人工智能05约束满足问题

Example: Map-Coloring
变量 WA, NT, Q, NSW, V, SA, T 值域 Di = {red,green,blue} 约束: adjacent regions must have different colors e.g., WA ≠ NT, or (if the language allows this), or (WA,NT) ∈ {(red,green),(red,blue),(green,red), (green,blue), … }
Degree heuristic（度启发式）
在MRV无法抉择时启动度启发式 度启发式: 通过选择涉及对其它未赋值变量的约束数最大的变 量
提高回溯效率
General-purpose methods can give huge gains in speed:
1. 哪一个变量应该被下一个赋值? 2. 赋值应该以什么样的顺序被尝试?
第五章 约束满足问题
Review: Last Chapter
• Best-first search Heuristic functions estimate costs of shortest paths Good heuristics can dramatically reduce search cost
3. 能更早察觉到不可避免的失败吗?
4. Can we take advantage of problem structure?
Minimum remaining values
Minimum remaining values 最少剩余值(MRV): 选择“合法”取值最少的变量
Why min rather than max? 被称为“最受约束变量” 或“失败优先”启发式
Standard search problem: state is a “black box“ – any old data structure that supports goal test, eval, successor 任何可以由目标测试、评价函数、后继函数访问的数据结构 CSP: state is defined by variables Xi with values from domain（值域）Di goal test is a set of constraints specifying allowable combinations of values for subsets of variables 每个约束包括一些变量的子集，并指定这些子集的值之间允许进行的 合并 Simple example of a formal representation language（形式化表示方法） Allows useful general-purpose（通用的，而不是问题特定的） algorithms with more power than standard search algorithms
Greedy best-first search expands lowest h — incomplete and not always optimal A* search expands lowest g+ h — complete and optimal — also optimally efficient (up to tie-breaks, for forward search)
列举分配
指数时间 dn But complete can we be clever about exponential time algorithms?
形式化描述标准搜索 (incremental增量形式化)
从简单直白的方法开始，状态被定义为已被赋值的变量 • 初始状态: 空的赋值, { } • 后继函数: 给一个未赋值变量赋值使之不与当前状态冲突 → fail 如果没有合法赋值 • 目标测试: 检验当前赋值是否完全 1. This is the same for all CSPs! 2. Every solution appears at depth n with n variables → use depth-first search 3. Path is irrelevant, so can also use complete-state formulation （完全状态形式化） 4. b = (n - l )d at depth l, hence n! ·dn leaves !!!!
Binary CSP: 每个约束与2个变量有关 约束图: 节点是变量, 边是约束
General-purpose CSP algorithms（通用CSP算法） use the graph structure to speed up search. E.g., Tasmania is an independent subproblem!
结合以上启发式来解决1000 queens 是可行的
提高回溯效率
General-purpose methods can give huge gains in speed:
1. 哪一个变量应该被下一个赋值? 2. 赋值应该以什么样的顺序被尝试?
3. 能更早察觉到不可避免的失败吗?
4. Can we take advantage of problem structure?
Example: Map-Coloring
Solutions are assignments satisfying all constraints, e.g., {WA=red, NT =green, Q=red, NSW =green, V =red, SA=blue, T=green}
Constraint graph（约束图）
回溯搜索
Backtracking example
Backtracking example
Backtracking example
Backtracking example
提高回溯效率
General-purpose methods can give huge gains in speed:
1. 哪一个变量应该被下一个赋值? 2. 赋值应该以什么样的顺序被尝试?
Backtracking search回溯搜索
变量赋值具有可交换性 , 也就是说 [ WA = red then NT = green ] same as [ NT = green then WA = red ] 在搜索树的每个节点上只考虑单个变量的可能赋值 b = d and there are dn leaves Depth-first search for CSPs with single-variable assignments is called backtracking search 回溯搜索是处理CSP问题最基础的无信息搜索算法 Can solve n-queens for n ≈ 25
Admissible heuristics can be derived from exact solution of relaxed problems
Review: Last Chapter
Local search algorithms • the path to the goal is irrelevant; the goal state itself is the solution • keep a single "current" state, try to improve it
• 偏好约束 (soft constraints), e.g., red is better than green often representable by a cost for each variable assignment （个体变量赋值的耗散） →约束优化问题
Example: 密码算数
变量: F T U W R O X1 X2 X3 值域: {0,1,2,3,4,5,6,7,8,9} 约束: alldiff (F,T,U,W,R,O) O + O = R + 10 ·X1 X1 + W + W = U + 10 ·X2 X2 + T + T = O + 10 ·X3 X3 = F, T ≠ 0, F ≠ 0
3. 能更早察觉到不可避免的失败吗?
4. Can we take advantage of problem structure?
最少约束值
一个变量被选定, choose the least constraining value（最少约束值） : • 这个选择的值是在约束图中排除邻居变量的可选值最少的 • 需注意的是可能需要经过一些计算来确定这个值
CSP的种类
离散变量 • finite domains 有限值域: n 个变量, 值域大小d → O(dn) 完全赋值 e.g., Boolean CSPs/布尔CSP问题(NP-complete) • infinite domains 无限值域 (integers, strings, etc.) e.g., job scheduling, variables are start/end days for each job 不能通过枚举来描述值域，只能用约束语言 , e.g., 线性约束可解, 非线性约束不可解
本章大纲
• CSP examples
• Backtracking search for CSPs • Problem structure and problem decomposition
• Local search for CSPs
Constraint satisfaction problems (CSPs)
连续值域的变量 e.g., 哈勃望远镜观测的开始、结束时间 线性规划问题linear constraints solvable in polynomial time by linear programming(LP) methods