Problem Solving through State Space Search Jacques Robin Ontologies Reasoning Components Agents Simulations Outline Search Agent Formulating Decision Problems as Navigating State Space to Find Goal State Generic Search Algorithm Specific Algorithms Breadth-First Search Uniform Cost Search Depth-First Search Backtracking Search Iterative Deepening Search

Bi-Directional Search Comparative Table Limitations and Difficulties Repeated States Partial Information Search Agents Generic decision problem to be solved by an agent: Among all possible action sequences that I can execute, which ones will result in changing the environment from its current state, to another state that matches my goal? Additional optimization problem: Among those action sequences that will change the environment from its current state to a goal state which one can be executed at minimum cost? or which one leads to a state with maximum utility? Search agent solves this decision problem by a generate and test approach: Given the environment model, generate one by one (all) the possible states (the state space) of the environment reachable through all possible action sequences from the current state, test each generated state to determine whether it satisfies the goal or maximize utility Navigation metaphor: The order of state generation is viewed as navigating the entire environment state space

Example of Natural Born Search Problem Example of Applying Navigation Metaphor to Arbitrary Decision Problem N-Queens problem: how to arrange n queens on a NxN chess board in a pacific configuration where no queen can attack another? Full-State Formulation: local navigation in the full-State Space Q Q Q Q Q Q Q Move Queen Q Q Q Move Queen

Q Q Problem Navigation Metaphor Partial-State Formulation: global navigation in partial-State Space Q Q Q Q Insert Queen Insert Queen Algorithm Q Q Insert Queen

Q Q Insert Queen Q Q Search Problem Taxonomy Search Problem non-overlapping, complete overlapping, complete non-overlapping, complete Full-State Space Search Problem Partial-State Space Search Problem State Finding Search Problem Path Finding Search Problem

Goal-Satisfaction Search Problem Optimization Search Problem InitState: Full TypicalState: Full Action: ModifyState InitialState: Empty TypicalState: Partial GoalState: Full Action: RefineState Solution: State Solution: Path Test: Goal-Satisfaction Test: Utility-Maximization FSG FSO FPG

FPO PSG PSO PPG PPO Search Tree The state space can be represented as a search tree Each search tree node represents an environment state Each search tree arc represents an action changing the environment from its source node to its target node Each node or arc can be associated to a utility or cost Each path from the tree root to another tree node represents an action sequence Some search tree node leaves represent goal states The problem of navigating the state space then becomes one of generating and maintaining a search tree until a solution node or path is generated The problem search tree: An abstract concept that contains one node for each possible environment state Can be infinite The algorithm search tree: A concrete data structure

A subset of the problem search tree that contains only the nodes generated and maintained (i.e., explored) at the current point during the algorithm execution (always finite) Problem search tree branching factor: average number of actions available to the agent in each state Algorithm effective branching factor: average number of nodes effectively generated as successors of each node during search Search Tree Example: Vacuum Cleaner World u c s u d d u s d c s

u c,d d c,d s u u s d u d c,d d u c d s d

d d s c,d s c d c,d d u s d c d c,d d u c,d d

c,d d c d d s c d d c,d c d c d u d c,d d c,d

c,d d d c s c d d c,d c d c d d c,d c d d c,d

c d u d c d c,d d c,d d c,d c d c s c,d c,d d

c d d d c d c,d c,d d c d c,d d c c,d d d c d

c,d Search Methods Searching the entire space of the action sequence reachable environment states is called exhaustive search, systematic search, blind search or uninformed search Searching a restricted subset of that state space based on knowledge about the specific characteristics of the problem or problem class is called partial search A heuristic is an insight or approximate knowledge about a problem class or problem class family on which a search algorithm can rely to improve its run time and/or space requirement An ordering heuristic defines: In which order to generate the problem search tree nodes, Where to go next while navigating the state space (to get closer faster to a solution point) A pruning heuristic defines: Which branches of the problem search tree to avoid generating alltogether, What subspaces of the state space not to explore (for they cannot or are very unlikely to contain a solution point) Non-heuristic, exhaustive search is not scalable to large problem instances (worstcase exponential in time and/or space) Heuristic, partial search offers no warrantee to find a solution if one exist, or to find the best solution if several exists. Formulating a Agent Decision Problem as a Search Problem Define abstract format of a generic environment state, ex., a class C Define the initial state, ex., a specific object of class C Define the successor operation:

Takes as input a state or state set S and an action A Returns the state or state set R resulting from the agent executing A in S Together, these 3 elements constitute an intentional representation of the state space The search algorithm transforms this intentional representation into an extensional one, by repeatedly applying the successor operation starting from the initial state Define a boolean operation testing whether a state is a goal, ex., a method of C For optimization problems: additionally define a operation that returns the cost or utility of an action or state Problem Formulation Most Crucial Factor of Search Efficiency Problem formulation is more crucial than choice of search algorithm or choice of heuristics to make an agent decision problem effectively solvable by state space search 8-queens problem formulation 1: Initial state: empty board Action: pick column and line of one queen Branching factor: 64 State-space: ~864

8-queens problem formulation 2: Initial state: empty board Action: pre-assign one column per queen, pick only line in pre-assigned column Branching factor: 8 State-space: ~88 Generic Exhaustive Search Algorithm 1.Initialize the fringe to the root node representing the initial state 2.Until goal node found in fringe, repeat: a.Choose one node from fringe to expand by calling its successor operation b.Extend the current fringe with the nodes generated by this successor operation c. If optimization problem, update path cost or utility value 3.Return goal node or path from root node to goal node Specific algorithms differ in terms of the order in which they respectively expand the fringe nodes fringe Arad Sibiu Arad Fagaras Oradea

Timisoara R.Vilcea Arad Lugoj Zenrid Arad Oradea Generic Exhaustive Search Algorithm 1.Initialize the fringe to the root node representing the initial state 2.Until goal node found in fringe, repeat: a.Choose one node from fringe to expand by calling its successor operation b.Extend the current fringe with the nodes generated by this successor operation c. If optimization problem, update path cost or utility value 3.Return goal node or path from root node to goal node Specific algorithms differ in terms of the order in which they respectively expand the fringe nodes fringe Arad

Sibiu Arad Fagaras Oradea Timisoara R.Vilcea Arad Lugoj Zenrid Arad Oradea Generic Exhaustive Search Algorithm 1.Initialize the fringe to the root node representing the initial state 2.Until goal node found in fringe, repeat: a.Choose one node from fringe to expand by calling its successor operation b.Extend the current fringe with the nodes generated by this successor operation c. If optimization problem, update path cost or utility value

3.Return goal node or path from root node to goal node Specific algorithms differ in terms of the order in which they respectively expand the fringe nodes open-list Arad fringe Arad Sibiu Fagaras Oradea Timisoara R.Vilcea Arad Lugoj Zenrid Arad

Oradea Search Algorithms Characteristics and Performance Complete: guaranteed to find a solution if one exists Optimal (for optimization problem): guaranteed to find the best (highest utility or lowest cost) solution if one exists Input parameters to complexity metrics: b = problem search tree branching factor d = depth of highest solution (or best solution for optimization problems) in problem search tree m = problem search tree depth (can be infinite) Complexity metrics of algorithms: TimeComplexity(b,d,m) = number of expanded nodes SpaceComplexity(b,d,m) = maximum number of nodes needed in memory at one point during the execution of the algorithm Exhaustive Search Algorithms Breadth-First: Expand first most shallow node from fringe Uniform Cost: Expand first node from fringe of lowest cost (or highest utility) path from the root node Depth-First: Expand first deepest node from fringe Backtracking: Depth first variant with fringe limited to a single node Depth-Limited: Depth-first stopping at depth limit N. Iterative Deepening: Sequence of depth limited search at increasing depth Bi-directional: Parallel search from initial state and from goal state

Solution found when the two paths under construction intersect Breadth-First Search Fringe A B C D H E I J F K L G M

N O Breadth-First Search Fringe A Expanded B C D H E I J F K

L G M N O Breadth-First Search Fringe A Expanded B C D H E I J

F K L G M N O Breadth-First Search Fringe A Expanded B C D H

E I J F K L G M N O Breadth-First Search Fringe A Expanded B C

D H E I J F K L G M N O Breadth-First Search Fringe A

Expanded B C D H E I J F K L G M N O

Breadth-First Search Fringe A Expanded B C D H E I J F K L G

M N O Breadth-First Search Fringe A Expanded B C D H E I J F K

L G M N O Uniform Cost Search B 10 1 Problem Graph: 5 A 15 A A C

D 1 5 15 B E 5 D A A B 5 C C D

5 15 B C D 15 EB EB Ec 11 11 10 Depth-First Search A B

C D H E I J F K L G M N O Depth-First Search A

B C D H E I J F K L G M N O

Depth-First Search A B C D H E I J F K L G M N

O Depth-First Search A B C D H E I J F K L G

M N O Depth-First Search A B C D H E I J F K L

G M N O Depth-First Search A B C D H E I J F

K L G M N O Depth-First Search A B C D H E I J

F K L G M N O Depth-First Search A B C D H E

I J F K L G M N O Depth-First Search A B C D H

E I J F K L G M N O Backtracking Search A B C

D H E I J F K L G M N O Backtracking Search A B

C D H E I J F K L G M N O Depth-First Search

A B C D H E I J F K L G M N O

Backtracking Search A B C D H E I J F K L G M

N O Backtracking Search A B C D E I J F K L G M

N O Backtracking Search A B C E J F K L G M N

O Backtracking Search A B C E J F K L G M N O Backtracking Search

A B C E F K L G M N O Backtracking Search A C

F L G M N O Iterative Deepening L=0 A A L=1 A A A B

C A L=2 B C D E C D C E A B C E B

E A D C A B D C B A B B C A A

A D A B E D C E D B E D C E D E Bi-Directional Search

Two parallel searches: one from the current state and one from the goal state When they reach a common node: a path from current to goal has been found Time complexity halved: O(bd/2) + O(bd/2) = O(bd/2) << O(bd) But not always possible: Irreversible actions Large number of intentionally specified goal states Comparing Search Algorithms BreadthFirst UniformCost DepthFirst Backtracking Iterative Deepening Bi-Directional Complete

if b finite if all step costs positives no no if b finite Optimal if all steps share same cost yes no no if all steps share same cost

depends on search used on each direction Time Complexit y O(bd+1) O(bC*/e) O(bm) O(bm) O(bd) O(bd/2) Space Complexit y O(bd+1) O(bC*/e)

O(b.m ) C* = cost of optimal solution a actions(agent), cost(a) e O(m) O(b.d) O(bd/2) Searching Environments with Repeated States In some search problems, each environment state is only reachable through a single action sequence ex, such as the N-Queens problem in the pre-assigned column formulation, However, for most problems, the same environment state can be reached through several, many or even an infinity of distinct action sequences, This leads to the repeated generation of the same nodes in different branches of the search tree saa szb

sab sbb sba sab sza saa saz sza sba saa szz saz sbz sbb szb saa

sbb sbz saa sbz szz saa szb szz Effects of Repeated States on Exhaustive Search Algorithms They must be modified to include test comparing newly generated nodes with already generated ones and prune the repeated ones from the frontier Without such test: Breadth-first and uniform-cost search fill memory far sooner with repeated nodes and likely before reaching the depth of the first goal node Depth-first and backtracking search can enter in deepening loop, never returning even in the presence of shallow goal nodes With such test

Depth-first, backtracking and iterative deepening search loose their linear worst-case space complexity, for any guarantee to avoid all loops may require keeping an exponential number of expanded nodes in memory, in practice, the probability of loop occurrence is traded-off for size of the expanded node history Heuristic Search: Definition and Motivation Definition: Non-exhaustive, partial state space search strategy, based on approximate heuristic knowledge of the search problem class (ex, n-queens, Romania touring) or family (ex, unordered finite domain constraint solving) allowing to leave unexplored (prune) state space zones that are either guaranteed or unlikely to contain a goal state (or a utility maximizing state or cost minimizing path) Note: An algorithm that uses a fringe node ordering heuristic to generate a goal state faster, but is still ready to generate all state space states if necessary to find goal state i.e., an algorithm that does no pruning is not a heuristic search algorithm Motivation: exhaustive search algorithms do not scale up, neither theoretically (exponential worst case time or space complexity)

nor empirically (experimentally measured average case time or space complexity) Heuristic search algorithms do scale up to very large problem instances, in some cases by giving up completeness and/or optimality New data structure: heuristic function h(s) estimates the cost of the path from a fringe state s to a goal state Best-First Global Search Keep all expanded states on the fringe just as exhaustive breadth-first search and uniform-cost search Define an evaluation function f(s) that maps each state onto a number Expand the fringe in order of decreasing f(s) values Variations: Greedy Global Search (also called Greedy Best-First Search) defines f(s) = h(s) A* defines f(s) = g(s) + h(s) where g(s) is the real cost from the initial state to the state s, i.e., the value used to choose the state to expand in uniform cost search Greedy Local Search: Example Greedy Local Search: Example Greedy Local Search: Example Greedy Local Search: Example Greedy Local Search Characteristics

Strengths: simple Weaknesses: Not optimal because it relies only on estimated cost from current state to goal while ignoring confirmed cost from initial state to current state ex, misses better path through Riminicu and Pitesti Incomplete can enter in loop between two states that seem heuristically closer to the goal but are in fact farther away ex, from Iasi to Fagaras, it oscillates indefinitely between Iasi and Meant because the only road from either one to Fagaras goes through Valsui which in straight line is farther away to Fagaras than both A* Example h(s): straight-line distance to Bucharest: 449 75 + 374 239 239 + 178 140 + 253 220 417 393

118 + 329 447 220 + 193 413 317 418 366 415 455 496 336 + 160 317 + 98 A* Search Characteristics Strengths: Graph A* search is complete and Tree A* search is complete and optimal if h(s) is an admissible heuristic, i.e., if it never overestimates the real cost to a goal Graph A* search if optimal if h(s) admissible and in addition a monotonic (or consistent) heuristic h(s) is monotonic iff it satisfies the triangle inequality, i.e., s,s stateSpace (a actions, s = result(a,s)) h(s) cost(a) + h(s) A* is optimally efficient,

i.e., no other optimal algorithm will expand fewer nodes than A* using the same heuristic function h(s) Weakness: Runs out of memory for large problem instance because it keeps all generated nodes in memory Why? Worst-case space complexity = O(bd), Unless n, |h(n) c*(n)| O(log c*(n)) But very few practical heuristics verify this property A* Search Characteristics A* explores generates a contour around the best path The better the heuristic estimates the real cost, the narrower the contour Extreme cases: Perfect estimates, A* only generates nodes on the best path Useless estimates, A* generate all the nodes in the worst-case, i.e., degenerates into uniform-cost search Recursive Best-First Search (RBFS) Fringe limited to siblings of nodes along the path from the root to the current node For high branching factors, this is far smaller than A*s fringe which keeps all generated nodes At each step: Expand node n with lowest f(n) to generate successor(n) = {n 1, ..., ni} Store at n: A pointer to node n with the second lowest f(n) on the previous fringe Its cost estimate f(n)

Whenever f(n) f(nm) where f(nm) = min{f(n1), ..., f(ni)}: Update f(n) with f(nm) Backtrack to f(n) RBFS: Example RBFS: Example RBFS: Example RBFS: Characteristics Complete and optimal for admissible heuristics Space complexity O(bd) Time complexity hard to characterize In hard problem instance, can loose a lot of time swinging from one side of the tree to the other, regenerating over and over node that it had erased in the previous swing to that direction in such cases A* is faster Heuristic Function Design Desirable properties: Monotonic, Admissible, i.e., s stateSpace, h(s) c*(s), where c*(s) is the real cost of s High precision, i.e., as close as possible to c* Precision measure: effective branching factor b*(h) 1, N = average number of nodes generated by A* over a sample set of runs using h as

heuristic Obtained by solving the equation: b*(h) + (b*(h))2 + ... + (b*(h))d = N The closer b*(h) gets to 1, the more precise h is h2 dominates h1 iff: s stateSpace, h1(s) h2(s), i.e., if b*(h1) closer to 1 than b*(h2) hd(n) = max{h1(n), .., hk(n)} always dominates h1(n), .., hk(n) General heuristic function design principle: Estimated cost = actual cost of simplified problem computed using exhaustive search as a cheap pre-processing stage Problem can be simplified by: Constraint relaxation on its actions and/or goal state, which guarantees admissibility and monotonicity Decomposition in independent sub-problems Heuristic Function Design: Constraint Relaxation Example Relaxed problem 1: Ignore all constraints h1: number of tiles out of place Relaxed problem 2: Ignore only constraint 2 Constraints: 1. Tile cannot move diagonally

2. Tile cannot move in occupied location 3. Tile cannot move directly to nonneighboring locations h2: sum over all tiles t of the Manhattan distance between ts current and goal positions h2 dominates h1 Heuristic Function Design: Disjoint Pattern Databases Sub-problems: A: move tiles {1-4} B: move tiles {5-8} Preprocessing for one problem class amortized over many run of different instances of this class For each possible sub-problem instance: Use backward search from the goal to compute its cost, counting only the cost of the actions involving the entities of the sub-problem ex, moving tiles 1-4 for sub-problem 1, tiles 5-8 for sub-problem 2 store this cost in a disjoint pattern database During a given full problem run: Divide it into sub-problems

Look up their respective costs in the database Use the sum of these costs as heuristic Only work for domains where most actions involve only a small subsets of entities Local Heuristic Search Search algorithms for complete state formulated problems for which the solution is only a target state that satisfies a goal or maximizes a utility function and not a path from the current initial state to that target state; ex: Spatial equipment distribution (N-queens, VLSI, plant plan) Scheduling of vehicle assembly Task division among team members Only keeps a very limited fringe in memory,

often only the direct neighbors of the current node, or even only the current node. Far more scalable than global heuristic search, though generally neither complete nor optimal. Frequently used for multi-criteria optimization Hill-Climbing (HC) Fringe limited to current node, no explicit search tree Always expand neighbor node which maximizes heuristic function This is greedy local search Strengths: simple, very space scalable, works without modification for partial information and online search Weaknesses: incomplete, not optimal, "an amnesic climbing the Everest on a foggy day" Ridge Hill-Climbing Variations Stochastic HC: randomly chooses among uphill moves Slower convergence, but often better result First-Choice HC: generates random successors instead of all successors of current node Efficient for state spaces with very high branching factor HC with random restart to "pull out" from local maximum, plateau, ridges

Simulated annealing: Generates random moves Uphill moves always taken Downhill moves taken with a given probability that: is lower for the steeper downhill moves decreases over time Local Search Key parameter of local search algorithms: Step size around current node (especially for real valued domains) Can also decrease over time during the search Local beam search: Maintain a current fringe of k nodes Form the new fringe by expanding at once the k successors state with highest utility from this entire fringe to form the new fringe Stochastic beam search: At each step, pick successors semi-randomly with nodes with higher utility having a higher probability to be picked It is a form of genetic search with asexual reproduction In which Environments can a State Space Searching Agent (S3A) Act Successfully?

Fully observable? Partially observable? Sensorless? Deterministic or stochastic? Discrete or continuous? Episodic or non-episodic? Static? sequential? concurrent synchronous or concurrent asynchronous? Small or large? Lowly or highly diverse? In which Environments can a State Space Searching Agent (S3A) Act Successfully? Fully observable? Partially observable? Sensorless? An S3A can act successfully in an environment that is either partially observable or even sensorless, provided that it is deterministic, small, lowly diverse and that the agent possesses an action model Instead of searching in the space of states, it can search in the space of possible state sets (beliefs) to know which actions are potentially available Deterministic or stochastic? An S3A can act successfully in an environment that is stochastic provided that it is small and at least partially observable Instead of searching offline in the space of states, it can search offline in the space of possible state sets (beliefs) resulting of an action with stochastic effects It can also search online and generate action programs or conditional plans instead of mere action sequences, ex, [suck, right, {if right.dirty then suck}] Discrete or continuous?

An S3A using a global search algorithm can only act successfully in an discrete environment, for continuous variables imply infinite branching factor In which Environments can a State Space Searching Agent (S3A) Act Successfully? Episodic or non-episodic? An S3A is only necessary for a non-episodic environment which requires choosing action sequences, where the choice of one action conditions the action range subsequently available Static? sequential? concurrent synchronous or concurrent asynchronous? An S3A can only act successfully in a static or sequential environment, for there is no point in searching for goal-achieving or optimal action sequences in an environment that changes due to factors outside the control of the agent Small or large? An S3A can only act successfully in a medium-sized environment provided that it is deterministic and at least partially observable, or small environment that are either stochastic or sensorless Lowly or highly diverse? An S3A can only act successfully in low diversity environment for diversity affects directly the branching factor of the problem search tree space Searching in the Space of Possible State Sets