CS 8520: Artificial Intelligence

CS 8520: Artificial Intelligence Solving Problems by Searching Paula Matuszek Spring, 2013 Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are in turn based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Problem-Solving Agents • A goal-based agent is essentially solving a problem • Given some state and some goal, • Figure out how to get from the current state to the goal • By taking some sequence of actions. • The problem is defined as a state space and a goal. Solving the problem consists of searching the state space for the sequence of actions that leads to the goal. Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

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 Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Example: Romania Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Single-state problem formulation • A problem is defined by five items: • initial state e.g., "at Arad" • actions or successor function S(x) = set of action–state pairs • e.g., S(Arad) = {<Arad to Zerind, Zerind>, … } • transition model: new state resulting from action • 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, etc. • c(x,a,y) is the step cost, assumed to be ≥ 0 • A solution is a sequence of actions leading from the initial state to a goal state Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Selecting a state space • Real world is absurdly complex • therefore state space must be abstracted for problem solving • (Abstract) state = set of real states • (Abstract) action = complex combination of real actions • e.g., "Arad to Zerind" represents a complex set of possible routes, detours, rest stops, etc. • For guaranteed realizability, any real state "in Arad“ must get to some real state "in Zerind" • (Abstract) solution = • set of real paths that are solutions in the real world • Each abstract action should be "easier" than the original problem Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Vacuum world state space graph • States? Actions? Transition Models? Goal Test? Path Cost? Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Vacuum world state space graph • states? integer dirt and robot location • actions? Left, Right, Suck • transition models? In A, in B, clean • goal test? no dirt at all locations • path cost? 1 per action Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Example: The 8-puzzle • states? • actions? • transition model? • goal test? • path cost? Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Example: The 8-puzzle • states? locations of tiles • actions? move blank left, right, up, down • transition model? square with tile moved. • goal test? = goal state (given) • path cost? 1 per move [Note: optimal solution of n-Puzzle family is NP-hard] Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Search • The basic concept of search views the state space as a search tree • Initial state is the root node • Each possible action leads to a new node defined by the transition model • Some nodes are identified as goals • Search is the process of expanding some portion of the tree in some order until we get to a goal node • The strategy we use to choose the order to expand nodes defines the type of search Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Tree search algorithms • Basic idea: • offline, simulated exploration of state space by generating successors of already-explored states (a.k.a.~expanding states) Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Tree search example Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Implementation: general tree search Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Implementation: states vs. nodes • A state is a (representation of) a physical configuration • A node is a data structure constituting part of a search tree includes state, parent node, action, path cost g(x), depth • The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states. Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Search strategies • A search strategy is defined by picking the order of node expansion. (e.g., breadth-first, depth-first) • Strategies are evaluated along the following dimensions: • completeness: does it always find a solution if one exists? • time complexity: number of nodes generated • space complexity: maximum number of nodes in memory • optimality: does it always find a least-cost solution? • Time and space complexity are measured in terms of • b: maximum branching factor of the search tree • d: depth of the least-cost solution • m: maximum depth of the state space (may be infinite) Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Summary • Problem-solving agents search through a problem or state space for an acceptable solution. • A state space can be specified by • An initial state • Actions • A transition model or successor function S(x) describing the results of actions • A goal test or goal state • A path cost • A solution is a sequence of actions leading from the initial state to a goal state • The formalization of a good state space is hard, and critical to success. It must abstract the essence of the problem so that • It is easier than the real-world problem. • A solution can be found. • The solution maps back to the real-world problem and solves it. Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Uninformed search strategies • Uninformed search strategies use only the information available in the problem definition • Breadth-first search • Uniform-cost search • Depth-first search • Depth-limited search • Iterative deepening search Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Implementation: general tree search Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Breadth-first search • Expand shallowest unexpanded node • Implementation: • fringe is a FIFO queue, i.e., new successors go at end Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Properties of breadth-first search • Complete? Yes (if b is finite) • Time? 1+b+b2+b3+… +bd + b(bd-1) = O(bd+1) • Space? O(bd+1) (keeps every node in memory) • Optimal? Yes (if cost = 1 per step) • Space is the bigger problem (more than time) Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Uniform-cost search • Expand least-cost unexpanded node • Implementation: • fringe = queue ordered by path cost • Equivalent to breadth-first if step costs all equal • Complete? Yes, if step cost >= epsilon (otherwise can loop) • Time and space? O(b(ceiling(C*/ epsilon))) where C* is the cost of the optimal solution and epsilon is the smallest step cost. Can be much worse than breadth-first if many small steps not on optimal path • Optimal? Yes – nodes expanded in increasing order of g(n) Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Uniform Cost Search Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Properties of depth-first search • Complete? No: fails in infinite-depth spaces, spaces with loops • Modify to avoid repeated states along path • then complete in finite spaces • Time? O(bm): terrible if m is much larger than d • but if solutions are dense, may be much faster than breadth-first • Space? O(bm), i.e., linear space! • Optimal? No Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Depth-limited search • = depth-first search with depth limit l • Nodes at depth l have no successors • Solves problem of infinite depth • Incomplete • Recursive implementation: Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Iterative deepening search • Repeated Depth-Limited search, incrementing limit l until a solution is found or failure. • Repeats earlier steps at each new level, so inefficient -- but never more than doubles cost • No longer incomplete Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Iterative deepening search l =0 Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Properties of iterative deepening • Complete? Yes • Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd) • Space? O(bd) • Optimal? Yes, if step cost = 1 Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Summary of Algorithms for Uninformed Search Where: b is branching factor d is depth of shallowest solution, m is max depth of search tree C* is cost of optimal solution ε is the smallest step size Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

A Caution: Repeated States • Failure to detect repeated states can turn a linear problem into an exponential one, or even an infinite one. • For example: 8-puzzle • Simple repeat -- empty square simply moves back and forth • More complex repeats also possible. • Save list of expanded states -- the closed list. • Add new state to fringe only if it's not in closed list. Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

Summary: Uninformed Search • Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored • There exists a variety of uninformed search strategies • Search strategies can be evaluated by • whether they find a solution if one exists: completeness • how long they take: time complexity • how much memory they take: space complexity • whether they find the best solution: optimality • Iterative deepening search uses linear space and not much more time than other algorithms: usual choice Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are based on Russell, aima.eecs.berkeley.edu/slides-pdf. Diagrams are based on AIMA. CSC 8520 Spring 2013. Paula Matuszek

CS 8520: Artificial Intelligence