Last bit on A*-search.

8 Puzzle I

Heuristic: Number of tiles misplaced

Why is it admissible?

h(start) =

This is a relaxed-problem heuristic

8 Average nodes expanded when optimal path has length…

…4 steps …8 steps …12 steps

UCS 112 6,300 3.6 x 106

TILES 13 39 227

8 Puzzle II

What if we had an easier 8-puzzle where any tile could slide any direction at any time, ignoring other tiles?

Total Manhattan distance

Why admissible?

h(start) =

3 + 1 + 2 + …

= 18

Average nodes expanded when optimal path has length…

…4 steps …8 steps …12 steps

TILES 13 39 227

MANHATTAN 12 25 73[demo: eight-puzzle]

8 Puzzle III

How about using the actual cost as a heuristic? Would it be admissible? Would we save on nodes expanded? What’s wrong with it?

With A*: a trade-off between quality of estimate and work per node!

Trivial Heuristics, Dominance

Dominance: ha ≥ hc if

Heuristics form a semi-lattice: Max of admissible

Graph Search

Tree Search: Extra Work!

Failure to detect repeated states can cause exponentially more work. Why?

Graph Search

In BFS, for example, we shouldn’t bother expanding the circled nodes (why?)

S

a

b

d p

a

c

e

p

h

f

r

q

q c G

a

qe

p

h

f

r

q

q c G

a

Graph Search

Very simple fix: never expand a state twice

Graph Search

Idea: never expand a state twice

How to implement: Tree search + list of expanded states (closed list) Expand the search tree node-by-node, but… Before expanding a node, check to make sure its state is new

Python trick: store the closed list as a set, not a list

Can graph search wreck completeness? Why/why not?

How about optimality?

Consistency

A

CS

1

1h = 4

h = 1h = 6

What went wrong? Taking a step must not reduce f value!

B

h = 0

Gh = 11 2 3

Consistency

Stronger than admissability Definition:

C(A→C)+h(C) h(A)≧ C(A→C) h(A)-h(C)≧

Consequence: The f value on a path never

decreases A* search is optimal

A

G

1

h = 4

h = 0

Ch = 1 3

Optimality of A* Graph Search

Proof: New problem (with graph instead of tree search):

nodes on path to G* that would have been in queue aren’t, because some worse n’ for the same state as some n was dequeued and expanded first (disaster!)

Take the highest such n in tree Let p be the ancestor which was on the queue

when n’ was expanded Assume f(p) f(n) ≦ (consistency!) f(n) < f(n’) because n’ is suboptimal p would have been expanded before n’ So n would have been expanded before n’, too Contradiction!

Optimality

Tree search: A* optimal if heuristic is admissible (and non-negative) UCS is a special case (h = 0)

Graph search: A* optimal if heuristic is consistent UCS optimal (h = 0 is consistent)

Consistency implies admissibility

In general, natural admissible heuristics tend to be consistent

Summary: A*

A* uses both backward costs and (estimates of) forward costs

A* is optimal with admissible heuristics

Heuristic design is key: often use relaxed problems

Quiz 2 : A* search

T/F: A* is optimal with any h(). False T/F: A* is complete with any h(). True T/F: A* with the zero heuristic is UCS. True T/F: Admissibility implies consistency. False T/F: If n’ is a sub-optimal path to n, then f(n)<f(n’) for any

heuristic. True h(n)=h(n’) but g(n)<g(n’) T/F: Admissibility is necessary for Graph-Search to be

optimal. True

(All heuristics are are non-negative and bounded by some B, i.e. h(x) B.)≦

Search Recap

Uniform Cost Search / BFS

Greedy Search / DFS

A* Search

A* applications

http://pacman.elstonj.com/index.cgi?dir=videos&num=&perpage=&section=

http://www.youtube.com/watch?feature=player_embedded&v=JnkMyfQ5YfY#!

CSE 511A: Artificial IntelligenceSpring 2013

Lecture 4: Constraint Satisfaction Programming

Jan 29, 2012

Robert Pless, slides from Kilian Weinberger, Ruoyun Huang, adapted from Dan Klein, Stuart Russell or Andrew Moore

Announcements

Projects: Project 1 out. Due Thursday 10 days from now,

midnight.

21

1.What is a CSP?

What is Search For?

Models of the world: single agents, deterministic actions, fully observed state, discrete state space

Planning: sequences of actions The path to the goal is the important thing Paths have various costs, depths Heuristics to guide, fringe to keep backups

Identification: assignments to variables The goal itself is important, not the path All paths at the same depth (for some formulations) CSPs are specialized for identification problems

23

Search Problems

A search problem consists of:

A state space

A successor function

A start state and a goal test

A cost function (for now we set cost=1 for all steps) A solution is a sequence of actions (a plan)

which transforms the start state to a goal state

“N”, 1.0

“E”, 1.0

One Example

Schedule Lectures at Washburn University

Time-table:

Lop101 Jolley 305

9:0010:0011:0012:0013:0014:00

Courses:-A: Intro to AI-B: Bioinformatics-C: Computational Geometry-D: Discrete Math

Constraints:-Only one course per room / slot-Each course meets one two hours slot-A can only be in Jelly 204-B must be scheduled before noon-D must be in the afternoon

Search Problem

State Space: (partially filled in) time schedule

Successor Function: Assign time slot to unassigned course

Start State: Empty schedule

Goal test: All constraints are met All courses scheduled

Time-table:

Lap101 Jelly 204

9:0010:0011:0012:0013:0014:00

AABB

CCDD

Naïve Search Tree

AAAA

AABB

BBBB

BBBB

BBAA AA AATree is gigantic!Depth?Width?Where is a solution?

Constraint Satisfaction Problems Standard search problems:

State is a “black box”: arbitrary data structure Goal test: any function over states Successor function can be anything

Constraint satisfaction problems (CSPs): A special subset of search problems State is defined by variables Xi with values from a

domain D (sometimes D depends on i) Goal test is a set of constraints specifying

allowable combinations of values for subsets of variables

Path cost irrelevant!

Simple example of a formal representation language

Allows useful general-purpose algorithms with more power than standard search algorithms

28

CSPsSearch

Problems

Example 1: Map-Coloring Variables:

Domain:

Constraints: adjacent regions must have different colors

Solutions are assignments satisfying all constraints, e.g.:

29

Implicit:

Explicit:

-or-

Constraint Graphs

Binary CSP: each constraint relates (at most) two variables

Binary constraint graph: nodes are variables, arcs show constraints

General-purpose CSP algorithms use the graph structure to speed up search. E.g., Tasmania is an independent subproblem!

30

Example 2: Cryptarithmetic

Variables:

Domains:

Constraints (boxes):

31

Example 2: Cryptarithmetic II

Variables:

Domains:

Constraints:

32

T, W, O, F, U, R

(T * 100 + W * 10 + O) * 2= F * 1000 + O * 100 + U * 10 + R

Example 3: N-Queens

Formulation 1: Variables: Domains: Constraints

33

Example 3: N-Queens

Formulation 1.5: Variables:

Domains:

Constraints:

… there’s an even better way! What is it?

Implicit:

Explicit:

-or-

Example 3: N-Queens

Formulation 2: Variables:

Domains:

Constraints:

Implicit:

Explicit:

-or-

Example 4: Sudoku

Variables: Each (open) square

Domains: {1,2,…,9}

Constraints:

9-way alldiff for each row

9-way alldiff for each column

9-way alldiff for each region

Check out Peter Norvig’s homepage http://norvig.com/sudoku.html

Example 5: The Waltz Algorithm The Waltz algorithm is for interpreting line drawings of

solid polyhedra An early example of a computation posed as a CSP

First, look at all intersections

?

37

Four types of junctions:L, fork, T, arrow

Waltz on Simple Scenes

3 Types of Lines: Boundary line (edge of an

object) () with right hand of arrow denoting “solid” and left hand denoting “space”

Interior convex edge (+) Interior concave edge (-)

4 types of junctions L, fork, T, arrow

Adjacent intersections impose constraints on each other

38

All the valid conjunction labeling:

Waltz on Simple Scenes

Variable (edges): (AB, AC, AE, BA, BD, …, )

Domain: {+,-,,} Constraints (both ends have the same label):

arrow (AC, AE, AB), fork(BA, BF, BD), L(CA, CD), arrow(DG, DC, DB), …

L Junctions

fork Junctions

fork Junctions

arrow Junctions

All the valid conjunction labeling

Waltz on Simple Scenes

At least 4 valid labeling choices

Example 6: Temporal Logic Reasoning

Problem: The meeting ran non-stop the whole day. Each person stayed at the meeting for a continuous period of time. Ms White arrived after the meeting has began. In turn, Director Smith was also present but he arrived after Jones had left. Mr. Brown talked to Ms. White in presence of Smith. Could possibly Jones and White have talked during this meeting?

42

Possible Relations:before, meets, overlap,

start, during, finish, equal

truly-overlap()= overlaps + overlaps-by + starts + started-by + during + contains+ finishes+ finished-by + equal

43

Example 6: Temporal Logic Reasoning

Example 6: Temporal Logic Reasoning

1. Ms White arrived after the meeting has began.

during(W,M) or finishes(W,M) or equals(W,M)

2. Director Smith was also present but he arrived after Jones had left.

after(S, J)

3. Mr. Brown talked to Ms. White in presence of Smith.

truly-overlap(B,M), truly-overlap(B,S), truly-overlap(W,S)

4. Could possibly Jones and White have talked during this meeting?

truly-overlap(J,W)

Event(s) Variable(s)

The duration of the meeting M

The period Jones, Brown, Smith, White was present J, B, S, W

44

Real-World CSPs

Assignment problems: e.g., who teaches what class Timetabling problems: e.g., which class is offered when

and where? Hardware configuration/verification Transportation scheduling Factory scheduling Floorplanning Fault diagnosis … lots more!

Many real-world problems involve real-valued variables…

45

Varieties of CSPs Discrete Variables

Finite domains (Most cases!) Size d means O(dn) complete assignments E.g., Boolean CSPs, including Boolean satisfiability (NP-complete)

Infinite domains (integers, strings, etc.) E.g., our temporal logic reasoning Linear constraints solvable, nonlinear undecidable

Continuous variables E.g., start/end times for Hubble Telescope observations Linear constraints solvable in polynomial time by LP methods

(see ESE505 Operation Research for a bit of this theory)

There might be objective functions: Branch-and-Bound method

46

Varieties of Constraints Varieties of Constraints

Unary constraints involve a single variable (equiv. to shrinking domains):

Binary constraints involve pairs of variables:

Higher-order constraints involve 3 or more variables: e.g., 9-way different constraints in sudoku

Preferences (soft constraints): E.g., red is better than green Often representable by a cost for each variable assignment Gives constrained optimization problems (We’ll ignore these until we get to Bayes’ nets)

47

2. How can we solve CSPs fast?

By taking advantage of their specific structure!

Standard Search Formulation

Standard search formulation of CSPs (incremental)

Let's start with the straightforward, dumb approach, then fix it

States are defined by the values assigned so far Initial state: the empty assignment, {} Successor function: assign a value to an unassigned variable Goal test: the current assignment is complete and satisfies all

constraints

Simplest CSP ever: two bits, constrained to be equal

51

Search Methods

What does BFS do?

What does DFS do?

What’s the obvious problem here? The order of assignment does not matter.

What’s the other obvious problem? We are checking constraints too late.

52

Backtracking Search Idea 1: Only consider a single variable at each point

Variable assignments are commutative, so fix ordering I.e., [WA = red then NT = green] same as [NT = green then WA = red] Only need to consider assignments to a single variable at each step How many leaves are there?

Idea 2: Only allow legal assignments at each point I.e. consider only values which do not conflict previous assignments Might have to do some computation to figure out whether a value is ok “Incremental goal test”

Depth-first search for CSPs with these two improvements is called backtracking search (useless name, really)

Backtracking search is the basic uninformed algorithm for CSPs

Can solve n-queens for n 25 54

Backtracking Example

55 What are the choice points?

Improving Backtracking

General-purpose ideas 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?

56

WASA

NT Q

NSW

V

Which Variable: Minimum Remaining Values

Minimum remaining values (MRV): Choose the variable with the fewest legal values

Why min rather than max? Also called “most constrained variable” “Fail-fast” ordering

57

Which Variable: Degree Heuristic

Tie-breaker among MRV variables Degree heuristic:

Choose the variable participating in the most constraints on remaining variables

Why most rather than fewest constraints?58

Which Value: Least Constraining Value

Given a choice of variable: Choose the least constraining

value The one that rules out the fewest

values in the remaining variables Note that it may take some

computation to determine this!

Why least rather than most?

Combining these heuristics makes 1000 queens feasible

59

Better choice