Problem Solving AgentsCSL 302 ARTIFICIAL INTELLIGENCE

SPRING 2014

Goal Based Agents

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Representation Mechanisms (propositional/first order/probabilistic logic)

Search (blind and informed)PlanningInference

Learning Models

Example

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Problem Solving AgentsGoal FormulationoOrganize behavior of the agent

oGoal – set of states in the world where the goal is satisfied

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Example

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Goal

Initial

Problem Solving AgentsGoal FormulationoOrganize behavior of the agent

oGoal – set of states in the world where the goal is satisfied

Problem FormulationoWhat are the actions?

oWhat are the states?

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Assumptions about the Task EnvironmentObservable or partially observable?

Discrete or Continuous?

Deterministic or Stochastic?

Static or Dynamic?

Episodic or Sequential?

Multiple or Single Agent?

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Assumptions about the Task EnvironmentObservable or partially observable?

Discrete or Continuous?

Deterministic or Stochastic?

Static or Dynamic?

Episodic or Sequential?

Multiple or Single Agent?

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Assumptions about the Task EnvironmentObservable or partially observable?

Discrete or Continuous?

Deterministic or Stochastic?

Static or Dynamic?

Episodic or Sequential?

Multiple or Single Agent?

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Finding a sequence of actions – Search!

Problem Solving AgentsGoal FormulationoOrganize behavior of the agent

oGoal – set of states in the world where the goal is satisfied

Problem FormulationoWhat are the actions?

oWhat are the states?

SearchoFinding the sequence of actions

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Example

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Goal

States

Operator/Action

Initial

What is the solution?

Problem TypesDeterministic and Fully Observable: Single state problemoSolution is sequence

Non-observable: Conformant problemoSolution (if any) is a sequence

Stochastic and/or Partially Observable: Contingency problemoSolution is a contingency plan or a policy

Unknown state space: Exploration problem

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Problem Solving – Atomic Agents

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Atomic AgentsoStates are indivisible

oSearching through the states to reach the goal.

Single State Problem Formulation

Problem can be defined by 5 components1. Initial State: the state the agent starts

2. Actions: the set of operators that can be executed at a state

3. Transition model: returns the state that results from doing an action in a state

4. Goal test: determines whether a given state is a goal state

5. Path Cost: function that assigns a numeric cost to a path

Step cost: cost of taking a single action

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• State Space• Graph• Path

Example

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Initial State: AradActions: Drive(Sibiu),Drive(Timisora)Goal Test: In(Bucharest)Path Cost: ?

Example: Toy Vacuum Problem

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State: Robot and Dirt Locations

Initial State: Any State

Actions: Left, Right Suck

Goal Test: No Dirt

Path Cost: cost 1 per action?

Example: Eight Puzzle Problem

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State: Tile Locations

Initial State: A specific tile configuration

Actions: move the blank tile left, right, up or down

Goal Test: tiles are in the required configuration

Path Cost: cost 1 per move?

Note: Optimal solution for an n-puzzle family is NP hard.

Example: 8 Queens Problem

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State: Configuration of the Queens

Initial State: Empty board

Actions: Add a queen to the board

Goal Test: configuration with 8 queens on the board with none attacking another

Path Cost: time taken to solve?

Example: Missionaries and Cannibals

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State: number of missionaries and cannibals on the boat and each bank

Initial State: all objects one bank

Actions: move boat with x missionaries and y cannibals, no more cannibals than missionaries on the boat or the shore, a boat with a maximum capacity.

Goal Test: All objects on the opposite bank

Path Cost: 1 per river crossing

Example: Rubik’s Cube

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State: List of colors on each face

Initial State: A specific color pattern

Actions: rotate a row or column or a face

Goal Test: configuration has the same color on all tiles on every face

Path Cost: cost 1 per move?

Example: Rubik’s Cube

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Example: Real WorldTravelling Salesman Problem (TSP)

Robot Navigation

Protein folding

Graph Coloring

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Uninformed Search21/1

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Search - TreesBasic Principle:oOffline simulated exploration of search space

oGenerate successors of already explored states

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Search Space as a Tree

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Parent Root

Node

Children Children

Node

Initial StateActions

Solution

Goal State

State

Example

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Search Strategies Strategies vary in the order in which nodes are picked for expansion

Evaluating search strategiesoCompleteness – Does it always find a solution if one exists?oOptimality – Does it always find a least cost solution?oSpace complexity – How much memory is needed to perform

search?oTime complexity – How long does it take to find a solution?

Time and Space complexities are measuredob – maximum branching factor of the search treeod – shallowest depth of the least cost solutionom- maximum depth of the search space

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Uninformed search strategiesUse only the information available in the problem definition

Breadth-first search

Uniform-cost search

Depth-first search

Depth-limited search

Iterative deepening search

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Breadth-first search (BFS)Expand shallowest unexpanded node

Implementation: FIFO Queue; successors at the end of the queue

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BFS – AnalysisCompleteness: Yes (if b is finite)

Optimality: Not optimal; Yes- Uniform cost edges

Time Complexity: exponential in d1 + 𝑏 + 𝑏2 + 𝑏3 +⋯+ 𝑏𝑑 + 𝑏 𝑏𝑑 − 1 = 𝑂(𝑏𝑑+1)

Space Complexity: 𝑂(𝑏𝑑+1)

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𝑏 = 10 106𝑛𝑜𝑑𝑒𝑠 𝑠𝑒𝑐 103𝑏𝑦𝑡𝑒𝑠 𝑛𝑜𝑑𝑒

Uniform cost search (UCS)Expand least-cost (𝑔(𝑛))unexpanded node

Implementation: Priority queue – sort the nodes in the queue based on cost.

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UCS - AnalysisCompleteness: Yes; if step cost ≥ 𝜖

Optimality: Yes; nodes are expanded in increasing order of 𝑔(𝑛)

Time Complexity: # of nodes with 𝑔 ≤ cost of optimal solution(𝐶∗) - 𝑂(𝑏 𝐶∗ 𝜖 )

Space Complexity: # of nodes with 𝑔 ≤ cost of optimal solution - 𝑂(𝑏 𝐶∗ 𝜖 )

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Large subtrees with inexpensive steps may be explored before useful paths with costly steps

Depth-first search (DFS)Expand deepest unexpanded node

Implementation: LIFO queue; successors at the front of the queue

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DFS - AnalysisCompleteness: complete only in finite spaces; incomplete when there are loops and infinite spaces

Optimality: No

Time Complexity: 𝑂(𝑏𝑚); terrible when 𝑚 ≫ 𝑑;might be faster than BFS, when solutions are dense.

Space Complexity: 1 + 𝑏 + 𝑏 +⋯+ (𝑚𝑡ℎ𝑙𝑒𝑣𝑒𝑙)𝑏 =𝑂(𝑏𝑚); Linear space!!

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Depth # nodes MemoryBFS

Memory DFS

16 1016 10Eb 156Kb

Depth-limited search (DLS)Depth-first search with depth limit 𝑙

Implementation: nodes at depth 𝑙 have no successors.

Only finite space to be explored.

Completeness: Yes/No???

Optimality: Yes/No???

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Iterative deepening search(IDS)

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𝑑𝑒𝑝𝑡ℎ = 0

𝑑𝑒𝑝𝑡ℎ = 1

𝑑𝑒𝑝𝑡ℎ =2

𝑑𝑒𝑝𝑡ℎ =3

𝑑𝑒𝑝𝑡ℎ =4

IDS- AnalysisCompleteness: Yes!

Optimality: Yes for uniform cost edges; can be modified to explore uniform cost tree

Time Complexity: 𝑑𝑏 + 𝑑 − 1 𝑏2 +⋯+1 𝑏𝑑 = 𝑂(𝑏𝑑)

Space Complexity: 𝑂(𝑏𝑑)

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Asymptotic ratio of # nodes expanded by IDS vs DFS: (𝑏 + 1) 𝑏 − 1 ≈ 1for large values of 𝑏

Summary

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Graph SearchBFS-?

DFS-?

IDDFS-?

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