Lecture 2 Uninformed Search

Introduction to Planning Agents and Search Problems

Overview of Planning Agents

• The focus of the lecture series is to develop a formalism for agents to plan ahead, framing problems as search problems.

• Different algorithms will be explored, starting with depth-first, breadth-first, and uniform cost search in future lectures.

Contrast Between Reflex Agents and Planning Agents

• Reflex agents act based solely on the current state without considering consequences; for example, winking at an approaching bug.

• In contrast, planning agents simulate potential outcomes before deciding on actions, weighing different scenarios.

Example of Reflex Agent: Pac-Man

• Pac-Man operates as a reflex agent by moving towards the nearest dot without strategic planning.

• However, this approach can lead to suboptimal results if the environment changes unexpectedly.

Understanding Optimality and Completeness in Planning

Definitions of Optimality and Completeness

• An optimal algorithm guarantees finding the least-cost solution when one exists.

• Completeness ensures that if there is a solution available, it will be found by the algorithm.

Importance of Replanning

• Replanning may be necessary in dynamic environments where initial plans become invalid due to unforeseen circumstances.

Mastermind Pac-Man vs. Naive Replanning

Mastermind Pac-Man's Approach

• This version computes an optimal path before acting but requires significant computational time upfront.

Naive Replanning Strategy

• A simpler approach involves continuously recalculating paths toward the nearest food pellet while executing actions.

Components of Search Problems

Structure of Search Problems

• A search problem consists of a state space representing relevant aspects of the world being modeled.

Successor Function and Goal Test

• The successor function defines possible actions from a given state along with their associated costs.

Solution Definition

• A solution is defined as a sequence of actions transforming the start state into one or more goal states.

Real-world Applications: Pathing Problem Example

Formalizing Pathing Problems

• For instance, navigating from Arad to Bucharest can be framed as a search problem using cities as states.

State Space Considerations

• The choice between abstracting at city level versus GPS coordinates impacts complexity and detail in planning.

Detailed Examples in Pac-Man World

World State Representation

• The world state includes all variables necessary for simulating game dynamics like positions and food locations.

Simplifying State Spaces

• Fixed elements like walls should not be included in dynamic state spaces since they do not change during gameplay.

Designing Effective State Spaces

Key Variables for Simulation

• Essential variables include dynamic elements such as player position and food status rather than static obstacles.

Implications for Algorithm Efficiency

• Keeping state spaces small enhances algorithm performance by reducing complexity during searches.

Understanding State Space Graphs and Search Trees

Introduction to State Space Graphs

• A state space graph is a mathematical abstraction used to represent search problems, where each state occurs only once.

• The goal is to build parts of the state space graph dynamically during algorithm execution rather than constructing the entire graph, which is often impractical.

Concept of Search Trees

• A search tree grows from a start state (the root), with available actions leading to successor states depicted in subsequent layers.

• Each node in the search tree represents not just a state but an entire path taken from the start state through various actions.

Characteristics of Search Trees

• States can appear multiple times in a search tree due to different action sequences leading to the same outcome.

• The aim is not to construct the full search tree but only that part necessary for finding a solution.

Building and Analyzing Search Trees

Example of a Tiny State Space Graph

• A small example illustrates how one would build a search tree starting from an initial state and exploring its successors.

Infinite Size Consideration

• In some cases, such as with certain four-state graphs, the resulting search tree can be infinite, emphasizing the need for efficient exploration strategies.

Tree Search Algorithm Overview

Initial Setup for Tree Search

• The process begins by initializing the fringe (the set of nodes under consideration), starting with just the initial state.

Expansion Process

• If no candidates are left in the fringe, it indicates failure; otherwise, nodes are selected based on specific strategies for expansion.

Implementing Tree Search Strategies

Functionality of Tree Search

• The function checks if any node contains a goal state; if so, it returns that node as part of the solution path.

Node Selection Strategy

• Different algorithms will prioritize which fringe nodes to explore first based on their respective strategies.

Practical Application: Pathfinding Example

Executing Tree Search on Pathfinding Problem

• Starting at "Arod," successors are identified and added to the fringe. Nodes are then selected for expansion based on strategy until reaching "Bucharest."

Depth First vs. Breadth First Search Strategies

Depth First Search (DFS)

• DFS explores as deep as possible along branches before backtracking; it may lead down paths that do not yield solutions quickly.

Properties of DFS

• Completeness depends on avoiding cycles; optimality isn't guaranteed unless all paths have equal costs. Time complexity can reach O(b^m).

Breadth First Search (BFS)

BFS Mechanics

• BFS expands all nodes at present depth before moving deeper into subsequent levels, ensuring shallowest solutions are found first when they exist.

Properties of BFS

• Completeness: Yes, provided there’s a finite solution depth.

• Optimality: Only if all actions have equal cost; otherwise not guaranteed.

Comparative Analysis: When Do BFS or DFS Outperform Each Other?

Situational Advantages

• BFS outperforms DFS when searching for shallow solutions or when branching factors are high. Conversely, DFS may excel with smart tie-breaking or memory constraints.

Conclusion

• Iterative deepening combines benefits from both approaches by running depth-limited searches iteratively until finding shallowest solutions while maintaining lower memory usage compared to BFS.

Understanding Tree Depth and Search Complexity

Exponential Growth of Tree Size

• The size of a tree grows exponentially with depth, where each node at depth d has B^d nodes. This indicates that the work required to explore deeper layers increases significantly compared to shallower ones.

Cost Implications in Search Algorithms

• When considering search algorithms, the last search at maximum depth dominates the overall running time, making it crucial to evaluate costs associated with actions in the search process.

Introduction to Uniform Cost Search

Definition and Mechanism

• Uniform cost search is an algorithm designed for graphs with varying edge costs. It prioritizes expanding nodes based on their traversal cost rather than uniformity in costs across edges.

Expanding Nodes Based on Cost

• The algorithm begins by expanding the start state and continues to expand the cheapest node from the fringe until reaching a goal state, ensuring that it always selects paths with minimal cost.

Goal Achievement Criteria

• Even if a node achieving the goal appears on the fringe, uniform cost search will not select it until it is determined to be the cheapest option available. This ensures optimal path selection.

Properties of Uniform Cost Search

Node Expansion Characteristics

• The algorithm expands nodes based on their cost relative to an optimal plan's cost (C^). It only considers nodes whose costs are less than or equal to C^.

Effective Depth and Memory Complexity

• The effective depth of uniform cost search is defined as C^*/epsilon, where epsilon represents a minimum action cost. This influences both node expansion and memory requirements during searches.

Completeness and Optimality Conditions

Assumptions for Success

• Uniform cost search is complete as long as there exists a non-zero lower bound (epsilon > 0) for action costs. Under these conditions, it guarantees finding an optimal path.

Limitations of Uniform Cost Search

Exploration Direction Issues

• While uniform cost search is complete and optimal, it may not effectively explore desired directions within a tree structure. Future discussions will address heuristic methods to enhance exploration efficiency.