itcs 3153 artificial intelligence

ITCS 3153Artificial Intelligence

Lecture 3Lecture 3

Uninformed SearchesUninformed Searches

(mostly copied from Berkeley)(mostly copied from Berkeley)

Lecture 3Lecture 3

Uninformed SearchesUninformed Searches

(mostly copied from Berkeley)(mostly copied from Berkeley)

Outline

Problem Solving AgentsProblem Solving Agents• Restricted form of general agentRestricted form of general agent

Problem TypesProblem Types• Fully vs. partially observable, deterministic vs. stochasticFully vs. partially observable, deterministic vs. stochastic

Problem FormulationProblem Formulation• State space, initial state, successor function, goal test, path costState space, initial state, successor function, goal test, path cost

Example ProblemsExample Problems

Basic Search AlgorithmsBasic Search Algorithms

Problem Solving AgentsProblem Solving Agents• Restricted form of general agentRestricted form of general agent

Problem TypesProblem Types• Fully vs. partially observable, deterministic vs. stochasticFully vs. partially observable, deterministic vs. stochastic

Problem FormulationProblem Formulation• State space, initial state, successor function, goal test, path costState space, initial state, successor function, goal test, path cost

Example ProblemsExample Problems

Basic Search AlgorithmsBasic Search Algorithms

Problem Solving Agents

Restricted form of general agent:Restricted form of general agent:functionfunction SIMPLE-PROBLEM-SOLVING-AGENT( SIMPLE-PROBLEM-SOLVING-AGENT(perceptpercept) ) returnsreturns an action an action staticstatic: : seqseq, an action sequence, initially empty, an action sequence, initially empty statestate, some description of the current world state, some description of the current world state goalgoal, a goal, initially null, a goal, initially null problemproblem, a problem definition, a problem definition state state <- UPDATE-STATE(<- UPDATE-STATE(statestate, , perceptpercept)) ifif seqseq is empty is empty thenthen goalgoal <- FORMULATE-GOAL( <- FORMULATE-GOAL(statestate)) problemproblem <- FORMULATE-PROBLEM( <- FORMULATE-PROBLEM(statestate, , goalgoal)) seq <- SEARCH(seq <- SEARCH(problemproblem)) actionaction <- RECOMMENDATION( <- RECOMMENDATION(seq, state)seq, state) seqseq <- REMAINDER( <- REMAINDER(seq, stateseq, state)) returnreturn actionaction

Note: This is offline problem solving; solution with “eyes closed.” Note: This is offline problem solving; solution with “eyes closed.” Online problem solving involves acting without complete knowledgeOnline problem solving involves acting without complete knowledge

Restricted form of general agent:Restricted form of general agent:functionfunction SIMPLE-PROBLEM-SOLVING-AGENT( SIMPLE-PROBLEM-SOLVING-AGENT(perceptpercept) ) returnsreturns an action an action staticstatic: : seqseq, an action sequence, initially empty, an action sequence, initially empty statestate, some description of the current world state, some description of the current world state goalgoal, a goal, initially null, a goal, initially null problemproblem, a problem definition, a problem definition state state <- UPDATE-STATE(<- UPDATE-STATE(statestate, , perceptpercept)) ifif seqseq is empty is empty thenthen goalgoal <- FORMULATE-GOAL( <- FORMULATE-GOAL(statestate)) problemproblem <- FORMULATE-PROBLEM( <- FORMULATE-PROBLEM(statestate, , goalgoal)) seq <- SEARCH(seq <- SEARCH(problemproblem)) actionaction <- RECOMMENDATION( <- RECOMMENDATION(seq, state)seq, state) seqseq <- REMAINDER( <- REMAINDER(seq, stateseq, state)) returnreturn actionaction

Note: This is offline problem solving; solution with “eyes closed.” Note: This is offline problem solving; solution with “eyes closed.” Online problem solving involves acting without complete knowledgeOnline problem solving involves acting without complete knowledge

Example: Romania

On holiday in Romania; currently in Arad.On holiday in Romania; currently in Arad.

Flight leaves tomorrow from Bucharest.Flight leaves tomorrow from Bucharest.

Formulate Goal:Formulate Goal:• be in Bucharestbe in Bucharest

Formulate Problem:Formulate Problem:• statesstates: various cities: various cities

• actionsactions: drive between citites: drive between citites

Find Solution:Find Solution:• Sequence of cities, e.g., Arad, Sibiu, Fagaras, BucharestSequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

On holiday in Romania; currently in Arad.On holiday in Romania; currently in Arad.

Flight leaves tomorrow from Bucharest.Flight leaves tomorrow from Bucharest.

Formulate Goal:Formulate Goal:• be in Bucharestbe in Bucharest

Formulate Problem:Formulate Problem:• statesstates: various cities: various cities

• actionsactions: drive between citites: drive between citites

Find Solution:Find Solution:• Sequence of cities, e.g., Arad, Sibiu, Fagaras, BucharestSequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

Example: Romania

Problem Types

Deterministic, fully observable Deterministic, fully observable single-state problem single-state problem• Agent knows exactly what state it will be in; solution is a sequenceAgent knows exactly what state it will be in; solution is a sequence

Non-observable Non-observable conformant problem conformant problem• Agent may have no idea where it is; solution (if any) is a sequenceAgent may have no idea where it is; solution (if any) is a sequence

Non-deterministic and/or partially observableNon-deterministic and/or partially observable• Percepts provide Percepts provide newnew information about current state information about current state

• Solution is a Solution is a treetree or or policypolicy

• Often Often interleaveinterleave search, execution search, execution

Unknown state space Unknown state space exploration problem (“online”) exploration problem (“online”)

Deterministic, fully observable Deterministic, fully observable single-state problem single-state problem• Agent knows exactly what state it will be in; solution is a sequenceAgent knows exactly what state it will be in; solution is a sequence

Non-observable Non-observable conformant problem conformant problem• Agent may have no idea where it is; solution (if any) is a sequenceAgent may have no idea where it is; solution (if any) is a sequence

Non-deterministic and/or partially observableNon-deterministic and/or partially observable• Percepts provide Percepts provide newnew information about current state information about current state

• Solution is a Solution is a treetree or or policypolicy

• Often Often interleaveinterleave search, execution search, execution

Unknown state space Unknown state space exploration problem (“online”) exploration problem (“online”)

Example: Vacuum World

Single-state, start in #5.Single-state, start in #5.

• Solution??Solution??





• Solution: [Right, Suck]Solution: [Right, Suck]

Conformant, start in Conformant, start in {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8}

• E.g., right goes to {2,4,6,8}E.g., right goes to {2,4,6,8}



• Solution: [Right, Suck]Solution: [Right, Suck]

Conformant, start in Conformant, start in {1,2,3,4,5,6,7,8}{1,2,3,4,5,6,7,8}





• Solution: Solution: [Right, Suck][Right, Suck]

Conformant, start in {1,2,3,4,5,6,7,8}Conformant, start in {1,2,3,4,5,6,7,8}


• Solution: Solution: [Right, Suck, Left, Suck][Right, Suck, Left, Suck]

Contingency, start in #5Contingency, start in #5

• Murphy’s Law: Murphy’s Law: SuckSuck can dirty a clean can dirty a clean carpetcarpet

• Local sensing: dirt, location onlyLocal sensing: dirt, location only



• Solution: Solution: [Right, Suck][Right, Suck]



• Solution: Solution: [Right, Suck, Left, Suck][Right, Suck, Left, Suck]







• Solution: [Solution: [Right, SuckRight, Suck]]



• Solution: [Solution: [Right, Suck, Left, SuckRight, Suck, Left, Suck]]




• Solution: [Solution: [Right, Right, if if dirtdirt then then SuckSuck]]


• Solution: [Solution: [Right, SuckRight, Suck]]



• Solution: [Solution: [Right, Suck, Left, SuckRight, Suck, Left, Suck]]




• Solution: [Solution: [Right, Right, if if dirtdirt then then SuckSuck]]

Single-state problem formationA A problemproblem is defined by four items: is defined by four items:

Initial stateInitial state

• E.g., “at Arad”E.g., “at Arad”

Successor function S(x) = set of action-state pairsSuccessor function S(x) = set of action-state pairs

• E.g., S(Arad) = {<AradE.g., S(Arad) = {<AradZerind,Zerind>, <AradZerind,Zerind>, <AradSibiu,Sibiu>,…}Sibiu,Sibiu>,…}

Goal test, can beGoal test, can be

• Explicit, e.g., “at Bucharest”Explicit, e.g., “at Bucharest”

• Implicit, e.g., Implicit, e.g., NoDirt(x)NoDirt(x)

Path cost (additive)Path cost (additive)

• E.g., a sum of distances, number of actions executed, etc.E.g., a sum of distances, number of actions executed, etc.

• C(x,a,y) is the step cost, assumed to be non-negativeC(x,a,y) is the step cost, assumed to be non-negative

A A solutionsolution is a sequence of actions leading from the initial state to a goal state is a sequence of actions leading from the initial state to a goal state

A A problemproblem is defined by four items: is defined by four items:

Initial stateInitial state

• E.g., “at Arad”E.g., “at Arad”

Successor function S(x) = set of action-state pairsSuccessor function S(x) = set of action-state pairs

• E.g., S(Arad) = {<AradE.g., S(Arad) = {<AradZerind,Zerind>, <AradZerind,Zerind>, <AradSibiu,Sibiu>,…}Sibiu,Sibiu>,…}

Goal test, can beGoal test, can be

• Explicit, e.g., “at Bucharest”Explicit, e.g., “at Bucharest”

• Implicit, e.g., Implicit, e.g., NoDirt(x)NoDirt(x)

Path cost (additive)Path cost (additive)

• E.g., a sum of distances, number of actions executed, etc.E.g., a sum of distances, number of actions executed, etc.

• C(x,a,y) is the step cost, assumed to be non-negativeC(x,a,y) is the step cost, assumed to be non-negative

A A solutionsolution is a sequence of actions leading from the initial state to a goal state is a sequence of actions leading from the initial state to a goal state

State Space

Real world is absurdly complex Real world is absurdly complex state space must be state space must be abstracted for problem solvingabstracted for problem solving

• (Abstract) state = set of real states(Abstract) state = set of real states

• (Abstract) action = complex combination of real actions, e.g., (Abstract) action = complex combination of real actions, e.g., “Arad“AradZerind” represents a complex set of possible routes, detours, Zerind” represents a complex set of possible routes, detours, rest stops, etc.rest stops, etc.

• (Abstract) solution = set of real paths that are solutions in the real world(Abstract) solution = set of real paths that are solutions in the real world

Each abstract action should be “easier” than the original Each abstract action should be “easier” than the original problem!problem!

Real world is absurdly complex Real world is absurdly complex state space must be state space must be abstracted for problem solvingabstracted for problem solving

• (Abstract) state = set of real states(Abstract) state = set of real states

• (Abstract) action = complex combination of real actions, e.g., (Abstract) action = complex combination of real actions, e.g., “Arad“AradZerind” represents a complex set of possible routes, detours, Zerind” represents a complex set of possible routes, detours, rest stops, etc.rest stops, etc.

• (Abstract) solution = set of real paths that are solutions in the real world(Abstract) solution = set of real paths that are solutions in the real world

Each abstract action should be “easier” than the original Each abstract action should be “easier” than the original problem!problem!

Example: Vacuum World state space graph

States? Actions? Goal test? Path cost?States? Actions? Goal test? Path cost?States? Actions? Goal test? Path cost?States? Actions? Goal test? Path cost?

Example: Vacuum World state space graph

States:States:

• Integer dirt and robot locations Integer dirt and robot locations (ignore dirt amounts)(ignore dirt amounts)

Actions:Actions:

• Left, Right, Suck, NoOpLeft, Right, Suck, NoOp

Goal test:Goal test:

• No dirtNo dirt

Path cost:Path cost:

• 1 per action (0 for 1 per action (0 for NoOpNoOp))

States:States:

• Integer dirt and robot locations Integer dirt and robot locations (ignore dirt amounts)(ignore dirt amounts)

Actions:Actions:

• Left, Right, Suck, NoOpLeft, Right, Suck, NoOp

Goal test:Goal test:

• No dirtNo dirt

Path cost:Path cost:

• 1 per action (0 for 1 per action (0 for NoOpNoOp))

Other Examples

Eight puzzleEight puzzle

Robotic AssemblyRobotic Assembly

States? Actions? Goal test? Path cost?States? Actions? Goal test? Path cost?

Eight puzzleEight puzzle

Robotic AssemblyRobotic Assembly

States? Actions? Goal test? Path cost?States? Actions? Goal test? Path cost?

Tree Search Algorithms

Basic idea:Basic idea:

• Offline, simulated exploration of state space by generating successors of Offline, simulated exploration of state space by generating successors of already explored states (AKA already explored states (AKA expandingexpanding states) states)

functionfunction TREE-SEARCH( TREE-SEARCH(problem, strategyproblem, strategy) ) returnsreturns a solution, or failure a solution, or failure initialize the search tree using the initial state of initialize the search tree using the initial state of problemproblem loop doloop do ifif there are no more candidates for expansion there are no more candidates for expansion then returnthen return failure failure choose a leaf node for expansion according to choose a leaf node for expansion according to strategystrategy ifif the node contains a goal state the node contains a goal state then returnthen return the corresponding solution the corresponding solution elseelse expand the node and add the resulting nodes to the search tree expand the node and add the resulting nodes to the search tree endend

Basic idea:Basic idea:

• Offline, simulated exploration of state space by generating successors of Offline, simulated exploration of state space by generating successors of already explored states (AKA already explored states (AKA expandingexpanding states) states)

functionfunction TREE-SEARCH( TREE-SEARCH(problem, strategyproblem, strategy) ) returnsreturns a solution, or failure a solution, or failure initialize the search tree using the initial state of initialize the search tree using the initial state of problemproblem loop doloop do ifif there are no more candidates for expansion there are no more candidates for expansion then returnthen return failure failure choose a leaf node for expansion according to choose a leaf node for expansion according to strategystrategy ifif the node contains a goal state the node contains a goal state then returnthen return the corresponding solution the corresponding solution elseelse expand the node and add the resulting nodes to the search tree expand the node and add the resulting nodes to the search tree endend

Implementation: states vs. nodes

StateState

• (Representation of) a physical configureation(Representation of) a physical configureation

NodeNode

• Data structure constituting part of a search treeData structure constituting part of a search tree

– Includes Includes parent, children, depth, path costparent, children, depth, path cost g(x) g(x)

States do not have parents, children, depth, or States do not have parents, children, depth, or path cost!path cost!

StateState

• (Representation of) a physical configureation(Representation of) a physical configureation

NodeNode

• Data structure constituting part of a search treeData structure constituting part of a search tree

– Includes Includes parent, children, depth, path costparent, children, depth, path cost g(x) g(x)

States do not have parents, children, depth, or States do not have parents, children, depth, or path cost!path cost!

Implementation: general tree searchfunction function TREE-SEARCH (TREE-SEARCH (problemproblem, , fringefringe) ) returnsreturns a solution, or failure a solution, or failure fringefringe INSERT(MAKE-NODE(INITIAL-STATE[problem]), INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringefringe)) loop doloop do ifif fringe fringe is empty is empty then returnthen return false false nodenode REMOVE-FRONT( REMOVE-FRONT(fringefringe)) ifif GOAL-TEST[ GOAL-TEST[problemproblem] applied to STATE(] applied to STATE(nodenode) succeeds ) succeeds returnreturn nodenode fringefringe INSERTALL(EXPAND( INSERTALL(EXPAND(nodenode, , problemproblem), ), fringefringe))

function function EXPAND (EXPAND (nodenode, , problemproblem) ) returnsreturns a set of states a set of states successorssuccessors the empty set the empty set for each for each action, resultaction, result inin SUCCESSOR-FN[problem](STATE[ SUCCESSOR-FN[problem](STATE[nodenode]) ]) dodo ss a new NODE a new NODE PARENT-NODE[PARENT-NODE[ss] ] nodenode; ACTION[; ACTION[ss] ] actionaction; STATE(; STATE(ss) ) resultresult PATH-COST[PATH-COST[ss] ] PATH-COST[ PATH-COST[nodenode] + STEP-COST(] + STEP-COST(node, action, snode, action, s)) DEPTH[DEPTH[ss] ] DEPTH[ DEPTH[nodenode] + 1] + 1 add add ss to to successorssuccessorsreturn return successorssuccessors

Find the error on this page!Find the error on this page!

function function TREE-SEARCH (TREE-SEARCH (problemproblem, , fringefringe) ) returnsreturns a solution, or failure a solution, or failure fringefringe INSERT(MAKE-NODE(INITIAL-STATE[problem]), INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringefringe)) loop doloop do ifif fringe fringe is empty is empty then returnthen return false false nodenode REMOVE-FRONT( REMOVE-FRONT(fringefringe)) ifif GOAL-TEST[ GOAL-TEST[problemproblem] applied to STATE(] applied to STATE(nodenode) succeeds ) succeeds returnreturn nodenode fringefringe INSERTALL(EXPAND( INSERTALL(EXPAND(nodenode, , problemproblem), ), fringefringe))

function function EXPAND (EXPAND (nodenode, , problemproblem) ) returnsreturns a set of states a set of states successorssuccessors the empty set the empty set for each for each action, resultaction, result inin SUCCESSOR-FN[problem](STATE[ SUCCESSOR-FN[problem](STATE[nodenode]) ]) dodo ss a new NODE a new NODE PARENT-NODE[PARENT-NODE[ss] ] nodenode; ACTION[; ACTION[ss] ] actionaction; STATE(; STATE(ss) ) resultresult PATH-COST[PATH-COST[ss] ] PATH-COST[ PATH-COST[nodenode] + STEP-COST(] + STEP-COST(node, action, snode, action, s)) DEPTH[DEPTH[ss] ] DEPTH[ DEPTH[nodenode] + 1] + 1 add add ss to to successorssuccessorsreturn return successorssuccessors

Find the error on this page!Find the error on this page!

Search strategiesA strategy is defined by picking the A strategy is defined by picking the order of node expansionorder of node expansion

Strategies are evaluated along the following dimensions:Strategies are evaluated along the following dimensions:

• Completeness – does it always find a solution if one exists?Completeness – does it always find a solution if one exists?

• Time complexity – number of nodes generated/expandedTime complexity – number of nodes generated/expanded

• Space complexity – maximum nodes in memorySpace complexity – maximum nodes in memory

• Optimality – does it always find a least cost solution?Optimality – does it always find a least cost solution?

Time and space complexity are measured in terms of:Time and space complexity are measured in terms of:

• b – maximum branching factor of the search treeb – maximum branching factor of the search tree

• d – depth of the least-cost solutiond – depth of the least-cost solution

• m – maximum depth of the state space (may be infinite)m – maximum depth of the state space (may be infinite)

A strategy is defined by picking the A strategy is defined by picking the order of node expansionorder of node expansion

Strategies are evaluated along the following dimensions:Strategies are evaluated along the following dimensions:

• Completeness – does it always find a solution if one exists?Completeness – does it always find a solution if one exists?

• Time complexity – number of nodes generated/expandedTime complexity – number of nodes generated/expanded

• Space complexity – maximum nodes in memorySpace complexity – maximum nodes in memory

• Optimality – does it always find a least cost solution?Optimality – does it always find a least cost solution?

Time and space complexity are measured in terms of:Time and space complexity are measured in terms of:

• b – maximum branching factor of the search treeb – maximum branching factor of the search tree

• d – depth of the least-cost solutiond – depth of the least-cost solution

• m – maximum depth of the state space (may be infinite)m – maximum depth of the state space (may be infinite)

Uninformed Search Strategies

Uninformed strategies use only the information Uninformed strategies use only the information available in the problem definitionavailable in the problem definition

• Breadth-first searchBreadth-first search

• Uniform-cost searchUniform-cost search

• Depth-first searchDepth-first search

• Depth-limited searchDepth-limited search

• Iterative deepening searchIterative deepening search

Uninformed strategies use only the information Uninformed strategies use only the information available in the problem definitionavailable in the problem definition

• Breadth-first searchBreadth-first search

• Uniform-cost searchUniform-cost search

• Depth-first searchDepth-first search

• Depth-limited searchDepth-limited search

• Iterative deepening searchIterative deepening search

Breadth-first search

Expand shallowest unexpanded nodeExpand shallowest unexpanded node

Implementation:Implementation:

• FringeFringe is a FIFO queue, i.e., new successors go at end is a FIFO queue, i.e., new successors go at end

• Execute first few expansions of Arad to Bucharest using Execute first few expansions of Arad to Bucharest using Breadth-first searchBreadth-first search

Expand shallowest unexpanded nodeExpand shallowest unexpanded node


• FringeFringe is a FIFO queue, i.e., new successors go at end is a FIFO queue, i.e., new successors go at end

• Execute first few expansions of Arad to Bucharest using Execute first few expansions of Arad to Bucharest using Breadth-first searchBreadth-first search

Properties of breadth-first search

Complete?? Yes (if b is finite)Complete?? Yes (if b is finite)

Time?? 1 + b + bTime?? 1 + b + b22 + … + b + … + bdd + b(b + b(bdd-1) = O(b-1) = O(bd+1d+1), i.e., ), i.e., exp in dexp in d

Space?? O(bSpace?? O(bd+1d+1) (keeps every node in memory)) (keeps every node in memory)

Optimal?? If cost = 1 per step, not optimal in Optimal?? If cost = 1 per step, not optimal in generalgeneral

Space is the big problem; can easily generate Space is the big problem; can easily generate nodes at 10 MB/s, so 24hrs = 860GB!nodes at 10 MB/s, so 24hrs = 860GB!

Complete?? Yes (if b is finite)Complete?? Yes (if b is finite)

Time?? 1 + b + bTime?? 1 + b + b22 + … + b + … + bdd + b(b + b(bdd-1) = O(b-1) = O(bd+1d+1), i.e., ), i.e., exp in dexp in d

Space?? O(bSpace?? O(bd+1d+1) (keeps every node in memory)) (keeps every node in memory)

Optimal?? If cost = 1 per step, not optimal in Optimal?? If cost = 1 per step, not optimal in generalgeneral

Space is the big problem; can easily generate Space is the big problem; can easily generate nodes at 10 MB/s, so 24hrs = 860GB!nodes at 10 MB/s, so 24hrs = 860GB!

Uniform-cost searchExpand least-cost unexpanded nodeExpand least-cost unexpanded node


• FringeFringe = queue ordered by path cost = queue ordered by path cost

Equivalent to breadth-first if…Equivalent to breadth-first if…

Complete?? If step cost Complete?? If step cost ≥≥ εε

Time?? # of nodes with g Time?? # of nodes with g ≤≤ cost of optimal solution, O(b cost of optimal solution, O(bC/ C/ εε), where C is the ), where C is the cost of the optimal solutioncost of the optimal solution

Space?? # of nodes with Space?? # of nodes with g g ≤≤ cost of optimal solution, O(b cost of optimal solution, O(bC/ C/ εε))

Optimal?? Yes–nodes expanded in increasing order of Optimal?? Yes–nodes expanded in increasing order of g(n)g(n)

Execute first few expansions of Arad to Bucharest using Uniform-first searchExecute first few expansions of Arad to Bucharest using Uniform-first search

Expand least-cost unexpanded nodeExpand least-cost unexpanded node


• FringeFringe = queue ordered by path cost = queue ordered by path cost

Equivalent to breadth-first if…Equivalent to breadth-first if…

Complete?? If step cost Complete?? If step cost ≥≥ εε

Time?? # of nodes with g Time?? # of nodes with g ≤≤ cost of optimal solution, O(b cost of optimal solution, O(bC/ C/ εε), where C is the ), where C is the cost of the optimal solutioncost of the optimal solution

Space?? # of nodes with Space?? # of nodes with g g ≤≤ cost of optimal solution, O(b cost of optimal solution, O(bC/ C/ εε))

Optimal?? Yes–nodes expanded in increasing order of Optimal?? Yes–nodes expanded in increasing order of g(n)g(n)

Execute first few expansions of Arad to Bucharest using Uniform-first searchExecute first few expansions of Arad to Bucharest using Uniform-first search

Depth-first search

Expand deepest unexpanded nodeExpand deepest unexpanded node


• FringeFringe = LIFO queue, i.e., a stack = LIFO queue, i.e., a stack

• Execute first few expansions of Arad to Bucharest using Execute first few expansions of Arad to Bucharest using Depth-first searchDepth-first search

Expand deepest unexpanded nodeExpand deepest unexpanded node


• FringeFringe = LIFO queue, i.e., a stack = LIFO queue, i.e., a stack

• Execute first few expansions of Arad to Bucharest using Execute first few expansions of Arad to Bucharest using Depth-first searchDepth-first search

Depth-first search

Complete??Complete??

Time??Time??

Space??Space??

Optimal??Optimal??


Time??Time??

Space??Space??

Optimal??Optimal??

Depth-first searchComplete??Complete??

• No: fails in infinite-depth spaces, spaces with loops.No: fails in infinite-depth spaces, spaces with loops.

• Can be modified to avoid repeated states along path Can be modified to avoid repeated states along path complete in finite spaces complete in finite spaces

Time??Time??

• O(bO(bmm)): terrible if : terrible if mm is much larger than is much larger than dd, but if solutions are dense, may be much faster , but if solutions are dense, may be much faster than breadth-firstthan breadth-first

Space??Space??

• O(bm), i.e., linear space!O(bm), i.e., linear space!

Optimal??Optimal??

• NoNo


• No: fails in infinite-depth spaces, spaces with loops.No: fails in infinite-depth spaces, spaces with loops.

• Can be modified to avoid repeated states along path Can be modified to avoid repeated states along path complete in finite spaces complete in finite spaces

Time??Time??

• O(bO(bmm)): terrible if : terrible if mm is much larger than is much larger than dd, but if solutions are dense, may be much faster , but if solutions are dense, may be much faster than breadth-firstthan breadth-first

Space??Space??

• O(bm), i.e., linear space!O(bm), i.e., linear space!

Optimal??Optimal??

• NoNo

Depth-limited search

Depth-first search with depth limit lDepth-first search with depth limit l• i.e., nodes at depth i.e., nodes at depth ll have no successors have no successors

function function DEPTH-LIMITED-SEARCH (DEPTH-LIMITED-SEARCH (problem, limitproblem, limit) ) returns returns soln/fail/cutoffsoln/fail/cutoff RECURSIVE-DLS(MAKE-NODE(INITIAL-STATE[RECURSIVE-DLS(MAKE-NODE(INITIAL-STATE[problemproblem], ], problemproblem, , limitlimit))

function function RECURSIVE-DLS (RECURSIVE-DLS (node, problem, limitnode, problem, limit) ) returns returns soln/fail/cutoffsoln/fail/cutoff cutoff-occurred? cutoff-occurred? falsefalse ifif GOAL-TEST[ GOAL-TEST[problemproblem](STATE[](STATE[nodenode]) ]) then return then return nodenode else if else if DEPTH[DEPTH[nodenode] = ] = limitlimit then return then return cutoffcutoff else for each else for each successor successor in in EXPAND(EXPAND(nodenode, , problemproblem) ) dodo resultresult RECURSIVE-DLS( RECURSIVE-DLS(successor, problem, limitsuccessor, problem, limit)) if if resultresult = = cutoffcutoff then then cutoff-occurred? cutoff-occurred? truetrue else if else if resultresult ≠ ≠ failurefailure then return then return resultresult if if cutoff-occurred? tcutoff-occurred? then returnhen return cutoffcutoff else return else return failurefailure

Depth-first search with depth limit lDepth-first search with depth limit l• i.e., nodes at depth i.e., nodes at depth ll have no successors have no successors

function function DEPTH-LIMITED-SEARCH (DEPTH-LIMITED-SEARCH (problem, limitproblem, limit) ) returns returns soln/fail/cutoffsoln/fail/cutoff RECURSIVE-DLS(MAKE-NODE(INITIAL-STATE[RECURSIVE-DLS(MAKE-NODE(INITIAL-STATE[problemproblem], ], problemproblem, , limitlimit))

function function RECURSIVE-DLS (RECURSIVE-DLS (node, problem, limitnode, problem, limit) ) returns returns soln/fail/cutoffsoln/fail/cutoff cutoff-occurred? cutoff-occurred? falsefalse ifif GOAL-TEST[ GOAL-TEST[problemproblem](STATE[](STATE[nodenode]) ]) then return then return nodenode else if else if DEPTH[DEPTH[nodenode] = ] = limitlimit then return then return cutoffcutoff else for each else for each successor successor in in EXPAND(EXPAND(nodenode, , problemproblem) ) dodo resultresult RECURSIVE-DLS( RECURSIVE-DLS(successor, problem, limitsuccessor, problem, limit)) if if resultresult = = cutoffcutoff then then cutoff-occurred? cutoff-occurred? truetrue else if else if resultresult ≠ ≠ failurefailure then return then return resultresult if if cutoff-occurred? tcutoff-occurred? then returnhen return cutoffcutoff else return else return failurefailure

Depth-limited search


Time??Time??

Space??Space??

Optimal??Optimal??


Time??Time??

Space??Space??

Optimal??Optimal??

Iterative deepening searchfunction function ITERATIVE-DEEPENING-SEARCH(ITERATIVE-DEEPENING-SEARCH(problemproblem) ) returnsreturns a solution a solution

inputsinputs: : problemproblem, a problem, a problem

for for depthdepth 0 0 toto ∞ ∞ dodo

resultresult DEPTH-LIMITED-SEARCH( DEPTH-LIMITED-SEARCH(problemproblem, , depthdepth))

ifif result result ≠ ≠ cutoffcutoff then returnthen return resultresult

endend

function function ITERATIVE-DEEPENING-SEARCH(ITERATIVE-DEEPENING-SEARCH(problemproblem) ) returnsreturns a solution a solution

inputsinputs: : problemproblem, a problem, a problem

for for depthdepth 0 0 toto ∞ ∞ dodo

resultresult DEPTH-LIMITED-SEARCH( DEPTH-LIMITED-SEARCH(problemproblem, , depthdepth))

ifif result result ≠ ≠ cutoffcutoff then returnthen return resultresult

endend

Properties of iterative deepening


Time??Time??

Space??Space??

Optimal??Optimal??


Time??Time??

Space??Space??

Optimal??Optimal??

Properties of iterative deepening

Complete??Complete??• YesYes

Time??Time??• (d+1)b(d+1)b00 + db + db11 + (d-1)b + (d-1)b22 + … b + … bdd = O(b = O(bdd))

Space??Space??• O(bd)O(bd)

Optimal??Optimal??• If step cost = 1If step cost = 1

• Can be modified to explore uniform-cost treeCan be modified to explore uniform-cost tree

Complete??Complete??• YesYes

Time??Time??• (d+1)b(d+1)b00 + db + db11 + (d-1)b + (d-1)b22 + … b + … bdd = O(b = O(bdd))

Space??Space??• O(bd)O(bd)

Optimal??Optimal??• If step cost = 1If step cost = 1

• Can be modified to explore uniform-cost treeCan be modified to explore uniform-cost tree

SummaryAll tree searching techniques are more alike than differentAll tree searching techniques are more alike than different

Breadth-first has space issues, and possibly optimality issuesBreadth-first has space issues, and possibly optimality issues

Uniform-cost has space issuesUniform-cost has space issues

Depth-first has time and optimality issues, and possibly completeness Depth-first has time and optimality issues, and possibly completeness issuesissues

Depth-limited search has optimality and completeness issuesDepth-limited search has optimality and completeness issues

Iterative deepening is the best uninformed search we have exploredIterative deepening is the best uninformed search we have explored

Next class we study informed searchesNext class we study informed searches

All tree searching techniques are more alike than differentAll tree searching techniques are more alike than different

Breadth-first has space issues, and possibly optimality issuesBreadth-first has space issues, and possibly optimality issues

Uniform-cost has space issuesUniform-cost has space issues

Depth-first has time and optimality issues, and possibly completeness Depth-first has time and optimality issues, and possibly completeness issuesissues

Depth-limited search has optimality and completeness issuesDepth-limited search has optimality and completeness issues

Iterative deepening is the best uninformed search we have exploredIterative deepening is the best uninformed search we have explored

Next class we study informed searchesNext class we study informed searches

itcs 3153 artificial intelligence

Documents

state pairse

problem solvingabstract

vacuum worldsinglestate

complex state space

current world state

abstract solution

cititesfind solution

original problem