still relevant after the player and opponent move, pruning subtrees below moves not chosen by either player. We also show how to efficiently generate a class of incremental random game trees. On uniform random game trees, best-first minimax outperforms alpha-beta, when both algorithms are given the same amount of computation. On random trees with random branching factors, best-first outperforms alpha-beta for shallow depths, but eventually loses at greater depths. We obtain similar results in the game of Othello. Finally, we present a hybrid best-first extension algorithm that combines alpha-beta with best-first minimax, and performs significantly better than alpha-beta in both domains, even at greater depths. In Othello, it beats alpha-beta in two out of three games. 1. Introduction and overview The best chess machines, such as Deep-Blue [lo], are competitive with the best humans, but generate billions of positions per move. -
Prolog Lecture 6
Prolog lecture 6 ● Solving Sudoku puzzles ● Constraint Logic Programming ● Natural Language Processing Playing Sudoku 2 Make the problem easier 3 We can model this problem in Prolog using list permutations Each row must be a permutation of [1,2,3,4] Each column must be a permutation of [1,2,3,4] Each 2x2 box must be a permutation of [1,2,3,4] 4 Represent the board as a list of lists [[A,B,C,D], [E,F,G,H], [I,J,K,L], [M,N,O,P]] 5 The sudoku predicate is built from simultaneous perm constraints sudoku( [[X11,X12,X13,X14],[X21,X22,X23,X24], [X31,X32,X33,X34],[X41,X42,X43,X44]]) :- %rows perm([X11,X12,X13,X14],[1,2,3,4]), perm([X21,X22,X23,X24],[1,2,3,4]), perm([X31,X32,X33,X34],[1,2,3,4]), perm([X41,X42,X43,X44],[1,2,3,4]), %cols perm([X11,X21,X31,X41],[1,2,3,4]), perm([X12,X22,X32,X42],[1,2,3,4]), perm([X13,X23,X33,X43],[1,2,3,4]), perm([X14,X24,X34,X44],[1,2,3,4]), %boxes perm([X11,X12,X21,X22],[1,2,3,4]), perm([X13,X14,X23,X24],[1,2,3,4]), perm([X31,X32,X41,X42],[1,2,3,4]), perm([X33,X34,X43,X44],[1,2,3,4]). 6 Scale up in the obvious way to 3x3 7 Brute-force is impractically slow There are very many valid grids: 6670903752021072936960 ≈ 6.671 × 1021 Our current approach does not encode the interrelationships between the constraints For more information on Sudoku enumeration: http://www.afjarvis.staff.shef.ac.uk/sudoku/ 8 Prolog programs can be viewed as constraint satisfaction problems Prolog is limited to the single equality constraint: – two terms must unify We can generalise this to include other types of constraint Doing so leads -
Binary Search Trees! – These Are "Nonlinear" Implementations of the ADT Table
Data Structures Topic #8 Today’s Agenda • Continue Discussing Table Abstractions • But, this time, let’s talk about them in terms of new non-linear data structures – trees – which require that our data be organized in a hierarchical fashion Tree Introduction • Remember when we learned about tables? – We found that none of the methods for implementing tables was really adequate. – With many applications, table operations end up not being as efficient as necessary. – We found that hashing is good for retrieval, but doesn't help if our goal is also to obtain a sorted list of information. Tree Introduction • We found that the binary search also allows for fast retrieval, – but is limited to array implementations versus linked list. – Because of this, we need to move to more sophisticated implementations of tables, using binary search trees! – These are "nonlinear" implementations of the ADT table. Tree Terminology • Trees are used to represent the relationship between data items. – All trees are hierarchical in nature which means there is a parent-child relationship between "nodes" in a tree. – The lines between nodes are called directed edges. – If there is a directed edge from node A to node B -- then A is the parent of B and B is a child of A. Tree Terminology • Children of the same parent are called siblings. • Each node in a tree has at most one parent, starting at the top with the root node (which has no parent). • Parent of n The node directly above node n in the tree • Child of n The node directly below the node n in the tree Tree -
The Logic of Satisfaction Constraint
Artificial Intelligence 58 (1992) 3-20 3 Elsevier ARTINT 948 The logic of constraint satisfaction Alan K. Mackworth* Department of Computer Science, University of British Columbia, Vancouver, BC, Canada V6T 1 W5 Abstract Mackworth, A.K., The logic of constraint satisfaction, Artificial Intelligence 58 (1992) 3-20. The constraint satisfaction problem (CSP) formalization has been a productive tool within Artificial Intelligence and related areas. The finite CSP (FCSP) framework is presented here as a restricted logical calculus within a space of logical representation and reasoning systems. FCSP is formulated in a variety of logical settings: theorem proving in first order predicate calculus, propositional theorem proving (and hence SAT), the Prolog and Datalog approaches, constraint network algorithms, a logical interpreter for networks of constraints, the constraint logic programming (CLP) paradigm and propositional model finding (and hence SAT, again). Several standard, and some not-so-standard, logical methods can therefore be used to solve these problems. By doing this we obtain a specification of the semantics of the common approaches. This synthetic treatment also allows algorithms and results from these disparate areas to be imported, and specialized, to FCSP; the special properties of FCSP are exploited to achieve, for example, completeness and to improve efficiency. It also allows export to the related areas. By casting CSP both as a generalization of FCSP and as a specialization of CLP it is observed that some, but not all, FCSP techniques lift to CSP and thereby to CLP. Various new connections are uncovered, in particular between the proof-finding approaches and the alternative model-finding ap- proaches that have arisen in depiction and diagnosis applications. -
Branch and Bound
Branch and Bound • Used to find optimal solution to many optimization problems, especially in discrete and combinatorial optimization • Systematic enumeration of all candidate solutions, discarding large subsets of fruitless candidates by using upper and lower estimated bounds of quantity being optimized Terminology Definition 1 Live node is a node that has been generated but whose children have not yet been generated. Definition 2 E-node is a live node whose children are currently being explored. In other words, an E-node is a node currently being expanded. Definition 3 Dead node is a generated node that is not to be expanded or explored any further. All children of a dead node have already been expanded. Definition 4 Branch-and-bound refers to all state space search methods in which all children of an E-node are generated before any other live node can become the E-node. • Used for state space search – In BFS, exploration of a new node cannot begin until the node currently being explored is fully explored General method • Both BFS and DFS generalize to branch-and-bound strategies – BFS is an FIFO search in terms of live nodes ∗ List of live nodes is a queue – DFS is an LIFO search in terms of live nodes ∗ List of live nodes is a stack • Just like backtracking, we will use bounding functions to avoid generating subtrees that do not contain an answer node • Example: 4-queens – FIFO branch-and-bound algorithm ∗ Initially, there is only one live node; no queen has been placed on the chessboard ∗ The only live node becomes E-node ∗ Expand and