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## Outline: Problem solving and search

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**Outline: Problem solving and search**• Introduction to Problem Solving • Complexity • Uninformed search • Problem formulation • Search strategies: depth-first, breadth-first • Informed search • Search strategies: best-first, A* • Heuristic functions SE 420**9 l**5 l 3 l Example: Measuring problem! Problem: Using these three buckets, measure 7 liters of water. SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • (one possible) Solution: a b c 0 0 0 start 3 0 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • Another Solution: a b c 0 0 0 start 0 5 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • Another Solution: a b c 0 0 0 start 0 5 0 3 2 0 0 0 3 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • Another Solution: a b c 0 0 0 start 0 5 0 3 2 0 3 0 2 3 0 3 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • Another Solution: a b c 0 0 0 start 0 5 0 3 2 0 3 0 2 3 5 2 0 0 6 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**9 l**5 l 3 l Example: Measuring problem! • Another Solution: a b c 0 0 0 start 0 5 0 3 2 0 3 0 2 3 5 2 3 0 7goal 3 0 6 0 3 6 3 3 6 1 5 6 0 5 7 goal a b c SE 420**Which solution do we prefer?**• Solution 1: • a b c • 0 0 0 start • 3 0 0 • 0 0 3 • 3 0 3 • 0 0 6 • 3 0 6 • 0 3 6 • 3 3 6 • 1 5 6 • 0 5 7 goal • Solution 2: • a b c • 0 0 0 start • 0 5 0 • 3 2 0 • 3 0 2 • 3 5 2 • 3 0 7goal SE 420**action**Problem-Solving Agent // What is the current state? // From LA to San Diego (given curr. state) // e.g., Gas usage // If fails to reach goal, update Note: This is offline problem-solving. Online problem-solving involves acting w/o complete knowledge of the problem and environment SE 420**Example: Buckets**Measure 7 liters of water using a 3-liter, a 5-liter, and a 9-liter buckets. • Formulate goal: Have 7 liters of water in 9-liter bucket • Formulate problem: • States: amount of water in the buckets • Operators: Fill bucket from source, empty bucket • Find solution: sequence of operators that bring you from current state to the goal state SE 420**Remember (lntelligent Agents): Environment types**The environment types largely determine the agent design. SE 420**Problem types**• Single-state problem: deterministic, accessible Agent knows everything about world, thus can calculate optimal action sequence to reach goal state. • Multiple-state problem: deterministic, inaccessible Agent must reason about sequences of actions and states assumed while working towards goal state. • Contingency problem: nondeterministic, inaccessible • Must use sensors during execution • Solution is a tree or policy • Often interleave search and execution • Exploration problem: unknown state space Discover and learn about environment while taking actions. SE 420**Problem types**• Single-state problem: deterministic, accessible • Agent knows everything about world (the exact state), • Can calculate optimal action sequence to reach goal state. • E.g., playing chess. Any action will result in an exact state SE 420**Problem types**• Multiple-state problem: deterministic, inaccessible • Agent does not know the exact state (could be in any of the possible states) • May not have sensor at all • Assume states while working towards goal state. • E.g., walking in a dark room • If you are at the door, going straight will lead you to the kitchen • If you are at the kitchen, turning left leads you to the bedroom • … SE 420**Problem types**• Contingency problem: nondeterministic, inaccessible • Must use sensors during execution • Solution is a tree or policy • Often interleave search and execution • E.g., a new skater in an arena • Sliding problem. • Many skaters around SE 420**Problem types**• Exploration problem: unknown state space Discover and learn about environment while taking actions. • E.g., Maze SE 420**Example: Romania**• In Romania, on vacation. Currently in Arad. • Flight leaves tomorrow from Bucharest. • Formulate goal: • be in Bucharest • Formulate problem: • states: various cities • operators: drive between cities • Find solution: • sequence of cities, such that total driving distance is minimized. SE 420**Problem formulation**SE 420**Selecting a state space**• Real world is absurdly complex; some abstraction is necessary to allow us to reason on it… • Selecting the correct abstraction and resulting state space is a difficult problem! • Abstract states real-world states • Abstract operators sequences or real-world actions (e.g., going from city i to city j costs Lij actually drive from city i to j) • Abstract solution set of real actions to take in the real world such as to solve problem SE 420**Example: 8-puzzle**• State: • Operators: • Goal test: • Path cost: start state goal state SE 420**Example: 8-puzzle**• State: integer location of tiles (ignore intermediate locations) • Operators: moving blank left, right, up, down (ignore jamming) • Goal test: does state match goal state? • Path cost: 1 per move start state goal state SE 420**Example: 8-puzzle**Why search algorithms? • 8-puzzle has 362,800 states • 15-puzzle has 10^12 states • 24-puzzle has 10^25 states So, we need a principled way to look for a solution in these huge search spaces… start state goal state SE 420**Back to Vacuum World**SE 420**Back to Vacuum World**SE 420**Example: Robotic Assembly**SE 420**Real-life example: VLSI Layout**• Given schematic diagram comprising components (chips, resistors, capacitors, etc) and interconnections (wires), find optimal way to place components on a printed circuit board, under the constraint that only a small number of wire layers are available (and wires on a given layer cannot cross!) • “optimal way”?? • minimize surface area • minimize number of signal layers • minimize number of vias (connections from one layer to another) • minimize length of some signal lines (e.g., clock line) • distribute heat throughout board • etc. SE 420**Enter schematics;**do not worry about placement & wire crossing SE 420**Use automated tools to place components**and route wiring. SE 420**Search algorithms**Function General-Search(problem, strategy) returns a solution, or failure initialize the search tree using the initial state problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solution else expand the node and add resulting nodes to the search tree end Basic idea: offline, systematic exploration of simulated state-space by generating successors of explored states (expanding) SE 420**Finding a solution**Function General-Search(problem, strategy) returns a solution, or failure initialize the search tree using the initial state problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solution else expand the node and add resulting nodes to the search tree end Solution: is a sequence of operators that bring you from current state to the goal state Basic idea: offline, systematic exploration of simulated state-space by generating successors of explored states (expanding) Strategy: The search strategy is determined by the order in which the nodes are expanded. SE 420**4**4 3 A B C S 5 5 G S 4 3 4 D E F 3 2 A D 4 4 2 5 5 B D A E 4 5 5 2 4 4 C E E B B F 2 4 5 4 4 5 4 4 3 D F B F C E A C G 3 4 3 4 G C G F 3 G From problem space to search tree • Some material in this and following slides is from http://www.cs.kuleuven.ac.be/~dannyd/FAI/ check it out! Problem space Associated loop-free search tree SE 420**Denotes:**SA S A Denotes:SDA D B D A E C E E B B F D F B F C E A C G Denotes: SDEBA G C G F G Paths in search trees SE 420**General search example**SE 420**General search example**SE 420**General search example**SE 420**General search example**SE 420