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## Solving problem by search

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**Solving problem by search**2013/10/3**Outline**• Problem-solving agents • A kind of goal-based agent • Problem types • Single state (fully observable) • Search with partial information • Problem formulation • Example problems • Basic search algorithms • Uninformed**Problem-solving agent**• Four general steps in problem solving: • Goal formulation • What are the successful world states • Problem formulation • What actions and states to consider give the goal • Search • Determine the possible sequence of actions that lead to the states of known values and then choosing the best sequence. • Execute • Give the solution perform the actions.**Problem-solving agent**function SIMPLE-PROBLEM-SOLVING-AGENT(percept) return an action static: seq, an action sequence state, some description of the current world state goal, a goal problem, a problem formulation state UPDATE-STATE(state, percept) ifseq is empty then goal FORMULATE-GOAL(state) problem FORMULATE-PROBLEM(state,goal) seq SEARCH(problem) action FIRST(seq) seq REST(seq) return action**Example: Romania**• On holiday in Romania; currently in Arad • Flight leaves tomorrow from Bucharest • Formulate goal • Be in Bucharest • Formulate problem • States: various cities • Actions: drive between cities • Find solution • Sequence of cities; e.g. Arad, Sibiu, Fagaras, Bucharest, …**Problem types**• Deterministic, fully observable single state problem • Agent knows exactly which state it will be in; solution is a sequence. • Partial knowledge of states and actions: • Non-observable sensorless or conformant problem • Agent may have no idea where it is; solution (if any) is a sequence. • Nondeterministic and/or partially observable contingency problem • Percepts provide new information about current state; solution is a tree or policy; often interleave search and execution. • Unknown state space exploration problem (“online”) • When states and actions of the environment are unknown.**Example: vacuum world**• Single state, start in #5. Solution??**Example: vacuum world**• Single state, start in #5. Solution?? • [Right, Suck]**Example: vacuum world**• Single state, start in #5. Solution?? • [Right, Suck] • Sensorless: start in {1,2,3,4,5,6,7,8} e.g Right goes to {2,4,6,8}. Solution?? • Contingency: start in {1,3}. (assume Murphy’s law, Suck can dirty a clean carpet and local sensing: [location,dirt] only. Solution??**Problem formulation**• A problem is defined by: • An initial state, e.g. Arad • Successor functionS(X)= set of action-state pairs • e.g. S(Arad)={<Arad Zerind, Zerind>,…} intial state + successor function = state space • Goal test, can be • Explicit, e.g. x=‘at bucharest’ • Implicit, e.g. checkmate(x) • Path cost (additive) • e.g. sum of distances, number of actions executed, … • c(x,a,y) is the step cost, assumed to be >= 0 A solution is a sequence of actions from initial to goal state. Optimal solution has the lowest path cost.**Selecting a state space**• Real world is absurdly complex. State space must be abstracted for problem solving. • (Abstract) state = set of real states. • (Abstract) action = complex combination of real actions. • e.g. Arad Zerind represents a complex set of possible routes, detours, rest stops, etc. • The abstraction is valid if the path between two states is reflected in the real world. • (Abstract) solution = set of real paths that are solutions in the real world. • Each abstract action should be “easier” than the real problem.**Example: vacuum world**• States?? • Initial state?? • Actions?? • Goal test?? • Path cost??**Example: vacuum world**• States?? two locations with or without dirt: 2 x 22=8 states. • Initial state?? Any state can be initial • Actions?? {Left, Right, Suck} • Goal test?? Check whether squares are clean. • Path cost?? Number of actions to reach goal.**Example: 8-puzzle**• States?? • Initial state?? • Actions?? • Goal test?? • Path cost??**Example: 8-puzzle**• States?? Integer location of each tile • Initial state?? Any state can be initial • Actions?? {Left, Right, Up, Down} • Goal test?? Check whether goal configuration is reached • Path cost?? Number of actions to reach goal**Example: 8-queens problem**• States?? • Initial state?? • Actions?? • Goal test?? • Path cost??**Example: 8-queens problem**Incremental formulation vs. complete-state formulation • States?? • Initial state?? • Actions?? • Goal test?? • Path cost??**Example: 8-queens problem**Incremental formulation • States?? Any arrangement of 0 to 8 queens on the board • Initial state?? No queens • Actions?? Add queen in empty square • Goal test?? 8 queens on board and none attacked • Path cost?? None 3 x 1014 possible sequences to investigate**Example: 8-queens problem**Incremental formulation (alternative) • States?? n (0≤ n≤ 8) queens on the board, one per column in the n leftmost columns with no queen attacking another. • Actions?? Add queen in leftmost empty column such that is not attacking other queens 2057 possible sequences to investigate;**Example: robot assembly**• States?? • Initial state?? • Actions?? • Goal test?? • Path cost??**Example: robot assembly**• States?? Real-valued coordinates of robot joint angles; parts of the object to be assembled. • Initial state?? Any arm position and object configuration. • Actions?? Continuous motion of robot joints • Goal test?? Complete assembly • Path cost?? Time to execute**Basic search algorithms**• How do we find the solutions of previous problems? • Search the state space (remember complexity of space depends on state representation) • Here: search through explicit tree generation • ROOT= initial state. • Nodes and leafs generated through successor function. • In general search generates a graph (same state through multiple paths)**Simple tree search example**function TREE-SEARCH(problem, strategy) return a solution or failure Initialize search tree to the initial state of the problem do if no candidates for expansion thenreturnfailure choose leaf node for expansion according to strategy if node contains goal state thenreturnsolution else expand the node and add resulting nodes to the search tree enddo**Simple tree search example**function TREE-SEARCH(problem, strategy) return a solution or failure Initialize search tree to the initial state of the problem do if no candidates for expansion thenreturnfailure choose leaf node for expansion according to strategy if node contains goal state thenreturnsolution else expand the node and add resulting nodes to the search tree enddo**Simple tree search example**function TREE-SEARCH(problem, strategy) return a solution or failure Initialize search tree to the initial state of the problem do if no candidates for expansion thenreturnfailure choose leaf node for expansion according to strategy if node contains goal state thenreturnsolution else expand the node and add resulting nodes to the search tree enddo Determines search process!!**State space vs. search tree**• A state is a (representation of) a physical configuration • A node is a data structure belong to a search tree • A node has a parent, children, … and includes path cost, depth, … • Here node= <state, parent-node, action, path-cost, depth> • FRINGE= contains generated nodes which are not yet expanded. • White nodes with black outline**Tree search algorithm**function TREE-SEARCH(problem,fringe) return a solution or failure fringe INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringe) loop do if EMPTY?(fringe) then return failure node REMOVE-FIRST(fringe) if GOAL-TEST[problem] applied to STATE[node] succeeds then return SOLUTION(node) fringe INSERT-ALL(EXPAND(node, problem), fringe)**Tree search algorithm (2)**function EXPAND(node,problem) return a set of nodes successors the empty set for each <action, result> in SUCCESSOR-FN[problem](STATE[node]) do s a new NODE STATE[s] result PARENT-NODE[s] node ACTION[s] action PATH-COST[s] PATH-COST[node]+ STEP-COST(node, action,s) DEPTH[s] DEPTH[node]+1 add s to successors returnsuccessors**Outline**• Problem-solving agents • A kind of goal-based agent • Problem types • Single state (fully observable) • Search with partial information • Problem formulation • Example problems • Basic search algorithms • Uninformed**Search strategies**• A strategy is defined by picking the order of node expansion. • Problem-solving performance is measured in four ways: • Completeness; Does it always find a solution if one exists? • Optimality; Does it always find the least-cost solution? • Time Complexity; Number of nodes generated/expanded? • Space Complexity; Number of nodes stored in memory during search? • Time and space complexity are measured in terms of problem difficulty defined by: • b - maximum branching factor of the search tree • d - depth of the least-cost solution • m - maximum depth of the state space (may be )**Uninformed search strategies**• (a.k.a. blind search) = use only information available in problem definition. • When strategies can determine whether one non-goal state is better than another informed search. • Categories defined by expansion algorithm: • Breadth-first search • Uniform-cost search • Depth-first search • Depth-limited search • Iterative deepening search. • Bidirectional search**BF-search, an example**• Expand shallowest unexpanded node • Implementation: fringe is a FIFO queue A**BF-search, an example**• Expand shallowest unexpanded node • Implementation: fringe is a FIFO queue A B C**BF-search, an example**• Expand shallowest unexpanded node • Implementation: fringe is a FIFO queue A B C E D**BF-search, an example**• Expand shallowest unexpanded node • Implementation: fringe is a FIFO queue A C B D E F G**BF-search; evaluation**• Completeness: • Does it always find a solution if one exists? • YES • If shallowest goal node is at some finite depth d • Condition: If b is finite • (maximum num. Of succ. nodes is finite)**BF-search; evaluation**• Completeness: • YES (if b is finite) • Time complexity: • Assume a state space where every state has b successors. • root has b successors, each node at the next level has again b successors (total b2), … • Assume solution is at depth d • Worst case; expand all but the last node at depth d • Total numb. of nodes generated:**BF-search; evaluation**• Completeness: • YES (if b is finite) • Time complexity: • Total numb. of nodes generated: • Space complexity: • Idem if each node is retained in memory**BF-search; evaluation**• Completeness: • YES (if b is finite) • Time complexity: • Total numb. of nodes generated: • Space complexity: • Idem if each node is retained in memory • Optimality: • Does it always find the least-cost solution? • In general YES • unless actions have different cost.**BF-search; evaluation**• Two lessons: • Memory requirements are a bigger problem than its execution time. • Exponential complexity search problems cannot be solved by uninformed search methods for any but the smallest instances.**Uniform-cost search**• Extension of BF-search: • Expand node with lowest path cost • Implementation: fringe = queue ordered by path cost. • UC-search is the same as BF-search when all step-costs are equal.**Uniform-cost search**• Completeness: • YES, if step-cost > (smal positive constant) • Time complexity: • Assume C* the cost of the optimal solution. • Assume that every action costs at least • Worst-case: • Space complexity: • Idem to time complexity • Optimality: • nodes expanded in order of increasing path cost. • YES, if complete.**DF-search, an example**• Expand deepest unexpanded node • Implementation: fringe is a LIFO queue (=stack) A**DF-search, an example**• Expand deepest unexpanded node • Implementation: fringe is a LIFO queue (=stack) A B C**DF-search, an example**• Expand deepest unexpanded node • Implementation: fringe is a LIFO queue (=stack) A C B D E**DF-search, an example**• Expand deepest unexpanded node • Implementation: fringe is a LIFO queue (=stack) A B C D E H I**DF-search, an example**• Expand deepest unexpanded node • Implementation: fringe is a LIFO queue (=stack) A C B E D H I**DF-search, an example**• Expand deepest unexpanded node • Implementation: fringe is a LIFO queue (=stack) A B C D E H I**DF-search, an example**• Expand deepest unexpanded node • Implementation: fringe is a LIFO queue (=stack) A C B E D I J K H