Artificial Intelligence

1. Artificial Intelligence Search: 1 Ian Gent ipg@cs.st-and.ac.uk

2. Artificial Intelligence Search: 1 Part I : What is Search? Part II: Presenting search abstractly Part III: Basic search algorithms

3. What is AI? • Very hard to define Artificial Intelligence • not attempt at building machine to pass Turing Test • Perhaps, exploiting power of machines to do tasks normally considered intelligence? • Practical day to day answer is: • AI is what we don’t know how to do yet • Once we know how to do something, it’s not AI • e.g. optical character recognition, speech recognition • Many AI problems involve combinatorial search

4. Example Search Problem: SAT • We need to define problems and solutions • Propositional Satisfiability (SAT) • really a logical problem -- I’ll present as a letters game • Problem is a list of words • contains upper and lower case letters (order unimportant) • e.g. ABC, ABc, AbC, Abc, aBC, abC, abc • Solution is choice of upper/lower case letter • one choice per letter • each word to contain at least one of our choices • e.g. AbC is unique solution to above problem.

5. Why is SAT a search problem? • There is no efficient algorithm known for SAT • all complete algorithms are exponential time • 3-SAT is NP-Complete • 3-SAT = each word contains exactly 3 letters • NP-Complete • we can recognise solutions in polynomial time • easy to check letter choice satisfies each word • all other NP problems can be solved by translation to SAT • Many AI problems fall into NP-Complete class

6. Example: Travelling Salesperson • Problem: graph with an cost for each edge • e.g. • Solution: tour visiting all nodes returning to base • meeting some cost limit (or reaching minimum cost) • e.g. minimum cost is 21 above • TSP is NP-Complete • easy to check that tour costs no more than limit • (finding optimal cost in technically different complexity class) 2 • 4 4 1 2 3 2 5 4

7. Example (Not): Sorting • Problem, a list of numbers • e.g. 5 6 3 2 4 8 • Solution, list in ascending order • e.g. 2 3 4 5 6 8 • In NP (easy to check that result in ascending order) • Not NP-complete • cannot solve SAT via sorting • can be solved in O(n log n) time • We know how to do it efficiently, so it’s not AI

8. Final Example: Games • Problem: a position in a Chess/Go/… game • Solution: a strategy to guarantee winning game • Harder than NP problems • it is not easy to check that a strategy wins • can solve SAT via games • Technically, games usually PSPACE-complete • All NP-complete problems in PSPACE • Games are valid AI application • AI usually attacks NP-complete or harder search problems

9. Presenting Search Abstractly • Helps to understand the abstract nature of search • search states, search spaces, search trees… • know what particular search algorithms are trying to do • There are two kinds of search algorithm • Complete • guaranteed to find solution or prove there is none • Incomplete • may not find a solution even when it exists • often more efficient (or there would be no point) • e.g. Genetic Algorithms • For now concerned with complete algorithms

10. Search States • Search states summarises the state of search • A solution tells us everything we need to know • e.g. in SAT, whether each letter is UPPER or lower case • in TSP, route taken round nodes of graph • This is a (special) example of a search state • it contains complete information • it solves the problem • In general a search state may not do either of those • it may not specify everything about a possible solution • it may not solve the problem or extend to a solution

11. Search States • Search states summarise the state of search • E.g. in SAT • a search state might be represented by aB • E.g. in TSP • a search state might specify some of the order of visits • E.g. in Chess • a search state might be represented by the board position • (quiz for chessplayers: …and what else?)

12. Generalising Search • With search states we can generalise search • not just finding a solution to a problem • Generally, find a solution which extends search state • e.g. find solution to ABC, ABc, AbC, Abc, aBC, abC, abc which extends aB • there is no such solution though whole problem solvable • Original search problem is to extend null state • Search in AI by structured exploration of search states

13. Search Space and Search Trees • Search space is logical space composed of • nodes are search states • links are all legal connections between search states • e.g. in chess, no link between states where W castles having previously moved K. • always just an abstraction • think of search algorithms trying to navigate this extremely complex space

14. Search Trees • Search trees do not summarise all possible searches • instead an abstraction of one possible search • Root is null state • edges represent one choice • e.g. to set value of A first • child nodes represent extensions • children give all possible choices • leaf nodes are solutions/failures • Example in SAT • algorithm detects failure early • need not pick same variables everywhere

15. Why are search trees abstract? • Search trees are very useful concept • but as an abstraction • Search algorithms do not store whole search trees • that would need exponential space • we can discard nodes in search tree already explored • Search algorithms store frontier of search • I.e. nodes in search tree with some unexplored children • Very many search algorithms understandable in terms of search trees • and specifically how they explore the frontier

16. Some classic search algorithms • Depth-first search • I.e. explore all nodes in subtree of current node before any other nodes • pick leftmost and deepest element of frontier • Breadth-first search • explore all nodes at one height in tree before any other nodes • pick shallowest and leftmost element of frontier • Best-first search • pick whichever element of frontier seems most promising

17. More classic search algorithms • Depth-bounded depth first search • like depth first but set limit on depth of search in tree • Iterative Deepening search • use depth-bounded search but iteratively increase limit

18. Next week on Search in AI • Presentation of search algorithms in terms of lists • e.g. depth-first = stack, breadth-first = queue • Heuristics in search • how to pick which variable to set