Problem Representation

1. Introduction to Artificial Intelligence Problem Representation Alan M Frisch, Dimitar Kazakov kazakov@cs.york.ac.uk Last edited: 14 Jan 2013

2. Next two weeks:Problem Solving and Search • Use a computer to solve a problem by: • Formulating a formal (symbolic) representation of the problem • Program a computer to use the representation to search for a solution • This lecture covers point 1. • Next two lectures cover point 2. ARIN: Problem Representation

3. Example: Romania • On holiday in Romania; currently in Arad. Flight leaves from Bucharest. • Formulate goal: • be in Bucharest • Formulate problem: • states: various cities • actions: drive between two adjacent cities • Find solution: • sequence of actions that ends at Bucharest, e.g., “Drive from Arad to Sibiu, drive from Sibiu to Fagaras, drive from Fagaras to Bucharest.” ARIN: Problem Representation

4. Example: Romania ARIN: Problem Representation

5. ARIN: Problem Representation

6. Problem Representation or Formulation A problemrepresentation is defined by five items: • set of states • could be infinite • each state is a symbolic structure; it could reside inside a computer • initial state • one of the states • e.g., "at Arad“ • finite set of operators(aka actions) • each is a function from states to states. • may be partial function as some operators may not be applicable to some states. E.g.“drive to Bucharest” is only applicable if in a city adjacent to Bucharest

7. Problem Representation or Formulation 4. goal test • A Boolean function that applies to a state • can be explicit, when we know all possible goal states in advances, e.g., state = "at Bucharest" • can be implicit, specified by a property, e.g., Checkmate(state) 5. path cost • additive, so can be derived from operators’ costs • e.g., sum of distances, number of actions executed, etc. • assumed to be ≥ 0

8. Example of path cost ≠ ∑ all operators? • Snakes and ladders! • (This is a Sufi Muslim board from Cambridge Museum of Anthropology)

9. Solutions Asolution is asequence of operators leading from the initial state to a goal state.

10. Recap: Problem Representation • set of states • initial state • finite set of operators (aka actions) • goal test • path cost

11. Problem Representation (another way) set of statesfinite set of operators (aka actions) • initial state • Transition model: a function Result(s,a) • goal test • path cost

12. Alternative Terminology • Transition model: defined by Result(s,a) • Successor: any state such that ssucc = Oi(s) • Successor fn: produces the set of all successors. • Example: representation has 3 operators: • O1(s)=s1; O2(s)=s2; O3(s) is undefined • Then Successors(s)={s1,s2} • The set of states is fully defined by initial state + transition model

13. State Space A problem representation induces a state space: • A graph in which each state is a node. If operatoro maps state s1 to state s2then the graph has a directed arc from s1 to s2and the arc is labelled o. • Could also label the initial state and goal state(s). • Could also label arcs with operator costs.

14. Example: Romania Consider each undirected arc to be two directed arcs.

15. Vacuum world state space graph • states? • operators? • goal test? • path cost?

16. Vacuum world state space graph • states:2 integers to represent amount of dirt in each room and 1 value to represent robot location • operators:Left, Right, Suck • goal test: no dirt at all locations • path cost:1 per action

17. Example: The 8-puzzle • states: • operators: • goaltest: • pathcost:

18. Example: The 8-puzzle • states: locations of tiles • operators: move blank left, right, up, down (these are partial functions!) • goaltest: goal state (given) • pathcost: one per move • Q: how can you do this with 24 operators? • BTW: optimal solution of n-Puzzle is NP-hard

19. Example: Robotic Assembly • states: real-valued coordinates of robot joints (positions, angles); parts of the object to be assembled • operators: continuous motions of robot joints • goal test: complete assembly • path cost: time to execute

20. Direct Representations • Often a problem itself has a natural notion of configurations and actions. • In such cases we can build a direct represent-ation in which the state and operators correspond directly to the configurations and actions. • Examples: all the examples so far and most in AIMA book.

21. A More General View • Operators in the representation might not correspond directly to actions in the world • Example: an operator can correspond to an action or to the inverse of an action. Consider parsing. • If the world has a natural “direction” then we can speak about forward search and backward search.

22. A More General View: Example Problem: Construct a sequence of 3 actions such that … Direct representation: each action is an operator B A A B A B B A B A A B B B A

23. A More General View: Example An alternative representation: _ _ _ _ B _ _ A _ _ A A _ A B

24. Problems vs. Problem Representations • In direct representations there is a close correspondence between problems and their representations. • But don’t confuse them!!! • Especially because some representations are not direct. • Be clear: States are data structures (they reside inside a computer) and operators map data structures to data structures: NextState=op(State).

25. Search Trees • A problem representation implicitly defines a search tree. • Rootis a node containing the initial state. • If a node contains state s then for each state s’inSuccessors(s) the node has one child node containing state s’. • Each node in the search tree is called a search node. It can contain more than just a state: e.g. the path cost from the initial node. • Search trees are not state spaces. Two nodes in a tree can contain the same state.

26. Search Tree for Romania Problem Representation Observe that this tree contains infinite paths. How could you reformulate the problem to eliminate them?

27. Search Trees • Can contain infinite paths • Because there are an infinite number of states • If state space contains a cycle, search tree contains a path with repeating states • Always finitary:- each node has finite number of children since a problem representation has a finite number of operators.

28. Weak Methods vs. Strong Methods • Weak Methods: • Solve a problem by building a problem representation and searching for a solution. • Doesn’t require a deep understanding of the problem. • This is a general method; can be used whenever a new problem is confronted. • It may be too inefficient to solve difficult problem instances. • Strong Methods: • Analyse a problem and use the resulting understanding to develop an algorithm specialised to the problem. • Can result in powerful algorithms that can solve difficult problem instances.

29. Solving NP-Complete Problems(Assuming P ≠NP) • There is no algorithm thatcan decide what to do next in polynomial time. • Some form of search must be used. • By using a weak method, one can easily get a correct algorithm (e.g., search space of all possible non-deterministic choices.) • For n binary choices, 2n combinations must be considered. • A strong method would still need to search, but could consider fewer combinations: 2c.n where c<1.

30. Example: Speech Segmentation “thatsthewaythecookiecrumbles” (28 chars) • Where do I put the word boundaries to segment this into a list of tokens from a given lexicon? • Problem representation • Search space:naïve vs informed search?

31. Summary • A problem can be solved by building a representation and then using the representation to search for a solution. • A problem representation consists of 5 items: a (possibly infinite) set of states, one of which is identified as the initial state, a goal condition, a finite set of operators, and a cost function. • A problem representation implicitly defines a search space.