Introduction#

Planning is at the heart of intelligent systems: from autonomous robots navigating cluttered rooms to complex logistics pipelines that route millions of packages worldwide, AI agents constantly face the challenge of deciding how to move from a current state to a desired goal. The discipline of goal formulation and planning algorithms provides the mathematical and computational frameworks that turn high‑level intentions into concrete action sequences.

In this article we will:

• Define goal formulation and its role in autonomous decision‑making.
• Survey classical search‑based planners (DFS, BFS, Dijkstra, A*).
• Examine heuristic and hierarchical planning approaches.
• Discuss probabilistic planning (Markov Decision Processes, Partially Observable MDPs).
• Introduce learning‑based goal setting via reinforcement learning and policy search.
• Present real‑world case studies and practical implementation tips.
• Summarize trade‑offs and provide guidance for selecting the right algorithm stack.

By the end you will have a robust toolbox for designing effective planners and a deeper understanding of how AI agents reason about goals.

Foundations of Goal Formulation#

What is a Goal?#

A goal is a condition that signifies the successful completion of a planning episode. Formally, a goal ( G ) can be represented as a set of constraints or desired states ( S_d ) that must hold when the plan terminates:

[ G = { s \in \mathcal{S} \mid \text{constraints}(s) } ]

where ( \mathcal{S} ) is the state space.

Goal Types#

Goal Formulation Process#

1. Goal Extraction – Translating high‑level user intent into formal goal constraints.
2. Feasibility Analysis – Checking if the goal is achievable given domain knowledge.
3. Goal Prioritization – Ordering goals when multiple targets conflict.
4. Goal Decomposition – Breaking large goals into manageable sub‑goals.

These steps require knowledge representation (world models, action schemas) and reasoning capabilities to guarantee that goals are reachable.

Classical Goal‑Based Planning Algorithms#

Breadth‑First Search (BFS)#

• Search Strategy: Expands nodes level‑by‑level.
• Complexity: Exponential in branching factor ( b ) and depth ( d ).
• Use‑Case: Best when all actions cost the same and a shallow solution is desired.

Pseudocode#

BFS(start, goal):
    queue ← [start]
    visited ← {start}
    while queue not empty:
        node ← queue.pop_front()
        if node satisfies goal: return node.path
        for each successor in expand(node):
            if successor ∉ visited:
                visited.add(successor)
                queue.append(successor)

Depth‑First Search (DFS)#

• Search Strategy: Explores as far as possible along each branch before backtracking.
• Complexity: Linear in the size of the search tree if a solution exists; may get stuck in deep cycles.
• Use‑Case: Memory efficiency, when solution depth is shallow.

Dijkstra’s Algorithm#

• Weighted Graph Search: Guarantees optimal path for non‑negative edge weights.
• Complexity: ( O(|E| + |V| \log |V|) ) using a priority queue.
• Use‑Case: Planning with varying action costs but without heuristics.

A* Search#

• Heuristic‑Guided: Extends Dijkstra by adding heuristic estimate ( h(n) ).
• Optimality & Completeness: Guarantee if ( h(n) ) is admissible and consistent.
• Use‑Case: Real‑time robotics and path planning where speed matters.

Example Heuristic#

Heuristic Planning: Beyond Classic Search#

Heuristics convert planning into an optimization problem by estimating the remaining cost to reach the goal, reducing exhaustive search.

Informative Heuristics#

1. Misplaced Tile in 8‑Puzzle – counts tiles out of place.
2. Tower of Hanoi – number of moves needed for the smallest disk.
3. Planning Graphs – compute level sum or h_add heuristics.

Consistency Condition#

For all edges ( (u, v) ):

[ h(u) \leq c(u, v) + h(v) ]

Why consistency matters: Guarantees no node expansion order will increase the cost estimate, maintaining A*’s optimality.

Hierarchical Planning#

Complex tasks often benefit from hierarchy: high‑level abstract actions decompose into executable low‑level primitives.

Hierarchical Task Networks (HTNs)#

An HTN planner uses methods to decompose tasks:

method AssembleTable:
    decompose into: [stack legs, attach tabletop, adjust screws]

Pros:

• Scalable to large domains.
• Leverages domain expertise.

Cons:

• Requires handcrafted decomposition.

Domain‑Independent Hierarchical Planning#

• STRIPS+Hierarchy: Extending STRIPS with procedural knowledge.
• Polytree Planner: Uses causal links for partial ordering of tasks.

Probabilistic Planning Algorithms#

When the environment is uncertain, planners must reason over stochastic state transitions.

Markov Decision Processes (MDPs)#

Formalism:

Solving MDPs#

1. Value Iteration – updates value function until convergence.
2. Policy Iteration – alternates policy evaluation & improvement.
3. Linear Programming – solves for optimal policy via constraints.

Partially Observable MDPs (POMDPs)#

When actions yield noisy observations ( O ) and current state is hidden:

• Belief state representation ( b(s) ).
• Bellman backup on belief space.

State‑of‑Art Solvers: SARSOP, PBVI, Alpha‑Star.

Learning‑Based Goal Formulation#

Reinforcement Learning (RL) reframes planning as policy search, where the policy implicitly encodes goal fulfillment.

Model‑Free RL#

• Q‑Learning: Estimates action‑value function ( Q(s, a) ).
• Policy Gradient: Directly optimizes parameterized action probabilities.

Example – Goal‑Conditioned RL#

Define a goal vector ( g ) and learn a policy ( \pi(a \mid s, g) ). The reward is shaped via distance to the goal.

Model‑Based RL#

• Learns dynamics model ( \hat{T}(s, a) ).
• Uses planning algorithms (e.g., MCTS) on model predictions.

Practical Insight#

• Imitation Learning: Pre‑train with expert trajectories to bootstrap complex goal‑oriented behaviors.
• Curriculum Learning: Gradually increase goal difficulty.

Real‑World Case Studies#

Example Spotlight:
In the Kiva Systems warehouse setup, robots employ a layered planner: the high‑level HTN generates shelves and robot assignments, each robot runs a D planner within its local map. This duality scales to thousands of concurrent agents.

Practical Implementation Tips#

1. State Representation

   • Compact encodings (bitvectors for 3‑D blocks, occupancy grids for robots).
   • Sparse matrices for MDP transition tables reduce memory.

2. Graph Adjacency

   • Use adjacency lists for large sparse domains.
   • Store predecessor links to reconstruct paths efficiently.

3. Heuristic Pre‑computation

   • Store heuristic tables for fast lookup during search.
   • Cache heuristic expansions in HTNs for repeated sub‑tasks.

4. Parallelization

   • Parallel expansion in BFS for grid‑based exploration.
   • GPU‑accelerated value iteration for large MDPs.

5. Debugging Strategies

   • Visualize search tree depth and frontier.
   • Log open list size to detect inefficiencies.

6. Testing under Variability

   • Introduce simulated noise or dynamic obstacles.
   • Monitor success rate vs deterministic scenarios.

Algorithm Trade‑Offs#

Selecting an algorithm is a matching problem: domain constraints, available computational resources, and required guarantees must line up.

Conclusion#

Planning intertwines formal goal specification with algorithmic search or optimization to convert abstract wants into concrete actions. Whether you start from deterministic actions, probabilistic transitions, or learned policies, the same core principles apply: state representation, action modeling, and goal decomposition.

To build a robust planner:

1. Understand your domain and express goals with formal constraints.
2. Prototype with classic search to validate feasibility.
3. Augment with heuristics, hierarchies, or probabilistic models as problem scale and uncertainty grow.
4. Iterate—empirically evaluate performance, adjust heuristics or decompose goals, and consider learning‑based augmentation for highly dynamic scenarios.

By blending these perspectives, you can craft intelligent agents that not only think of what they must achieve but methodically chart the path to success—whether on a clean grid or a noisy, real‑world street.