Informed and Uninformed Search

Summary of Last Lecture• Practical Motivation– Voice recognition (i.e. checking flight status)– Written letter recognition (i.e. zip codes on letters)– Planning (i.e. a robot path planning)– Medical diagnosis systems• Academic Motivation– Game playing (i.e. Checkers, Rubik’s cube, Chess)– Robotics, Theorem proving• Designing algorithms to accomplish the above, algorithmswill be rational.– (i.e. minimize # moves, maximize expected # of wins)• In defense of my simple examples through the course,assignments will be more pragmatic.CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search1Chapter 2 – Russell and Norvig(not on exam)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search21

Environment Properties - 3• Static versus Dynamic– Semi-dynamic• Discrete versus continuous– Applies to actions, states• Single agent versus multiple agents– Co-operative and competitive agentsCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search5Name the Environment Properties• Chess• Monopoly• Robocupsimulation league• Email crawlerCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search63

Goal Oriented Search – Ex:1CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search9Four Step Process• Goal formulation, problem formulation,search, execute• What does this tell you about theenvironment?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search105

Problem TypesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search11Single State Problem DefinitionCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search126

State Space DefinitionCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search13Eight State PuzzleCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search147

Eight State PuzzleCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search15Converting the Problem to a GraphCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search168

What is a State and a Node?Why? ThinkAbout GoalStatesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search17Search Strategies and EffectivenessCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search189

Basic Search AlgorithmCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search19Uninformed SearchCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search2010

Breadth First SearchComplete?Time?Space?Optimal?What queue type should the fringe be?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search21Uniform SearchFringe queue order?Complete?Time?Space?Optimal?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search2211

Depth First and Depth LimitedComplete?Time?Space?Optimal?Fringe queue order?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search23Classic Problem #1: Eight QueenProblemState space,Successor func.Start StateTarget Stateb, d, mBest searchtechniqueCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search2412

CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search25Iterative Deepening and BidirectionalSearchComplete?Time?Space?Optimal?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search2613

Hmmm. What’s the Overhead inVisiting Node Multiple TimesInstead of Just Once• Expand nodes just once:– 1+b^2+b^3+b^4+b^5– For b = 10, d = 5, 111,111 expansions• Expand nodes multiple times– d+1 + d(b) + (d-1)b^2 + (d-2)b^3 + (d-3)b^4 +(d-4)b^5 = 123,456CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search27Formalizing the State Space - 1• A state space is a graph, (V, E), where V is a setof nodes and E is a set of arcs, where each arc isdirected from a node to another node• Each node is a data structure, represents aparticular state• Each arc corresponds to an instance of one of theoperators. When the operator is applied to the stateassociated with the arc's source node, then theresulting state is the state associated with the arc'sdestination node• Each arc has a fixed, positive cost associated withit corresponding to the cost of the operator.CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search2814

Formalizing the State Space - 2• Each node has a set of successor nodes = all legal operators.• One or more nodes are designated as start nodes• A goal test predicate is applied to a state to determine if itsassociated node is a goal node• A solution is a sequence of operators that is associated with a pathin a state space from a start node to a goal node• The cost of a solution is the sum of the arc costs on the solutionpath• State-space search is searching through a state space for a solutionby making explicit a sufficient portion of an implicit state-spacegraph to include a goal node. Hence, initially V={S}, where S isthe start node; when S is expanded and so forth until …• Each node implicitly or explicitly represents a partial solutionpath (and cost of the partial solution path) from the start nodeCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search29Basic Search AlgorithmCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search3015

Uninformed SearchCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search31Simple Example - 1Non-monotonic decreaseMesses up BFCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search3216

Simple Example - 2Depth first LIFO queueUniform cost ordered queueBreadth first FIFO queueCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search33Classic Problem #1: Eight QueenProblemCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search3417

Two Key Issues in This Module• 1) Representing a goal search problem as atree or graph search• 2) Efficient representation of a graphCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search35Basic Search AlgorithmCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search3618

Simple Example - 2Depth first LIFO queueUniform cost ordered queueBreadth first FIFO queueCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search37Simple Example - 1CompleteOptimalTime ComplexitySpace ComplexityCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search3819

Classic Problem #2CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search39More Classic ProblemsCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search4020

Water Jug SolutionCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search41Uniform Cost SearchCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search4221

Shortest Path Planning• Consider planning a path from a start stateto a goal state with geometric obstacles• Formulate this problem as tree searchCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search43Problem TypesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search4422

Four Step Process• Goal formulation, problem formulation,search, execute• What does this tell you about theenvironment?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search45Three Tile Problem!CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search4623

Three Tile Problem!• Sensor-less problems– How do we formulate as a tree search solvingthe three tile problem … if we don’t know theinitial start state, but we know the goal state!CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search47Three Tile Problem!• Sensor-less problems– How do we formulate as a tree search solvingthe three tile problem … if we don’t know theinitial start state, but we know the goal state!• Partial sensors– How do we formulate as a tree search solvingthe tree tile problem … if we can observepartial information after each step.CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search4824

Motivation for Informed Search• What we have covered so far is effectivelyblind/uninformed search.• At best we have used the cost to get to a particularsolution• We only have a predicate to test for the goal state.• But in many problems, if we know what state weare in, we can also estimate the cost to the goalstate.• A* requires this estimate to be an under-estimatedor admissible heuristic. Common admissibleheuristic???CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search49Homework #1• Formulate this problem as a tree search. List the statespace description, start state, goal state and successorfunction(s).• Using the above, draw the search tree and solve theproblem.• What is the complexity of each search algorithm for thisparticular problem.• What is the best search algorithm for this problem.CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search5025

Next Class• This class uninformed search• Next two classes are informed (heuristic)search (i.e. A*, Chapter 4, R&N)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search51Summary of Last Class• We can convert single state problems(complete observability and deterministicenvironment) into tree searches.CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search5226

More Classic ProblemsCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search53Water Jug SolutionCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search5427

A More Complicated ProblemHow do weDescribe all possibleStates (including thisone) to a machineCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search55A More Complicated ProblemHow do weDescribe all possibleStates (including thisone) to a machineCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search5628

Actions? So You Can Get FromAny State to Any Other State?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search57Actions? So You Can Get FromAny State to Any Other State?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search5829

Actions? So You Can Get FromAny State to Any Other State?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search59Variety of Uninformed SearchTechniquesSearchCompleteTimeSpaceOptimalBFSYesO(b d )O(b d )No*DFSYes*O(b d )O(bd)No*IDSYesO(b d )O(bd)NoUCYesO(b d )O(b d )YesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search6030

Simple Example - 1CompleteOptimalTime ComplexitySpace ComplexityCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search61Motivation for Informed Search• What we have covered so far is effectivelyblind/uninformed search.• At best we have used the cost to get to a particularsolution. (i.e. only the problem definition, nointelligence or background information).• We only have a predicate to test for the goal state.• But in many problems, if we know what state weare in, we can also estimate the cost to the goalstate. Can we use this information to come up witha more efficient algorithm?• A* requires this estimate to be an under-estimatedor admissible heuristic.CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search6231

Informed Search – Chapter 4CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search63Lecture 4 and 5• Chapter 4 – making use of problem details beyondproblem specification• More efficient that uninformed search• Uninformed search approaches can enhance basicA* algorithm• This lecture, basic A* with examples– Robot navigation, Towers of Hanoi, Rubik’s cube• Next lecture, advanced A* and iterativealgorithmsCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search6432

Best First Search• Take the basic tree search algorithm• f(n) evaluation function determines expansion order• f(n)=g(n)+h(n)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search65Simplest Informed Search• f(n) = h(n)– h is some heuristic function that estimates costfrom n to goal, h(n) = 0, if n = goal• Results in a greedy search technique thatfinds local minimaCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search6633

Greedy Algorithm and PropertiesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search67CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search6834

A* Search???A* complete and optimal, proof of optimality is trivial for treesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search69CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search7035

CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search71Example on GraphsCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search7236

Informed Search – Chapter 4CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search73CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search7437

CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search75CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search7638

Up to HereCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search77Trees vs Graphs: AvoidingRepeating StatesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search7839

Use Graph Search InsteadCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search79A* Graph Search InsteadCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search8040

A* Graph Search for Shortest PathProblem – What’s the Trend?OpenA(366)S(393) T(447) Z(449)RV(413) F(415) Z(449) *A(646)* O(671)F(415) P(417) Z(449) C(526) S(553) O(671)P(417) Z(449) B(450) C(526) S(553) S(591)O(671)B(418) Z(449) B(450) C(526) S(553) S(591)RV(607) C(615) O(671)ClosedA(366)A(366) S(393)A(366) S(393) RV(413)A(366) S(393) RV(413) F(415)A(366) S(393) RV(413) F(415) P(417)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search81That is every node with f(n) < f(G*) isexpanded. Why?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search8241

A Star as BFS with f-ContoursCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search83Robot NavigationAdmissible Heuristic?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search8442

Greedy Search?Robot Navigationf(N) = h(N), with h(N) = Manhattan distance to the goal8 76543 2 34 5 6754356 3 210 1 24765876 5 4 3 2 3 4 56CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search85Robot Navigationf(N) = h(N), with h(N) = Manhattan distance to the goal8 76543 2 34 5 6754356 3 210 1 24765876 5 4 3 2 3 4 56CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search8643

Robot Navigationf(N) = g(N)+h(N), with h(N) = Manhattan distance to goal8+3 8 7+4 7 6+5 6+3 65+6 54+7 43+8 3 2+9 2 32+9 3+104 5 67+2 75+6 5 4+7 43+8 356+1 6 3 2+9 2 1+10 11+10 0+110 1 27+0 76+1 68+1 87+2 76+3 6 5+4 5 4+5 4 3+6 32+7 2 3+8 3 4 5456CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search87Robot navigationf(N) = g(N) + h(N), with h(N) = straight-line distance from N to goalCost of one horizontal/vertical step = 1Cost of one diagonal step = √2CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search8844

A* - Is Admissible HeuristicsGood Enough• In our examples we found that items on theclosed lists need to be updated as there weremultiple paths to the same node?• Can we overcome this?CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search89Consistent Heuristic• The admissible heuristic h is consistent (orsatisfies the monotone restriction) if for everynode N and every successor N’ of N:h(N) ≤ c(N,N’) + h(N’)• i.e. its always quicker to go• Directly to the goal• (triangular inequality)Nc(N,N’)N’ h(N)h(N’)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search9045

Some Proofs of A* Behavior - 1• In A*, for any node n, n is extracted from OPEN iff f(n) (If part)• Consider any node m in the optimal path in the state searchgraph: f(m) = g(m)+h(m) = g*(m)+h(m)

Some Proofs of A* Behavior - 3• FACT: Any consistent heuristic h isadmissible.• Proof: We need to show that for all n, h(n)

8-Puzzle581 2 34214 5 673N67 8goal• h1(N) = number of misplaced tiles• h2(N) = sum of distances of each tile to goalare both consistentCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search95Claims• If h is consistent, then the function f alongany path is non-decreasing:f(N) = g(N) + h(N)f(N’) = g(N) +c(N,N’) + h(N’)h(N) ≤ c(N,N’) + h(N’)f(N) ≤ f(N’)• If h is consistent, then whenever A* expands anode it has already found an optimal path to thestate associated with this nodeNc(N,N’)N’ h(N)h(N’)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search9648

Avoiding Repeated States in A*If the heuristic h is consistent, then:• Let CLOSED be the list of states associatedwith expanded nodes• When a new node N is generated:– If its state is in CLOSED, then discard N– If it has the same state as another node in thefringe, then discard the node with the largest fCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search97Complexity of Consistent A*• Let s be the size of the state space• Let b be the maximal number of states thatcan be attained in one step from any state• Assume that the time needed to test if astate is in CLOSED is O(1)• The time complexity of A* is O(sblogs)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search9849

A* Search PropertiesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search99Inventing Admissible Heuristics• List constraints• Relax constraints to obtain admissibleheuristics (constraints increase cost)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search10050

Examples• For each, specify the state space, operators,constraints of the problem and admissibleheuristics– Towers of Hanoi– Robot path planning– TSP– Rubik’s CubeCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search101Towers of HanoiCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search10251

A*Consider the optimal path.Every node on this path has what propertyWhat about the start state. Therefore A* has what property wrt nodeexpansion? . If A* is optimal what other property wrt nodeexpansion must hold.CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search103Iterative Deepening A* (IDA*)• Use f(N) = g(N) + h(N) with admissible h• Each iteration is depth-first with cutoff onthe value of f of expanded nodesCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search10452

8-Puzzlef(N) = g(N) + h(N)with h(N) = number of misplaced tiles4Cutoff=46CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search1058-Puzzle4Cutoff=4466CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search10653

8-Puzzle4Cutoff=44566CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search1078-Puzzle54Cutoff=44566CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search10854

8-Puzzle654Cutoff=44566CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search1098-Puzzle4Cutoff=56CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search11055

8-Puzzle4Cutoff=5466CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search1118-Puzzle4Cutoff=54566CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search11256

8-Puzzle4Cutoff=545766CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search1138-Puzzle4Cutoff=5455766CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search11457

8-Puzzle4Cutoff=5455 5766CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search1158-Puzzle4Cutoff=5455 5766CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search11658

Examples• For each, specify the state space, operators,constraints of the problem and admissibleheuristics– Towers of Hanoi– Robot path planning– TSP– Rubik’s CubeCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search117TSP Example42BA48What is the admissibleHeuristic?23D20C40LConsider the constraintsof the problem4229EF2010G42K20H4041M10I3223JCSI 435/535 - Introduction to A.I. : Informedand Uninformed Search11859

CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search119N-Queens (Again)CSI 435/535 - Introduction to A.I. : Informedand Uninformed Search12060