Problems in Artificial Intelligence

Transcription

1 Problems in Artificial Intelligence Constraint Satisfaction Problems Florent Madelaine Office in the first floor of CS building Florent Madelaine Problems in AI Durham University p.1/45

2 Overview Yesterday: Adversarial search. Today: CSP. Definition. Complexity. Backtracking Search. Forward Checking. Consistency. Decomposition Methods. Florent Madelaine Problems in AI Durham University p.2/45

3 Definition Florent Madelaine Problems in AI Durham University p.3/45

4 Decision Problem Input: a domain D, a set of variables V a set C C 1 C m x 1 x2 C2 x n of constraints ; and, Question: does there exists an assignment T : V D such that all constraints are satisfied simultaneously? Florent Madelaine Problems in AI Durham University p.4/45

5 Constraint and Satisfaction A constraint C is a pair, where, R D k is the constraint relation ; and, s x i1 xi2 x ik R s is the constraint scope. An assignment T satisfies C if, and only if, T s T x i1 T x i2 T x ik R That is, the constraint relation specifies the allowable combinations of values. Florent Madelaine Problems in AI Durham University p.5/45

6 Example: map colouring Western Australia Northern Territory South Australia Queensland New South Wales Victoria V D WA red NT Q NSW V SA green blue constraints: adjacent regions must have different colours T Tasmania e.g., WA NT red green red blue green red Florent Madelaine Problems in AI Durham University p.6/45

7 Map Colouring: solution Western Australia Northern Territory Queensland Solutions are assignments satisfying all constraints; e.g., South Australia New South Wales Victoria Tasmania WA NT SA Q V NSW T red green blue Florent Madelaine Problems in AI Durham University p.7/45

8 Finite Domain i.e., D CSP is V and C are finite. -complete. CSP with domain size 2 and only one ternary constraint relation is -complete. CSP with binary constraints only is -complete. CSP with only one binary constraint relation and domain size at least 3 is -complete. Florent Madelaine Problems in AI Durham University p.8/45

9 Infinite Domains It is no longer possible to describe constraints by enumeration. Instead a constraint language must be used: e.g., linear programming x 5 y. Some efficient methods exists for Linear programming. Sometimes, we can transform a continuous problem into a discrete one (e.g. Allen s interval algebra). In this course we stick to the finite case. Florent Madelaine Problems in AI Durham University p.9/45

10 CSP as a search problem Florent Madelaine Problems in AI Durham University p.10/45

11 Incremental formulation INITIAL STATE: all variables unassigned. STATES: (possibly partial) assignments. SUCCESSORS: a value is assigned to an unassigned variable, provided that it does not conflict with previously assigned variables. Florent Madelaine Problems in AI Durham University p.11/45

12 A naive algorithm Depth-first search for CSPs with single-variable assignments is called backtracking search. Backtracking search is the basic uninformed algorithm for CSPs. Can solve n-queens for n 25. Florent Madelaine Problems in AI Durham University p.12/45

13 Backtracking example Florent Madelaine Problems in AI Durham University p.13/45

14 Backtracking example Florent Madelaine Problems in AI Durham University p.13/45

15 Backtracking example Florent Madelaine Problems in AI Durham University p.13/45

16 Backtracking example Florent Madelaine Problems in AI Durham University p.13/45

17 Improving backtracking efficiency General purpose methods can give huge gains in speed: Which variable should be assigned next? In what order should its values be tried? Can we detect inevitable failure early? Can we take advantage of problem structure? Florent Madelaine Problems in AI Durham University p.14/45

18 Most constrained variable Choose the variable with the fewest legal values Florent Madelaine Problems in AI Durham University p.15/45

19 Most constraining variable Tie-breaker among most constrained variables. Choose the variable with the most constraints on remaining variables. Florent Madelaine Problems in AI Durham University p.16/45

20 Least constraining value Given a variable, choose the least constraining value. i.e., the one that rules out the fewest values in the remaining variables. Allows 1 value for SA Allows 0 values for SA Combining these heuristics makes 1000 queens feasible. Florent Madelaine Problems in AI Durham University p.17/45

21 Forward Checking Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values Florent Madelaine Problems in AI Durham University p.18/45

22 Forward Checking Example WA NT Q NSW V SA T Florent Madelaine Problems in AI Durham University p.19/45

23 Forward Checking Example WA NT Q NSW V SA T Florent Madelaine Problems in AI Durham University p.19/45

24 Forward Checking Example WA NT Q NSW V SA T Florent Madelaine Problems in AI Durham University p.19/45

25 Forward Checking Example WA NT Q NSW V SA T Florent Madelaine Problems in AI Durham University p.19/45

26 Limits of Forward Checking Forward checking propagates information from assigned to unassigned variables. But it doesn t provide early detection for all failures. Florent Madelaine Problems in AI Durham University p.20/45

27 Example WA NT Q NSW V SA T NT and SA cannot both be blue! More advanced forms of Constraint propagation repeatedly enforces constraints locally. Florent Madelaine Problems in AI Durham University p.21/45

28 Arc consistency In the case of Binary CSP. The simplest form of propagation makes each arc consistent. x 1 x 2 is consistent if, and only if, for every value v 1 of x 1 there is some allowed value v 2 for x 2. Florent Madelaine Problems in AI Durham University p.22/45

29 Enforcing arc consistency WA NT Q NSW V SA T Florent Madelaine Problems in AI Durham University p.23/45

30 Enforcing arc consistency WA NT Q NSW V SA T Florent Madelaine Problems in AI Durham University p.23/45

31 Enforcing arc consistency WA NT Q NSW V SA T If x loses a value, the neighbours of x need to be rechecked. Florent Madelaine Problems in AI Durham University p.23/45

32 Enforcing arc consistency WA NT Q NSW V SA T Arc consistency detects failure earlier than forward checking. Florent Madelaine Problems in AI Durham University p.23/45

33 Using arc consistency Can be run as a preprocessor or after each assignment. Is more expensive than forward checking but still polynomial. Implementation of arc consistency for non-binary CSP? Also Higher order i-consistency. In the case of binary CSP: path consistency. Florent Madelaine Problems in AI Durham University p.24/45

34 La Pub Le Beaucor et le Nareu (Ange Edal Tainefon) Tremai Beaucor, sur un brare chéper, Naite en son quaib un magefro. Tremai Nareu, par l eurdo chéallé, Lui tint à peu près ce gagelan: "Hé! jourbon, sieurmo du Beaucor. Que vous êtes lijo! que vous me blessan beau! Sans tirmen, si votre magera Se rapporte à votre mageplu, Vous êtes le Nixphé des hôtes de ces bois." Florent Madelaine Problems in AI Durham University p.25/45

35 Plus de Pub A ces mots le Beaucor ne se sent pas de joie; Et pour traimon sa belle voix, Il ouvre un large quaib, laisse béton sa proie. Le Nareu s en saisit, et dit : "Mon bon sieurmo, Apprenez que tout teurfla Vit aux pensedé de celui qui t écoule: Cette sonle vaut bien un magefro, sans doute. Le Beaucor, honteux et fucon, Raju, mais un tard peu, qu on ne l y prendrait plus. OuLiPo. Florent Madelaine Problems in AI Durham University p.26/45

36 Problem structure WA NT Q Tasmania and mainland are independent subproblems SA V Victoria NSW Identifiable as connected components of constraint graph T Florent Madelaine Problems in AI Durham University p.27/45

37 Practical Consequence Suppose each subproblem has c variables out of n total Worst-case solution cost is n c e.g., n 80, d 2, c 20 d c 2 80 = 4 billion years at 10 million nodes/sec = 0.4 seconds at 10 million nodes/sec May make huge problem feasible (divide to conquer). O n, Florent Madelaine Problems in AI Durham University p.28/45

38 Tree-structured CSPs A C B D E F Theorem : if the constraint graph has no loops, the CSP can be solved in O nd 2 time Compare to general CSPs, where worst-case time is O d n Florent Madelaine Problems in AI Durham University p.29/45

39 Algorithm for tree-structured CSPs 1. Choose a variable as root, order variables from root to leaves such that every node s parent precedes it in the ordering 2. For j from n down to 2, apply RemoveInconsistent Parent X j Xj 3. For j from 1 to n, assign X j consistently with Parent A C B X j D E F A B C D E F Florent Madelaine Problems in AI Durham University p.30/45

40 Conditioning Conditioning : instantiate a variable, prune its neighbours domains. NT Q NT Q WA WA SA NSW NSW V Victoria V Victoria T T Florent Madelaine Problems in AI Durham University p.31/45

41 Cutset Conditioning Cutset conditioning : instantiate (in all ways) a set of variables such that the remaining constraint graph is a tree Cutset size c fast for small c. runtime O d c n c d 2, very Florent Madelaine Problems in AI Durham University p.32/45

42 CSP as a local search problem Florent Madelaine Problems in AI Durham University p.33/45

43 Complete formulation INITIAL STATE: some assignment. STATES: any assignment. SUCCESSORS: a variable s value is changed. Florent Madelaine Problems in AI Durham University p.34/45

44 Local Search for CSP Every local search method studied in this course: e.g., Hill Climbing, Simulated Annealing. To apply to CSPs: we allow states with unsatisfied constraints and operators reassign variable values. Variable selection : randomly select any conflicted variable Value selection by min-conflicts heuristic: choose a value that violates the fewest constraints i.e., hillclimb with h n = total number of violated constraints. Florent Madelaine Problems in AI Durham University p.35/45

45 To go a bit further Florent Madelaine Problems in AI Durham University p.36/45

46 Other results Problems can be decomposed efficiently even when they are not trees: e.g., Bounded tree width (for small bound). One can take the dual approach and consider the type of constraints to be taken from a fixed set Γ of relations. There are 3 important theoretical results related to the complexity of CSP. Γ Florent Madelaine Problems in AI Durham University p.37/45

47 Dichotomy Hell and Nešetřil for H-COLOURING (1990). Shaeffer for SAT (1978). partially extended by Jeavons et al. for larger domain size (1994-now). Florent Madelaine Problems in AI Durham University p.38/45

48 Shaeffer Build up on work by E.Post. Found 6 classes of maximal tractable subproblems of SAT: HORNSAT and its dual, 2-SAT, LINEAR EQUATION and two trivial classes. Everything else is intractable. Florent Madelaine Problems in AI Durham University p.39/45

49 Hell and Nešetřil tractable if H is bipartite intractable otherwise. Florent Madelaine Problems in AI Durham University p.40/45

50 Jeavons et al. Generalised partially Shaeffer s result for domain size greater than 2. Uses extensively results from Universal Algebra. Florent Madelaine Problems in AI Durham University p.41/45

51 Summary CSPs are a very general kind of problem: states defined by values of a fixed set of variables goal test defined by constraints on variable values. Backtracking : depth-first search with one variable assigned per node. Variable ordering and value selection heuristics help significantly. Forward checking prevents assignments that guarantee later failure. Florent Madelaine Problems in AI Durham University p.42/45

52 Summary Constraint propagation (e.g., arc consistency) does additional work to constrain values and detect inconsistencies. The CSP representation allows analysis of problem structure. Tree-structured CSPs can be solved in polynomial time. min-conflicts evaluation for local search can also be effective. Florent Madelaine Problems in AI Durham University p.43/45

53 Next Week Mock exam on Monday at 2:15pm in portakabin; Thursday: correction of Mock exam. Friday: (setting up) a Tsp contest. Florent Madelaine Problems in AI Durham University p.44/45

54 La Fin Excellent exercice intellectuel, la contrepéterie est bien autre chose qu une frivole amusette. Elle est objet de science, mieux, elle est oeuvre d art. Pour cette science ou pour cet art, assez analogue à celui du Contrepoint, nous avons forgé le nom de Contrepet. Luc Etienne Florent Madelaine Problems in AI Durham University p.45/45