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Problem Solving and Search ►Let’S Configure This to Be an AI Problem

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Problem Solving and Search ►Let’S Configure This to Be an AI Problem Let’sLet’s havehave HolidayHoliday inin RomaniaRomania ☺☺ ►On holiday in Romania; currently in Arad. 2 ►Flight leaves tomorrow from Bucharest at 1:00. 2 Problem Solving and Search ►Let’s configure this to be an AI problem. Problem Solving and Search Problem Artificial Intelligence 1 Artificial Intelligence 2 WhatWhat isis thethe Problem?Problem? IsIs therethere more?more? ►Accomplish a goal ►Do it well… according to a performance measure 2 2 • Reach Bucharest by 1:00 – Minimize distance traveled ►So this is a goal-based problem Problem Solving and Search Problem Problem Solving and Search Problem Artificial Intelligence 3 Artificial Intelligence 4 ExamplesExamples ofof aa non-goal-basednon-goal-based problem?problem? WhatWhat QualifiesQualifies asas aa Solution?Solution? ►Live longer and prosper ►You can/cannot reach Bucharest by 1:00 2 2 ►Maximize the happiness of your trip to Romania ►You can reach Bucharest in x hours ►Don’t get hurt too much ►The shortest path to Bucharest passes through these cities ►The sequence of cities in the shortest path from Arad to Bucharest is ________ ►The actions one takes to travel from Arad to Bucharest along the shortest path Problem Solving and Search Problem Problem Solving and Search Problem Artificial Intelligence 5 Artificial Intelligence 6 WhatWhat additionaladditional informationinformation doesdoes oneone need?need? ProblemProblem DefinitionDefinition anandd SolutionSolution ProcessProcess ►A map ►The first task when solving any problem is the precise 2 2 definition of the problem ►Well-defined Problems – A solution exists – It is unique – The solution does not change with initial state ►Formal definition of a problem – Initial state – Description of possible and available Actions – Goal Test which determines whether a given state is the goal Problem Solving and Search Problem Problem Solving and Search Problem – Path Cost assigns a numeric cost to each path Artificial Intelligence 7 Artificial Intelligence 8 AA simplifiedsimplified RoadRoad MapMap AbstractionAbstraction ► A solution to problem is a path from the initial state to goal state ► Proposed formulation of the problem seems reasonable ► Solution quality is measured by the path cost 2 2 ► But in actual cross-country trip, the states of the world includes so cost many things: initial – the traveling companions, state next – what is on the radio, state – how far away it is to the next rest stop, acti on – any traffic law enforcements, – the weather and road conditions, ... ► All these considerations are left out of our state description of the road because they are irrelevant to the problem of finding a route to Problem Solving and Search Problem Problem Solving and Search Problem Bucharest Goal ► The process of removing detail from a representation is called Test abstraction Artificial Intelligence 9 Artificial Intelligence 10 AbstractionAbstraction LevelLevel ExampleExample ProblemsProblems ►Can we more precise about defining the appropriate level ►The problem-solving approach has been applied to a vast 2 2 of abstraction? array of task environments ►The abstraction is valid if we can expand any abstract ►We list some of the best known here solution into a solution in the more detailed world. ►Toy Problems ►The abstraction is useful if carrying out each of the actions in the solution is easier than the original problem – Vacuum world ►The choice of a good abstraction involves removing as – 8-puzzle Problem much detail as possible while retaining validity and – 8-queens Problem ensuring that the abstract actions are easy to carry out. ►Real-world Problems –TSP Problem Solving and Search Problem Problem Solving and Search Problem – VLSI Layout – Robot Navigation Artificial Intelligence 11 Artificial Intelligence 12 VacuumVacuum WorldWorld VacuumVacuum WorldWorld ►A vacuum-cleaner world with just two locations ►States: The cleaner is in one of two locations, each of 2 2 which might or might not contain dirt – Thus there are 2×22=8 possible world states ►Initial State: Any state can be designated as the initial state ►Actions: Left, Right, Suck ►Goal Test: All the squares are clean ►Path cost: Each step costs 1, so the path cost is the number of steps in the path. Problem Solving and Search Problem Problem Solving and Search Problem Artificial Intelligence 13 Artificial Intelligence 14 TheThe StateState SpaceSpace forfor thethe VacuumVacuum WorldWorld 8-Puzzle8-Puzzle ► The 8-puzzle consists of 3×3 board with eight numbered tiles and a blank space 2 2 ► A tile adjacent to the blank space can slide into blank space Problem Solving and Search Problem Problem Solving and Search Problem Initial State Goal Artificial Intelligence 15 Artificial Intelligence 16 8-Puzzle8-Puzzle AbstractionAbstraction RevisitedRevisited ►States: location of each of the eight tiles and the location ►States ►What abstraction have we included here? of the blank space 2 2 – Thus there are 9!/2=181,440 reachable states – we ignore the intermediate locations where the block is ►Initial State: Any state can be designated as the initial sliding state – we abstract away actions such as shaking the board ►Actions: Left, Right, Up, Down when pieces get stuck, or extracting pieces with a knife ►Goal Test: and putting them back again ►we avoid all the details of physical manipulations Problem Solving and Search Problem Problem Solving and Search Problem ►Path cost: Each step costs 1, so the path cost is the number of steps in the path. Artificial Intelligence 17 Artificial Intelligence 18 MoreMore onon 8-Puzzle8-Puzzle ProblemProblem 8-Queens8-Queens ProblemProblem ►The 8-puzzle belongs to the family of sliding-block ►The goal of 8-Queens problem is to place eight queens on 2 2 puzzle, which are often used to test problems for new a chessboard such that no queen attacks any other search algorithms in AI ►This general class is known to be NP-complete – One does not expect to find methods significantly better in the worst case than the search algorithms that we will study in the course ►The 8-puzzle (3×3 board): 181,440 states ►The 15-puzzle (4×4 board): 1,3 trillion states – can be solved in milliseconds by the best search algorithm Problem Solving and Search Problem Problem Solving and Search Problem ►The 24-puzzle (5×5 board): ~1025 states – quite difficult to solve with current machines Artificial Intelligence 19 Artificial Intelligence 20 8-Queens8-Queens ProblemProblem AnotherAnother RepresentationRepresentation ►States: Any arrangement of 0 to 8 queens on the board ►States: Arrangement of n queens, one per column in the 2 2 – Thus there are 64×63...57≈3×1014 possible sequences leftmost n columns, with no queen attacking another ►Initial State: No queens on the board – Thus there are just 2057 possible sequences ►Actions: Add a queen to any empty square ►Initial State: No queens on the board ►Goal Test: 8 queens are on the board, none attacked ►Actions: Add a queen to any square in the leftmost empty column such that it is not attacked by any other queen ►Goal Test: 8 queens are on the board, none attacked Problem Solving and Search Problem Problem Solving and Search Problem Artificial Intelligence 21 Artificial Intelligence 22 Real-worldReal-world ProblemsProblems Real-worldReal-world ProblemsProblems (con’t) (con’t) ►Traveling Salesperson Problem (TSP) ►VLSI Layout Problem 2 2 – a touring problem in which each city must be visited exactly – Cell Layout Problem once • Each cell has a fixed footprint (size and shape) – the aim is to find the shortest tour • certain number of connection to other cells – the problem is known to be NP-Hard • the aim is to place the cells on the chip so that they ►VLSI Layout Problem do not overlap and there is room for the connecting – requires positioning millions of components an connections on a wires to be placed between cells chip to minimize area, minimizing circuit delays, minimize stray capacitances, maximizing manufacturing yield – Channel Routing Problem – comes after logical design • finding a specific route for each wire through the Problem Solving and Search Problem Problem Solving and Search Problem – split into two parts: cell layout and channel routing gaps between the cells Artificial Intelligence 23 Artificial Intelligence 24 CellCell LayoutLayout ProblemProblem ChannelChannel RoutingRouting 2 2 Random Initial Final After Detailed Placement Placement Routing Problem Solving and Search Problem Problem Solving and Search Problem Artificial Intelligence 25 Artificial Intelligence 26 Real-worldReal-world ProblemsProblems (con’t) (con’t) SearchingSearching forfor SolutionsSolutions ►Robot Navigation ►We have formulated some problems, we now need to 2 2 – generalization of the route-finding problem described earlier solve them – rather than a discrete set of routes, a robot can move in a ►This is done by a search through the state space continuous space with an infinite set of possible actions and states ►Here we deal with search techniques that use an explicit search tree ►Search tree is generated by the initial state and actions that together define the state space Problem Solving and Search Problem Problem Solving and Search Problem The 25-pound, six-wheeled robotic explorer Artificial Intelligence 27 Artificial Intelligence 28 SearchSearch NodeNode anandd ExpandingExpanding TreeTree SearchSearch NodeNode andand EExpendingxpending TreeTree (Con’t) (Con’t) ►The root of the search tree is a search node corresponding ►After expanding Arad 2 2 to the initial state ►The first step is to test whether this is a goal
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