1. Classical Planning

Classical Planning

State Model $S(P)$

• Finite and discrete state space $S$
• a known initial state $\ s_0 \in S$
• a set $S_G \subseteq S$ of goal states
• actions $A(s) \subseteq A$ applicable in each $s \in S$
• a deterministic transition function $s' = f(a,s)$ for $a \in A(s)$
• positve action cost $c(a,s)$

Goals states can be more than one

Search Algorithm

• b: maximal branching factor
• d: goal depth
• m: the maximal depth reached

Breadth First Search Data structure: FIFO (Queue)

Breadth First Search explores node level by level, and a queue (FIFO) allows nodes to be processed in the order that they are discovered, which ensures all nodes at a given depth are expanded before moving to the next level.

Depth first Search

Data structure: LIFO (Stack)

Depth FiRst Search explores as deep as possible along each branch before backtracking, and a stack (LIFO) enables this by always processing the most recently discovered node that still has unexplored successors.

Uniform Cost Search

Data structure: Priority Queue

Uniform Cost Search expands nodes based on t6he lowest cumulaitve cost, and a priority queue efficiently retrieves the node with the minimal cost at each step, allowing UCS for finding the optimal path.

	DFS	BFS	ID	A*	HC	IDA*
Complete	No	Yes	Yes	Yes	No	Yes
Optimal	No	Yes*	Yes*	Yes	No	Yes
Time	$b^m$	$b^d$	$b^d$	$b^d$	$∞$	$b^d$
Space	$bm$	$b^d$	$bd$	$b^d$	$b$	$bd$

A search algorithm is considered complete if it guarantees to find a solution, if one exists.

BFS and IDS are optimal only all nodes (edges) has equal costs(uniform).

Astar Search

$f(n) = g(n) + h(n)$
$g(n)$ : The actual cost from the initial state to the current state
$h(n)$ : The heuristic value from the current state to goal state

$h^*(s)$ Actual optimal value from current state s to goal state

Property of heuristic

1. Admissble: never over-estimate. If $h(s) \leq h^*(s)$ for all $s \in S$
2. Safe: for all $h(s) = \infty \rightarrow h^*(s) = \infty$
3. Goal aware： $h(s) = 0$ for all $s\in S_G$
4. Consistent: $h(s) \leq cost(a) + h(s')$

consistent + goal aware $\rightarrow$ admissible
admissible $\rightarrow$ goal aware + safe

Optimality for A star search

• A star search without re-open mechanism requires both consistent and admissible to ensure finding an optimal path.
• With re-open mechanism, it only requies admissible

Example for a non-optimal solution by inconsistent but adimissble for A* search without re-open nodes.

     (8)
      A
     / \
+1  /   \ +3
   /     \   
  B ----- C ----- D
(7)  +1  (0)  +6  (0)

Reference: Why does A* with admissible non consistent heuristic find non optimal solution?

Greedy best-first search

• Using priority queue expand node by h(s)
• Completeness: For safe heuristic, Yes
• Optimality: No

STRIPS And Relaxation

A STRIPS problem P = $\langle F, O, I, G \rangle$ : Propositions, Initial state, Goal, Actions.

h+: The Optimal Delete Relaxation Heuristic

Default STRIP model: s' = s + add(s) - del(s)
after delete relaxation: s' = s + add(s)

Defination $h^{add}$

$$\begin{cases} 0 & g \subseteq s \\ \min_{a \in A, g \in \text{add}_a} c(a) + h^{\text{add}}(s, \text{pre}_a) & |g| = 1 \\ \sum_{g' \in g} h^{\text{add}}(s, \{g'\}) & |g| > 1 \\ \end{cases}$$

如果目标大于1: $h^{add}$ 等于每个小目标的$h^{add}$ 之和

Defination $h^{max}$

$$\begin{cases} 0 & g \subseteq s \\ \min_{a \in A, g \in \text{add}_a} c(a) + h^{\text{max}}(s, \text{pre}_a) & |g| = 1 \\ \max_{g' \in g} h^{\text{max}}(s, \{g'\}) & |g| > 1 \\ \end{cases}$$

如果目标大于1: $h^{max}$ 等于每个小目标的$h^{max}$ 的最大值

$h^{add}$ 和 $h^{max}$：如果一个动作有多个precondition，则视为多个g：$|g| > 1$

Bellman-Ford Table

$h^{max}$

$h^{add}$

• $h^{max}$ is admissible, but is typically far too optimistic

• $h^{add}$ is not admissible, but is typically a lot more informed than $h^{max}$

Relaxed Plan Extraction $h^{ff}$

A heuristic function is called a relaxed plan heuristic, denoted $h^{FF}$, if, given a state $s$, it returns $\infty$ if no relaxed plan exists, and otherwise returns $\sum_{a \in RPlan} c(a)$ where $RPlan$ is the action set returned by relaxed plan extraction on a closed well-founded best-supporter function for $s$.

• Plan set means no duplicate actions.

• $h^{FF}$ never over count sub plans due to $\text{RPlan} := \text{RPlan} \cup \{ b_s(g) \}$
• $h^{FF}$ may be inadmissible, just like $h^{add}$, but for more subtle reasons.
• In practice, $h^{FF}$ typically does not over-estimate $h^*$

For relaxed plan extraction to make sense, it requires a closed well-founded best-supporter function:

The $h^{max}$ supporter function:

$$b^{\text{max}}_s(p) := \argmin_{a \in A, p \in \text{add}_a} c(a) + h^{\text{max}}(s, \text{pre}_a).$$

The $h^{add}$ supporter function:

$$b^{\text{add}}_s(p) := \argmin_{a \in A, p \in \text{add}_a} c(a) + h^{\text{add}}(s, \text{pre}_a).$$

Proposition : ($h^{FF}$ is Pessimistic and Agrees with $h^{*}$ on $\infty$). For all STRIPS planning tasks $\Pi$, $h^{FF} \geq h^{+}$ for all states, $h^{+}(s) = \infty$ if and only if $h^{FF}(s) = \infty$. There exists STRIPS planning tasks $\Pi$ and $s$ so that $h^{FF}(s) > h^{*}(s)$.

Width-based search

novelty w(s)