Introduction to Artificial Intelligence

Lecture 2: Solving problems by searching

Prof. Gilles Louppe
g.louppe@uliege.be

Today

Planning agents
Search problems
Uninformed search methods
- Depth-first search
- Breadth-first search
- Uniform-cost search
Informed search methods
- A*
- Heuristics ] .kol-1-2[ .center.width-100[]

] ]

Planning agents

Reflex agents

select actions on the basis of the current percept;
may have a model of the world current state;
do not consider the future consequences of their actions;
consider only how the world is now.

.grid[ .kol-1-2[ .width-100[] ] .kol-1-2[ .width-100[] ] ] .caption[For example, a simple reflex agent moves to a dot if there is one in its neighborhood.
No planning is involved to take this decision. ]

.Q[[Q]] Can a reflex agent be rational?

???

Yes, provided the correct decision can be made on the basis of the current percept. That is, if the environment is fully observable, deterministic and known.

In the figure, the sequence of actions is clearly suboptimal.

Problem-solving agents

Assumptions:

Observable, deterministic (and known) environment.

Problem-solving agents

take decisions based on (hypothesized) consequences of actions;
must have a model of how the world evolves in response to actions;
formulate a goal, explicitly;
consider how to world would be.

.grid[ .kol-1-2[ .width-100[] ] .kol-1-2[ .width-100[] ] ] .caption[A planning agent looks for sequences of actions to eat all the dots.]

Offline vs. Online solving

Problem-solving agents are offline. The solution is executed "eyes closed", ignoring the percepts.
Online problem solving involves acting without complete knowledge. In this case, the sequence of actions might be recomputed at each step.

Search problems

A search problem consists of the following components:

The initial state of the agent.
A description of the actions available to the agent given a state $s$, denoted $\text{actions}(s)$.
A transition model that returns the state $s' = \text{result}(s, a)$ that results from doing action $a$ in state $s$.
- We say that $s'$ is a successor of $s$ if there is an acceptable action from $s$ to $s'$.

Together, the initial state, the actions and the transition model define the state space of the problem, i.e. the set of all states reachable from the initial state by any sequence of action.
- The state space forms a directed graph:
  - nodes = states
  - links = actions
- A path is a sequence of states connected by actions.
A goal test which determines whether the solution of the problem is achieved in state $s$.
A path cost that assigns a numeric value to each path.
- We may also assume that the path cost corresponds to a sum of positive step costs $c(s,a,s')$ associated to the action $a$ in $s$ leading to $s'$.

A solution to a problem is an action sequence that leads from the initial state to a goal state.

A solution quality is measured by the path cost function.
An optimal solution has the lowest path cost among all solutions.

[Q] What if the environment is partially observable? non-deterministic?

???

With partial observability, the agent needs to keep in which states it might be in.
- Percepts narrow down the set of possible states.
If stochastic, the agent will need to consider what to do for each contingency that its percepts may reveal.
- Percepts reveal which the outcomes has actually occurred.

See 4.3 and 4.4 for more details.

???

Search problems are models, i.e. abstract mathematical abstractions. These models omit details that not relevant for solving the problem.

The process of removing details from a representation is called abstraction.

Example: Traveling in Romania

.center.width-100[![](figures/lec2/romania.svg)]

Initial state = the city we start in.
- $s_0 = \text{in}(\text{Arad})$
Actions = Going from the current city to the cities that are directly connected to it.
- $\text{actions}(s_0) = \{ \text{go}(\text{Sibiu}), \text{go}(\text{Timisoara}), \text{go}(\text{Zerind}) \}$
Transition model = The city we arrive in after driving to it.
- $\text{result}(\text{in}(Arad), \text{go}(Zerind)) = \text{in}(Zerind)$
Goal test: whether we are in Bucharest.
- $s \in \{ \text{in}(\text{Bucharest}) \}$
Step cost: distances between cities.

Selecting a state space

Real world is absurdly complex.
- The world state includes every last detail of the environment.
- State space must be abstracted for problem solving.
A search state keeps only the details needed for planning.
- Example: eat-all-dots
  - States: $\{ (x, y), \text{dot booleans}\}$
  - Actions: NSEW
  - Transition: update location and possibly a dot boolean
  - Goal test: dots all false

State space size

World state:
- Agent positions: 120
- Found count: 30
- Ghost positions: 12
- Agent facing: NSEW
How many?
- World states?
  - $120 \times 2^{30} \times 12^2 \times 4$
- States for eat-all-dots?
  - $120 \times 2^{30}$ ] .kol-1-2[ .width-100[] ] ]

Search trees

The set of possible acceptable sequences starting at the initial state form a search tree:

Nodes correspond to states in the state space, where the initial state is the root node.
Branches correspond to applicable actions.
- Child nodes correspond to successors.
For most problems, we can never actually build the whole tree.

Tree search algorithms

Important ideas:

Fringe (or frontier) of partial plans under consideration
Expansion
Exploration

[Q] Which fringe nodes to explore? How to expand as few nodes as possible, while achieving the goal?

.center.width-90[]

Uninformed search strategies

Uninformed search strategies use only the information available in the problem definition.

They do not know whether a state looks more promising than some other.

Strategies:

Depth-first search
Breadth-first search
Uniform-cost search
Iterative deepening

Depth-first search

Strategy: expand the deepest node in the fringe.
Implementation: fringe is a LIFO stack.

.width-80.center[]

Properties of search strategies

A strategy is defined by picking the order of expansion.
Strategies are evaluated along the following dimensions:
- Completeness: does it always find a solution if one exists?
- Optimality: does it always find the least-cost solution?
- Time complexity: how long does it take to find a solution?
- Space complexity: how much memory is needed to perform the search?
Time and complexity are measured in terms of
- $b$: maximum branching factor of the search tree
- $d$: depth of the least-cost solution
  - the depth of $s$ is defined as the number of actions from the initial state to $s$.
- $m$: maximum length of any path in the state space (may be $\infty$)

[Q] Number of nodes in a tree?

???

Number of nodes = $\frac{b^{d+1}-1}{b-1}$

.center.width-80[]

Properties of DFS

Completeness:
- $m$ could be infinite, so only if we prevent cycles (more on this later).
Optimality:
- No, DFS finds the leftmost solution, regardless of depth or cost.
Time complexity:
- May generate the whole tree (or a good part of it, regardless of $d$). Therefore $O(b^m)$, which might much greater than the size of the state space!
Space complexity:
- Only store siblings on path to root, therefore $O(bm)$.
- When all the descendants of a node have been visited, the node can be removed from memory.

.center.width-80[]

Breadth-first search

Strategy: expand the shallowest node in the fringe.
Implementation: fringe is a FIFO queue.

.width-80.center[]

Properties of BFS

Completeness:
- If the shallowest goal node is at some finite depth $d$, BFS will eventually find it after generating all shallower nodes (provided $b$ is finite).
Optimality:
- The shallowest goal is not necessarily the optimal one.
- BFS is optimal only if the path cost is a non-decreasing function of the depth of the node.
Time complexity:
- If the solution is a depth $d$, then the total number of nodes generated before finding this node is $b+b^2+b^3+...+b^d = O(b^d)$
Space complexity:
- The number of nodes to maintain in memory is the size of the fringe, which will be the largest at the last tier. That is $O(b^d)$

.center.width-80[]

Iterative deepening

Idea: get DFS's space advantages with BFS's time/shallow solution advantages.
- Run DFS with depth limit 1.
- If no solution, run DFS with depth limit 2.
- If no solution, run DFS with depth limit 3.
- ...

[Q] Isn't this process wastefully redundant? ] .kol-1-2[ .center.width-100[] ] ]

Uniform-cost search

Strategy: expand the cheapest node in the fringe.
Implementation: fringe is a priority queue, using the cumulative cost $g(n)$ from the initial state to node $n$ as priority.

Properties of UCS

Completeness:
- Yes, if step cost are all such that $c(s,a,s') \geq \epsilon > 0$.
Optimality:
- Yes, sinces UCS expands nodes in order of their optimal path cost.
Time complexity:
- Assume $C^*$ is the cost of the optimal solution and that step costs are all $\geq \epsilon$.
- The "effective depth" is then roughly $C^*/\epsilon$.
- The worst-case time complexity is $O(b^{C^*/\epsilon})$.
Space complexity:
- The number of nodes to maintain is the size of the fringe, so as many as in the last tier $O(b^{C^*/\epsilon})$.

???

R: explain why positive step cost is required for completeness.

.center.width-70[]

Informed search strategies

One of the issues of UCS is that it explores the state space in every direction, without exploiting information about the (plausible) location of the goal node.

.center.width-70[]

Informed search strategies aim to solve this problem by expanding nodes in the fringe in decreasing order of desirability.

Greedy search
A*

Greedy search

Heuristics

A heuristic (or evaluation) function $h(n)$ is:

a function that estimates the cost of the cheapest path from node $n$ to a goal state;
- $h(n) \geq 0$ for all nodes $n$
- $h(n) = 0$ for a goal state.
is designed for a particular search problem.

.center.width-70[![](figures/lec2/heuristic-pacman.png)]

Greedy search

Strategy: expand the node $n$ in the fringe for which $h(n)$ is the lowest.
Implementation: fringe is a priority queue, using $h(n)$ as priority.

$h(n)$ = straight line distance to Bucharest.

Properties of greedy search

Completeness:
- No, unless we prevent cycles (more on this later).
Optimality:
- No, e.g. the path via Sibiu and Fagaras is 32km longer than the path through Rimnicu Vilcea and Pitesti.
Time complexity:
- $O(b^m)$, unless we have a good heuristic function.
Space complexity:
- $O(b^m)$, unless we have a good heuristic function.

.center.width-90[]

.center[At best, greedy search takes you straight to the goal.
At worst, it is like a badly-guided BFS.]

A*

Shakey the Robot

A* was first proposed in 1968 to improve robot planning.
Goal was to navigate through a room with obstacles. ] .kol-1-2[ .center.width-80[] ] ]

A*

Uniform-cost orders by path cost, or backward cost $g(n)$
Greedy orders by goal proximity, or forward cost $h(n)$
A* combines the two algorithms and orders by the sum $$f(n) = g(n) + h(n)$$
$f(n)$ is the estimated cost of cheapest solution through $n$.

.center.width-80[]

.center.width-80[]

[Q] Why doesn't A* stop at step (e), since Bucharest is in the fringe?

Admissible heuristics

A heuristic $h$ is admissible if $$0 \leq h(n) \leq h^*(n)$$ where $h^*(n)$ is the true cost to a nearest goal.

.center.width-80[] .caption[The Manhattan distance is admissible]

???

$h$ is admissible if it underestimates the true cost towards the goal.

Optimality of A* (tree search)

Assumptions

$A$ is an optimal goal node
$B$ is a suboptimal goal node
$h$ is admissible

Claim

$A$ will exit the fringe before $B$. ] .kol-1-3[ .width-100[] ] ]

Claim

$n$ will be expanded before $B$.

Proof

Assume $B$ is on the fringe. Some ancestor $n$ of $A$ is on the fringe too.

$f(n) \leq f(A)$
- $f(n) = g(n) + h(n)$ (by definition)
- $f(n) \leq g(A)$ (admissibility of $h$)
- $f(A) = g(A) + h(A) = g(A)$ ($h=0$ at a goal)
$f(A) < f(B)$
- $g(A) < g(B)$ ($B$ is suboptimal)
- $f(A) < f(B)$ ($h=0$ at a goal)
$n$ expands before $B$
- since $f(n) \leq f(A) < f(B)$ ] .kol-1-3[ .width-100[] ] ] All ancestors of $A$ expand before $B$, including $A$. Therefore A is optimal*.

A* contours

Assume $f$-costs are non-decreasing along any path.
We can define contour levels $t$ in the state space, that include all nodes $n$ for which $f(n) \leq t$.

.center[ ] .grid[ .kol-1-2[ For UCS ($h(n)=0$ for all $n$), bands are circular around the start. ] .kol-1-2[ For A* with accurate heuristics, bands stretch towards the goal. ] ]

.grid[ .kol-1-3[ .width-100[] ] .kol-1-3[ .width-100[] ] .kol-1-3[ .width-100[] ] ] .center.grid[ .kol-1-3[ Greedy search ] .kol-1-3[ UCS ] .kol-1-3[ A* ] ]

???

A* finds the shortest path.

Creating admissible heuristics

Most of the work in solving hard search problems optimally is in finding admissible heuristics.

Admissible heuristics can be derived from the exact solutions to relaxed problems, where new actions are available.

.center.width-80[]

Dominance

If $h_1$ and $h_2$ are both admissible and if $h_2(n) \geq h_1(n)$ for all $n$, then $h_2$ dominates $h_1$ and is better for search.
Given any admissible heuristics $h_a$ and $h_b$, $$h(n) = \max(h_a(n), h_b(n))$$ is also admissible and dominates $h_a$ and $h_b$.

Learning heuristics from experience

Assuming an episodic environment, an agent can learn good heuristics by playing the game many times.
Each optimal solution $s^*$ provides training examples from which $h(n)$ can be learned.
Each example consists of a state $n$ from the solution path and the actual cost $g(s^*)$ of the solution from that point.
The mapping $n \to g(s^*)$ can be learned with supervised learning algorithms.
- Linear models, Neural networks, etc.

Graph search

The failure to detect repeated states can turn a linear problem into an exponential one!

Redundant paths and cycles can be avoided by keeping track of the states that have been explored. This amounts to grow a tree directly on the state-space graph.

[Q] What are the properties of DFS/GFS/UCS/GS/A* based on graph search?

Recap example: Super Mario

.center.width-50[]

Task environment?
- performance measure, environment, actuators, sensors?
Type of environment?
Search problem?
- initial state, actions, transition model, goal test, path cost?
Good heuristic?

???

performance measure = score, the further right, the better; coins; killed enemies, ...
environment: the Mario world
actuators: left, right, up, down, jump, speed
sensors: the screen
type of environment: partially observable, deterministic, episodic, dynamic, discrete/continuous, multi-agent, known
search problem:
- state = Mario's position (x, y), map, score, time
- initial state = start of the game
- transition model = given by the game engine (assume we know that!)
- goal test = have we reached the flag?
- path cost = shortest path, the better; malus if killed, bonus if coin and killed enemies

A* in action

???

Comment on the actions taken at any frame (right, jump, speed) shown in red.

Summary

Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored.
Variety of uninformed search strategies (DFS, BFS, UCS, Iterative deepening)
Heuristic functions estimate costs of shortest paths.
Good heuristic can dramatically reduce search cost.
Greedy best-first search expands lowest $h$.
- incomplete and not always optimal.
A* search expands lower $f=g+h$
- complete and optimal
Admissible heuristics can be derived from exact solutions of relaxed problems or learned from training examples.
Graph search can be exponentially more efficient than tree search.

The end.

References

Hart, Peter E., Nils J. Nilsson, and Bertram Raphael. "A formal basis for the heuristic determination of minimum cost paths." IEEE transactions on Systems Science and Cybernetics 4.2 (1968): 100-107.

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Introduction to Artificial Intelligence

Today

Planning agents

Reflex agents

Problem-solving agents

Offline vs. Online solving

Search problems

Search problems

Example: Traveling in Romania

Selecting a state space

State space size

Search trees

Tree search algorithms

Uninformed search strategies

Depth-first search

Properties of search strategies

Properties of DFS

Breadth-first search

Properties of BFS

Iterative deepening

Uniform-cost search

Properties of UCS

Informed search strategies

Greedy search

Heuristics

Greedy search

Properties of greedy search

A*

Shakey the Robot

A*

Admissible heuristics

Optimality of A* (tree search)

Assumptions

Claim

Claim

Proof

A* contours

Creating admissible heuristics

Dominance

Learning heuristics from experience

Graph search

Recap example: Super Mario

Summary

References