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Problem Solving and Search Andrea Danyluk September 9, 2013

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Announcements Progamming Assignment 0: Python Tutorial – Should have received my about it – Due Wednesday at 10am Should not take long to complete Talk to me if the due date/time is a real problem – Turnin should be fixed today – Grading for this one is mainly a sanity check – Any trouble with accounts: let me know!

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Today’s Lecture Agents Goal-directed problem-solving and search Uninformed search – Breadth-first – Depth-first and variants – Uniform-cost search (Wednesday)

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Rational Agents An agent perceives and acts. “Doing the right thing” captured by a performance measure that evaluates a given sequence of environment states. A rational agent: selects an action that is expected to maximize the performance measure, given evidence provided by the percept sequence and whatever built-in knowledge the agent has.

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Rational ≠ omniscient – Percepts may not supply all relevant information Rational ≠ clairvoyant – Action outcomes may not be as expected Rational ≠ successful [ Adapted from Russell]

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Reflex Agents Act on the basis of the current percept (and possibly what they remember from the past). May have memory or a model of the world’s state. Do not consider future consequences of their actions. [Adapted from CS 188 UC Berkeley]

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Goal-based Agents Plan ahead Ask “what if” Decisions based on (hypothesized) consequences of actions Have a model of how the world evolves in response to actions [Adapted from CS 188 UC Berkeley]

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Building a goal-based agent Determine the percepts available to the agent Select/devise a representation for world states Determine the task knowledge the agent will need Clearly articulate the goal Select/devise a problem-solving technique so that the agent can decide what to do

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Search as a Fundamental Problem- Solving Technique Originated with Newell and Simon’s work on problem solving in the late 60s.

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Search Problems A search problem consists of A state space – A set of states – As set of actions – A transition model that specifies results of applying actions to states Successor function: Result(s, a) An initial state A goal test

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An Example: the 8-Puzzle Initial StateGoal State # possible distinct states = 9! = 362,880 (but only 9!/2 reachable)

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Real world examples Navigation Vehicle parking Parsing (natural and artificial languages) – The old dog slept on the porch – The old dog the footsteps of the young

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And back to the 8-Puzzle States: – Puzzle configurations Actions: – Move blank N, S, E, or W Start state: – As given Goal test: – Is current state = specified goal state?

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State Space Graph S S G

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Finding a solution in a problem graph Solving the puzzle = finding a path through the graph from initial state to goal state Simple graph search algorithms: – Breadth-first search – Depth-first search

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Breadth-first Graph Search: Review a a b b h h g g c c f f e e d d i i Fringe is a FIFO queue Start: a Goal: i Explore vertices (really edges closest to the start state first.

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Depth-first Graph Search: Review a a b b h h g g c c f f e e d d i i Fringe is a LIFO stack Start: a Goal: i Explore deepest vertex (really edge) first.

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State Space Search in the AI World Rarely given a graph to begin with We don’t build the graph before doing the search – Our search problems are BIG

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Search Trees {N, E, S, W} Search tree for BFS White nodes = explored (i.e., expanded) Orange nodes = frontier Maintains an “explored” set of states to avoid redundant search

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Search Tree A “what if” tree of plans and outcomes Start state at the root node Children correspond to successors Nodes contain states; correspond to plans to those states Aim to build as little as possible Because we build the tree “on the fly” the representations of states and actions matter! [Adapted from CS 188 UC Berkeley]

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Formalizing State Space Search A state space is a graph (V, E), where V is the set of states and E is a set of directed edges between states. The edges may have associated weights (costs). Our exploration of the state space using search generates a search tree. [Adapted from Eric Eaton]

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Formalizing State Space Search A node in a search tree typically contains: – A state description – A reference to the parent node – The name of the operator that generated it from its parent – The cost of the path from the initial state to itself The node that is the root of the search tree typically represents the initial state

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Formalizing State Space Search Child nodes are generated by applying legal operators to a node – The process of expanding a node means to generate all of its successor nodes and to add them to the frontier. A goal test is a function applied to a state to determine whether its associated node is a goal node

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Formalizing State Space Search A solution is either – A sequence of operators that is associated with a path from start state to goal or – A state that satisfies the goal test The cost of a solution is the sum of the arc costs on the solution path – If all arcs have the same (unit) cost, then the solution cost is just the length of the solution (i.e., the length of the path)

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Evaluating Search Strategies Completeness – Is the strategy guaranteed to find a solution when there is one? Optimality – Does the strategy find the highest-quality solution when there are several solutions? Time Complexity – How long does it take (in the worst case) to find a solution? Space Complexity – How much memory is required (in the worst case)?

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Evaluating BFS Complete? – Yes Optimal? – Not in general. When is it optimal? Time Complexity? – How do we measure it? Space Complexity?

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Time and Space Complexity Let b = branching factor (i.e., max number of successors) m = maximum depth of the search tree d = depth of shallowest solution For BFS Time: O(b d ) If we check for goal as we generate a node! Not if we check as we get ready to expand! Space: O(b d )

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Evaluating DFS Complete? – Not in general – Can be if space is finite and we modify tree search to account for loops and redundant paths Optimal? – No Time Complexity? – O(b m ) Space Complexity? – O(mb)

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Depth-Limited DFS? Let l be the depth limit Complete? – No Optimal? – No Time Complexity? – O(b l ) Space Complexity? – O(m l )

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Iterative Deepening? Complete? – Yes (if b is finite) Optimal? – Yes (if costs of all actions are the same) Time Complexity? – O(b d ) Space Complexity? – O(bd)

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