Roli, Andrea

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Almae Matris Studiorum Campus

Solving problems by searchingUninformed searchSlides from Russell & Norvig book, revised by Andrea RoliOutline♦ Problem-solving agents♦ Problem types♦ Problem formulation♦ Example problems♦ Basic search algorithmsPrologue• Is there a general strategy for solving problems such as‘Wolf, goat and cabbage’, ‘Cryptoarithmetic’, ‘8-puzzle’,etc.?• What are the entities that have to be formalized?• Is it possible to design a machine that can solve theseproblems?• What are the assumptions on the (real) world that we haveto formulate?Example: RomaniaOn holiday in Romania; currently in Arad. Flight leavestomorrow from Bucharest. Suppose we do not have a map, butwe only know which cities we can reach from the city we are in.From Arad: Zerind, Sibiu, TimisoaraExample: RomaniaOn holiday in Romania; currently in Arad. Flight leavestomorrow from Bucharest.Formulate goal:be in BucharestFormulate problem:states: various citiesactions: drive between citiesFind solution:sequence of cities, e.g., Arad, Sibiu, Fagaras, BucharestExample: RomaniaGiurgiuUrziceniHirsovaEforieNeamtOradeaZerindAradTimisoaraLugojMehadiaDobretaCraiovaSibiu FagarasPitestiVasluiIasiRimnicu VilceaBucharest71751181117075120151140998097101211138146 859098142928786Problem-solving agentsRestricted form of general agent:function SIMPLE-PROBLEM-SOLVING-AGENT( percept) returns anactionstatic: seq, an action sequence, initially emptystate, some description of the current world stategoal, a goal, initially nullproblem, a problem formulationstate←UPDATE-STATE(state, percept)if seq is empty thengoal← FORMULATE-GOAL(state)problem← FORMULATE-PROBLEM(state, goal)seq←SEARCH( problem)action←RECOMMENDATION(seq, state)seq←REMAINDER(seq, state)return actionNote: this is offline problem solving; solution executed “eyes closed.” Online problemsolving involves acting without complete knowledge.Problem types• Deterministic, fully observable =⇒ single-state problemAgent knows exactly which state it will be in; solution is asequence• Non-observable =⇒ sensorless problemAgent may have no idea where it is; solution (if any) is asequence• Nondeterministic and/or partially observable =⇒contingency problempercepts provide new information about current statesolution is a contingent plan or a policyoften interleave search, execution• Unknown state space =⇒ exploration problem (“online”)Example: vacuum worldSingle-state, start in #5. Solution??[Right , Suck ]Sensorless, start in{1, 2, 3, 4, 5, 6, 7, 8}e.g., Right goes to {2, 4, 6, 8}.Solution??[Right , Suck , Left , Suck ]Contingency, start in #5Murphy’s Law: Suck can dirty a cleancarpetLocal sensing: dirt, location only.Solution??[Right , loop{if dirt then Suck}]1 23 45 67 8Example: vacuum worldSingle-state, start in #5. Solution??[Right , Suck ]Sensorless, start in{1, 2, 3, 4, 5, 6, 7, 8}e.g., Right goes to {2, 4, 6, 8}.Solution??[Right , Suck , Left , Suck ]Contingency, start in #5Murphy’s Law: Suck can dirty a cleancarpetLocal sensing: dirt, location only.Solution??[Right , loop{if dirt then Suck}]1 23 45 67 8Example: vacuum worldSingle-state, start in #5. Solution??[Right , Suck ]Sensorless, start in{1, 2, 3, 4, 5, 6, 7, 8}e.g., Right goes to {2, 4, 6, 8}.Solution??[Right , Suck , Left , Suck ]Contingency, start in #5Murphy’s Law: Suck can dirty a cleancarpetLocal sensing: dirt, location only.Solution??[Right , loop{if dirt then Suck}]1 23 45 67 8Example: vacuum worldSingle-state, start in #5. Solution??[Right , Suck ]Sensorless, start in{1, 2, 3, 4, 5, 6, 7, 8}e.g., Right goes to {2, 4, 6, 8}.Solution??[Right , Suck , Left , Suck ]Contingency, start in #5Murphy’s Law: Suck can dirty a cleancarpetLocal sensing: dirt, location only.Solution??[Right , loop{if dirt then Suck}]1 23 45 67 8Example: vacuum worldSingle-state, start in #5. Solution??[Right , Suck ]Sensorless, start in{1, 2, 3, 4, 5, 6, 7, 8}e.g., Right goes to {2, 4, 6, 8}.Solution??[Right , Suck , Left , Suck ]Contingency, start in #5Murphy’s Law: Suck can dirty a cleancarpetLocal sensing: dirt, location only.Solution??[Right , loop{if dirt then Suck}]1 23 45 67 8Example: vacuum worldSingle-state, start in #5. Solution??[Right , Suck ]Sensorless, start in{1, 2, 3, 4, 5, 6, 7, 8}e.g., Right goes to {2, 4, 6, 8}.Solution??[Right , Suck , Left , Suck ]Contingency, start in #5Murphy’s Law: Suck can dirty a cleancarpetLocal sensing: dirt, location only.Solution??[Right , loop{if dirt then Suck}]1 23 45 67 8Single-state problem formulationA problem is defined by four items:• initial state e.g., “at Arad”• successor function S(x) = set of action–state pairse.g., S(Arad) = {〈Arad → Zerind , Zerind〉, . . .}• goal test, can beexplicit, e.g., x = “at Bucharest”implicit, e.g., NoDirt(x)• path cost (additive)e.g., sum of distances, number of actions executed, etc.c(x , a, y) is the step cost, assumed to be ≥ 0A solution is a sequence of actions leading from the initial stateto a goal stateSelecting a state space• Real world is absurdly complex ⇒ state space must beabstracted for problem solving• (Abstract) state = set of real states• (Abstract) action = complex combination of real actionse.g., “Arad → Zerind” represents a complex setof possible routes, detours, rest stops, etc.• For guaranteed realizability, any real state “in Arad” mustget to some real state “in Zerind”• (Abstract) solution = set of real paths that are solutions inthe real world• Each abstract action should be “easier” than the originalproblem!Example: vacuum world state spacegraphRLS SS SRLRLRLSSSSLLLL RRRRstates??actions??goal test??path cost??Example: vacuum world state spacegraphRLS SS SRLRLRLSSSSLLLL RRRRstates??: integer dirt and robot locations (ignore dirt amountsetc.)actions??goal test??path cost??Example: vacuum world state spacegraphRLS SS SRLRLRLSSSSLLLL RRRRstates??: integer dirt and robot locations (ignore dirt amountsetc.)actions??: Left , Right , Suck , NoOpgoal test??path cost??Example: vacuum world state spacegraphRLS SS SRLRLRLSSSSLLLL RRRRstates??: integer dirt and robot locations (ignore dirt amountsetc.)actions??: Left , Right , Suck , NoOpgoal test??: no dirtpath cost??Example: vacuum world state spacegraphRLS SS SRLRLRLSSSSLLLL RRRRstates??: integer dirt and robot locations (ignore dirt amountsetc.)actions??: Left , Right , Suck , NoOpgoal test??: no dirtpath cost??: 1 per action (0 for NoOp)Example: The 8-puzzle2Start State Goal State51 34 67 8512346785states??actions??goal test??path cost??Example: The 8-puzzle2Start State Goal State51 34 67 8512346785states??: integer locations of tiles (ignore intermediatepositions)actions??goal test??path cost??Example: The 8-puzzle2Start State Goal State51 34 67 8512346785states??: integer locations of tiles (ignore intermediatepositions)actions??: move blank left, right, up, down (ignore unjammingetc.)goal test??path cost??Example: The 8-puzzle2Start State Goal State51 34 67 8512346785states??: integer locations of tiles (ignore intermediatepositions)actions??: move blank left, right, up, down (ignore unjammingetc.)goal test??: = goal state (given)path cost??Example: The 8-puzzle2Start State Goal State51 34 67 8512346785states??: integer locations of tiles (ignore intermediatepositions)actions??: move blank left, right, up, down (ignore unjammingetc.)goal test??: = goal state (given)path cost??: 1 per move[Note: optimal solution of n-Puzzle family is NP-hard]Example: robotic assemblyRRRPR Rstates??: real-valued coordinates of robot joint anglesparts of the object to be assembledactions??: continuous motions of robot jointsgoal test??: complete assembly with no robot included!path cost??: time to executeOther famous problems• Missionaires and cannibals problem• Hanoi tower• Monkey and banana problem• Puzzles and logical gamesTree search algorithmsBasic idea:offline, simulated exploration of state spaceby generating successors of already-explored states(a.k.a. expanding states)function TREE-SEARCH( problem, strategy) returns a solution, orfailureinitialize the search tree using the initial state of problemloop doif there are no candidates for expansion then return failurechoose a leaf node for expansion according to strategyif the node contains a goal state then return the correspondingsolutionelse expand the node and add the resulting nodes to thesearch treeendTree search exampleRimnicu Vilcea LugojZerindSibiuArad Fagaras OradeaTimisoaraAradArad OradeaAradTree search exampleRimnicu Vilcea LugojArad Fagaras Oradea AradArad OradeaZerindAradSibiu TimisoaraTree search exampleLugoj AradArad OradeaRimnicu VilceaZerindAradSibiuArad Fagaras OradeaTimisoaraImplementation: states vs. nodesA state is a (representation of) a physical configurationA node is a data structure constituting part of a search treeincludes parent, children, depth, path cost g(x)States do not have parents, children, depth, or path cost!12345678State Node depth = 6g = 6stateparent, actionThe EXPAND function creates new nodes, filling in the variousfields and using the SUCCESSORFN of the problem to createthe corresponding states.Implementation: general tree searchfunction TREE-SEARCH( problem, fringe) returns a solution, or failurefringe← INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringe)loop doif fringe is empty then return failurenode←REMOVE-FRONT(fringe)if GOAL-TEST(problem, STATE(node)) then return nodefringe← INSERTALL(EXPAND(node, problem), fringe)function EXPAND( node, problem) returns a set of nodessuccessors← the empty setfor each action, result in SUCCESSOR-FN(problem, STATE[node]) dos← a new NODE;PARENT-NODE[s]← node;ACTION[s]← action;STATE[s]← result;PATH-COST[s]← PATH-COST[node] + STEP-COST(STATE[node], action,result)DEPTH[s]←DEPTH[node] + 1add s to successorsreturn successorsSearch strategies• A strategy is defined by picking the order of nodeexpansion• Strategies are evaluated along the following dimensions:completeness—does it always find a solution if oneexists?time complexity—number of nodes generated/expandedspace complexity—maximum number of nodes inmemoryoptimality—does it always find a least-cost solution?• Time and space complexity are measured in terms ofb—maximum branching factor of the search treed—depth of the least-cost solutionm—maximum depth of the state space (may be ∞)Uninformed search strategiesUninformed strategies use only the information availablein the problem definition• Breadth-first search• Uniform-cost search• Depth-first search• Depth-limited search• Iterative deepening searchBreadth-first searchExpand shallowest unexpanded nodeImplementation:fringe is a FIFO queue, i.e., new successors go at endAB CD E F GBreadth-first searchExpand shallowest unexpanded nodeImplementation:fringe is a FIFO queue, i.e., new successors go at endAB CD E F GBreadth-first searchExpand shallowest unexpanded nodeImplementation:fringe is a FIFO queue, i.e., new successors go at endAB CD E F GBreadth-first searchExpand shallowest unexpanded nodeImplementation:fringe is a FIFO queue, i.e., new successors go at endAB CD E F GProperties of breadth-first search• Complete: Yes (if b is finite)• Time: 1 + b + b2 + b3 + . . . + bd + b(bd − 1) = O(bd+1),i.e., exp. in d• Space: O(bd+1) (keeps every node in memory)• Optimal: Yes (if cost is a nondecreasing function of nodedepth)Space is the big problem; can easily generate nodes at100MB/sec, so 24hrs = 8640GB.Uniform-cost searchExpand least-cost unexpanded nodeImplementation:fringe = queue ordered by path cost, lowest first• Complete and optimal• Equivalent to breadth-first if step costs all equalDepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N ODepth-first searchExpand deepest unexpanded nodeImplementation:fringe = LIFO queue, i.e., put successors at frontAB CD E F GH I J K L M N OProperties of depth-first search• Complete: No: fails in infinite-depth spaces, spaces withloopsModify to avoid repeated states along path⇒ complete in finite spaces• Time: O(bm): terrible if m is much larger than dbut if solutions are dense, may be much faster thanbreadth-first• Space: O(bm), i.e., linear space!• Optimal: NoChronological backtracking• Variant of DFS• Successors generated one at a time• Reduced space complexity wrt DFS: O(b)Depth-limited search• depth-first search with depth limit l , i.e., nodes at depth lhave no successors• Not complete if l < d.Iterative deepening searchfunction ITERATIVE-DEEPENING-SEARCH( problem) returns a solu-tioninputs: problem, a problemfor depth← 0 to∞ doresult←DEPTH-LIMITED-SEARCH( problem, depth)if result 6= cutoff then return resultendIterative deepening search l = 0Limit = 0 A AIterative deepening search l = 1Limit = 1 AB CAB CAB CAB CIterative deepening search l = 2Limit = 2 AB CD E F GAB CD E F GAB CD E F GAB CD E F GAB CD E F GAB CD E F GAB CD E F GAB CD E F GIterative deepening search l = 3Limit = 3AB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH I J K L M N OAB CD E F GH J K L M N OIAB CD E F GH I J K L M N OProperties of iterative deepeningsearch• Complete: Yes• Time: (d + 1)b0 + db1 + (d − 1)b2 + . . . + bd = O(bd)• Space: O(bd)• Optimal: Yes, if step cost is a nondecreasing function ofnode depth.◮ IDS preferred uninformed strategy when search space islarge and solution depth not known.Bidirectional search• Run two simultaneous searches• one forward from the initial state• and the other backward from the goal• stop when they meetProblems:• How to compute predecessors?• Sometimes the goal state is only implicitly definedRepeated statesFailure to detect repeated states can turn a linear problem intoan exponential one!ABCDABBCCCCGraph searchfunction GRAPH-SEARCH( problem, fringe) returns a solution, orfailureclosed← an empty setfringe← INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringe)loop doif fringe is empty then return failurenode←REMOVE-FRONT(fringe)if GOAL-TEST(problem, STATE[node]) then return nodeif STATE[node] is not in closed thenadd STATE[node] to closedfringe← INSERTALL(EXPAND(node, problem), fringe)endSummary• Problem formulation usually requires abstracting awayreal-world details to define a state space that can feasiblybe explored• Variety of uninformed search strategies• Iterative deepening search uses only linear space and notmuch more time than other uninformed algorithms• Graph search can be exponentially more efficient than treesearch

03a - Uninformed search

http://campus.unibo.it/2839/1/03a-uninformed_search.1x.pdf

03a - Uninformed search

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