Sessions 1

1
Sessions 1 2Introduction to AIPlanning
Search

• CSE 592
• Applications of Artificial Intelligence
• Henry Kautz
• Winter 2003

2
What is Intelligence?What is Artificial
Intelligence?
3
What is Artificial Intelligence?

• The study of the principles by which natural or
  artificial machines manipulate knowledge
• how knowledge is acquired
• how goals are generated and achieved
• how concepts are formed
• how collaboration is achieved

4
. . . Exactly what the computer provides is the
ability not to be rigid and unthinking but,
rather, to behave conditionally. That is what it
means to apply knowledge to action It means to
let the action taken reflect knowledge of the
situation, to be sometimes this way, sometimes
that, as appropriate. . . . -Allen Newell
5

• Classical AI
• Disembodied Intelligence
• Autonomous Systems
• Embodied Intelligence

6
Classical AI

• The principles of intelligence are separate from
  any hardware / software / wetware implementation
• logical reasoning
• probabilistic reasoning
• strategic reasoning
• diagnostic reasoning
• Look for these principles by studying how to
  perform tasks that require intelligence

7
Success Story Medical Expert Systems

• Mycin (1980)
• Expert level performance in diagnosis of blood
  infections
• Today 1,000s of systems
• Everything from diagnosing cancer to designing
  dentures
• Often outperform doctors in clinical trials
• Major hurdle today non-expert part
  doctor/machine interaction

8
Success StoryChess

• I could feel I could smell a new kind of
  intelligence across the table- Kasparov

• Examines 5 billion positions / second
• Intelligent behavior emerges from brute-force
  search

9
Autonomous Systems

• In the 1990s there was a growing concern that
  work in classical AI ignored crucial scientific
  questions
• How do we integrate the components of
  intelligence (e.g. learning planning)?
• How does perception interact with reasoning?
• How does the demand for real-time performance in
  a complex, changing environment affect the
  architecture of intelligence?

10

• Provide a standard problem where a wide range of
  technologies can be integrated and examined
• By 2050, develop a team of fully autonomous
  humanoid robots that can win against the human
  world champion team in soccer.

11
Started January 1996 Launch October 15th,
1998 Experiment May 17-21
courtesy JPL
12
Speed Capacity
13
Not Speed Alone

• Speech Recognition
• Word spotting feasible today
• Continuous speech rapid progress
• Turns out that low level signal not as
  ambiguous as we once thought
• Translation / Interpretation / Question-answering
• Very limited progress
• The spirit is willing but the flesh is weak.
  (English)
• The vodka is good but the meat is rotten.
  (Russian)

14
Varieties of Knowledge

• What kinds of knowledge required to understand
• Time flies like an arrow.
• Fruit flies like a banana.
• Fruit flies like a rock.

15
Historic Perspective

• 1940's - 1960's Artificial neural networks
• McCulloch Pitts 1943
• 1950's - 1960's Symbolic information processing
• General Problem Solver Simon Newell
• "Weak methods for search and learning
• 1969 - Minsky's Perceptrons
• 1940s 1970s Control theory for adaptive
  (learning) systems
• USSR Cybernetics Norbert Weiner
• Japan Fuzzy logic
• 1970's 1980s Expert systems
• Knowledge is power" Ed Feigenbaum
• Logical knowledge representation
• AI Boom
• 1985 2000 A million flowers bloom
• Resurgence of neural nets backpropagation
• Control theory OR Pavlovian conditioning
  reinforcement learning
• Probabilistic knowledge representation Bayesian
  Nets Judea Pearl
• Statistical machine learning
• 2000s Towards a grand unification

16
In sum, technology can be controlled
especially if it is saturated with intelligence
to watch over how it goes, to keep accounts, to
prevent errors, and to provide wisdom to each
decision. -Allen Newell
17
Course Mechanics

• Topics
• What is AI?
• Search, Planning, and Satisfiability
• Bayesian Networks
• Statistical Natural Language Processing
• Decision Trees and Neural Networks
• Data Mining Pattern Discovery in Databases
• Planning under Uncertainty and Reinforcement
  Learning
• Autonomous Systems Case Studies
• Project Presentations
• Assignments
• 4 homeworks
• Significant project presentation
• Information
• http//www.cs.washington.edu/education/courses/592
  /03wi/

18
Planning Search

• Search the foundation for all work in AI
• Deduction
• Probabilistic reasoning
• Perception
• Learning
• Game playing
• Expert systems
• Planning

RN Ch 3, 4, 5, 11
19
Planning

• Input
• Description of set of all possible states of the
  world (in some knowledge representation language)
• Description of initial state of world
• Description of goal
• Description of available actions
• May include costs for performing actions
• Output
• Sequence of actions that convert the initial
  state into one that satisfies the goal
• May wish to minimize length or cost of plan

20
Classical Planning

• Simplifying assumptions
• Atomic time
• Actions have deterministic effects
• Agent knows complete initial state of the world
• Agent knows the effects of all actions
• States are either goal or non-goal states, rather
  than numeric utilities or rewards
• Agent is sole cause of change
• All these assumptions can be relaxed, as we will
  see by the end of the course

21
Example Route Planning

• Input
• State set
• Start state
• Goal state test
• Operators
• Output

22
Example Robot Control (Blocks World)

• Input
• State set
• Start state
• Goal state test
• Operators (and costs)
• Output

23
Implicitly Generated Graphs

• Planning can be viewed as finding paths in a
  graph, where the graph is implicitly specified by
  the set of actions
• Blocks world
• vertex relative positions of all blocks
• edge robot arm stacks one block

How many states for K blocks?
stack(blue,table)
stack(green,blue)
stack(blue,red)
stack(green,red)
stack(green,blue)
24
Missionaries and Cannibals

• 3 missionaries M1, M2, M3
• 3 cannibals C1, C2, C3
• Cross in a two person boat, so that missionaries
  never outnumber cannibals on either shore
• What is a state? How many states?

M1 M2 M3
C1 C2 C3
25
STRIPS Representation

• Description of initial state of world
• Set of propositions that completely describes a
  world
• (block a) (block b) (block c) (on-table a)
  (on-table b) (clear a) (clear b) (clear c)
  (arm-empty)
• Description of goal (i.e. set of desired worlds)
• Set of propositions that partially describes a
  world
• (on a b) (on b c)
• Description of available actions

26
How Represent Actions?

• World set of propositions true in that world
• Actions
• Precondition conjunction of propositions
• Effects propositions made true propositions
  made false (deleted from the state description)

operator stack_B_on_R precondition (on B Table)
(clear R) effect (on B R) (not (clear R))
27
Action Schemata

• Compact representation of a large set of actions

(operator pickup parameters
((block ?ob1)) precondition (and (clear
?ob1) (on-table ?ob1)
(arm-empty)) effect
(and (not (on-table ?ob1)) (not
(clear ?ob1)) (not
(arm-empty)) (holding ?ob1)))
28
Search Algorithms

• Backtrack Search
• DFS
• BFS / Dijkstras Algorithm
• Iterative Deepening
• Best-first search
• A
• Constraint Propagation
• Forward Checking
• k-Consistency
• DPLL Resolution
• Local Search
• Hillclimbing
• Simulated annealing
• Walksat

29
Depth First Search

• Maintain stack of nodes to visit
• Evaluation
• Complete?
• Time Complexity?
• Space Complexity?

Not for infinite spaces
a
O(bd)
b
c
O(d)
g
h
d
e
f
30
Breadth First Search

• Maintain queue of nodes to visit
• Evaluation
• Complete?
• Time Complexity?
• Space Complexity?

Yes
a
O(bd)
b
c
O(bd)
g
h
d
e
f
31
Iterative Deepening Search

• DFS with limit incrementally grow limit
• Evaluation
• Complete?
• Time Complexity?
• Space Complexity?

Yes
a
a
O(bd)
b
b
c
c
O(d)
g
h
d
e
f
32
Dijkstras Shortest Path Algorithm

• Like breadth-first search, but uses a priority
  queue instead of a FIFO queue
• Always select (expand) the vertex that has a
  lowest-cost path from the initial state
• Correctly handles the case where the lowest-cost
  path to a vertex is not the one with fewest edges
• Handles actions planning with costs, with same
  advantages / disadvantages of BFS

33
Pseudocode for Dijkstra

• Initialize the cost of each vertex to ?
• costs 0
• heap.insert(s)
• While (! heap.empty())
• n heap.deleteMin()
• For (each vertex a which is adjacent to n along
  edge e)
• if (costn edge_coste lt costa) then
• cost a costn edge_coste
• previous_on_path_toa n
• if (a is in the heap) then heap.decreaseKey(a)
• else heap.insert(a)

34
Important Features

• Once a vertex is removed from the head, the cost
  of the shortest path to that node is known
• While a vertex is still in the heap, another
  shorter path to it might still be found
• The shortest path itself from s to any node a can
  be found by following the pointers stored in
  previous_on_path_toa

35
Edsger Wybe Dijkstra (1930-2002)

• Invented concepts of structured programming,
  synchronization, weakest precondition, and
  semaphores
• 1972 Turing Award
• In their capacity as a tool, computers will be
  but a ripple on the surface of our culture. In
  their capacity as intellectual challenge, they
  are without precedent in the cultural history of
  mankind.

36
Heuristic Search

• A heuristic is
• Function from a state to a real number
• Low number means state is close to goal
• High number means state is far from the goal

Designing a good heuristic is very
important! (And often hard! Later we will see
how some heuristics can be created automatically)
37
An Easier Case

• Suppose you live in Manhattan what do you do?

S
52nd St
G
51st St
50th St
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave
38
Best-First Search

• The Manhattan distance (? x ? y) is an estimate
  of the distance to the goal
• a heuristic value
• Best-First Search
• Order nodes in priority to minimize estimated
  distance to the goal h(n)
• Compare BFS / Dijkstra
• Order nodes in priority to minimize distance from
  the start

39
Best First in Action

• Suppose you live in Manhattan what do you do?

S
52nd St
G
51st St
50th St
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave
40
Problem 1 Led Astray

• Eventually will expand vertex to get back on the
  right track

S
G
52nd St
51st St
50th St
10th Ave
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
3rd Ave
2nd Ave
41
Problem 2 Optimality

• With Best-First Search, are you guaranteed a
  shortest path is found when
• goal is first seen?
• when goal is removed from priority queue (as with
  Dijkstra?)

42
Sub-Optimal Solution

• No! Goal is by definition at distance 0 will be
  removed from priority queue immediately, even if
  a shorter path exists!

(5 blocks)
S
52nd St
h2
h5
G
h4
51st St
h1
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
43
Synergy?

• Dijkstra / Breadth First guaranteed to find
  optimal solution
• Best First often visits far fewer vertices, but
  may not provide optimal solution
• Can we get the best of both?

44
A (A star)

• Order vertices in priority queue to minimize
• (distance from start) (estimated distance to
  goal)
• f(n) g(n) h(n)
• f(n) priority of a node
• g(n) true distance from start
• h(n) heuristic distance to goal

45
Optimality

• Suppose the estimated distance (h) is always
  less than or equal to the true distance to the
  goal
• heuristic is a lower bound on true distance
• heuristic is admissible
• Then when the goal is removed from the priority
  queue, we are guaranteed to have found a shortest
  path!

46
Problem 2 Revisited
vertex g(n) h(n) f(n)
52nd 9th 0 5 5


S
(5 blocks)
52nd St
G
51st St
50th St
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
47
Problem 2 Revisited
vertex g(n) h(n) f(n)
52nd 4th 5 2 7
51st 9th 1 4 5

S
(5 blocks)
52nd St
G
51st St
50th St
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
48
Problem 2 Revisited
vertex g(n) h(n) f(n)
52nd 4th 5 2 7
51st 8th 2 3 5
50th 9th 2 5 7
S
(5 blocks)
52nd St
G
51st St
50th St
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
49
Problem 2 Revisited
vertex g(n) h(n) f(n)
52nd 4th 5 2 7
51st 7th 3 2 5
50th 9th 2 5 7
50th 8th 3 4 7
S
(5 blocks)
52nd St
G
51st St
50th St
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
50
Problem 2 Revisited
vertex g(n) h(n) f(n)
52nd 4th 5 2 7
51st 6th 4 1 5
50th 9th 2 5 7
50th 8th 3 4 7
50th 7th 4 3 7
S
(5 blocks)
52nd St
G
51st St
50th St
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
51
Problem 2 Revisited
vertex g(n) h(n) f(n)
52nd 4th 5 2 7
51st 5th 5 0 5
50th 9th 2 5 7
50th 8th 3 4 7
50th 7th 4 3 7
S
(5 blocks)
52nd St
G
51st St
50th St
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
52
Problem 2 Revisited
vertex g(n) h(n) f(n)
52nd 4th 5 2 7
50th 9th 2 5 7
50th 8th 3 4 7
50th 7th 4 3 7
S
(5 blocks)
52nd St
G
51st St
50th St
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
DONE!
53
What Would Dijkstra Have Done?
S
(5 blocks)
52nd St
G
51st St
50th St
49th St
48th St
47th St
9th Ave
8th Ave
7th Ave
6th Ave
5th Ave
4th Ave
54
Proof of A Optimality

• A terminates when G is popped from the heap.
• Suppose G is popped but the path found isnt
  optimal
• priority(G) gt optimal path length c
• Let P be an optimal path from S to G, and let N
  be the last vertex on that path that has been
  visited but not yet popped.
• There must be such an N, otherwise the optimal
  path would have been found.
• priority(N) g(N) h(N) ? c
• So N should have popped before G can pop.
  Contradiction.

non-optimal path to G
S
G
undiscovered portion of shortest path
portion of optimal path found so far
N
55
What About Those Blocks?

• Distance to goal is not always physical
  distance
• Blocks world
• distance number of stacks to perform
• heuristic lower bound number of blocks out of
  place

out of place 1, true distance to goal 3
56
3-Blocks State Space Graph
ABCh2
CABh3
BACh2
ABCh1
CBAh3
A CBh2
BCAh1
BC Ah3
CB Ah3
C A Bh3
AC Bh3
BA Ch2
AB Ch0
start
goal
57
3-Blocks Best First Solution
ABCh2
CABh3
BACh2
ABCh1
CBAh3
A CBh2
BCAh1
BC Ah3
CB Ah3
C A Bh3
AC Bh3
BA Ch2
AB Ch0
start
goal
58
3-Blocks BFS Solution
ABCh2
expanded, but not in solution
CABh3
BACh2
ABCh1
CBAh3
A CBh2
BCAh1
BC Ah3
CB Ah3
C A Bh3
AC Bh3
BA Ch2
AB Ch0
start
goal
59
3-Blocks A Solution
ABCh2
expanded, but not in solution
CABh3
BACh2
ABCh1
CBAh3
A CBh2
BCAh1
BC Ah3
CB Ah3
C A Bh3
AC Bh3
BA Ch2
AB Ch0
start
goal
60
Maze Runner Demo
61
Other Real-World Applications

• Routing finding computer networks, airline
  route planning
• VLSI layout cell layout and channel routing
• Production planning just in time optimization
• Protein sequence alignment
• Many other NP-Hard problems
• A class of problems for which no exact polynomial
  time algorithms exist so heuristic search is
  the best we can hope for

62
Importance of Heuristics

• h1 number of tiles in the wrong place
• h2 sum of distances of tiles from correct
  location

D IDS A(h1) A(h2) 2 10
6 6 4 112 13 12 6
680 20 18 8 6384 39 25 10
47127 93 39 12 364404 227
73 14 3473941 539 113 18
3056 363 24 39135 1641
63
A STRIPS Planning

• Is there some general way to automatically create
  a heuristic for a given set of STRIPS operators?
• Count number of false goal propositions in
  current state
• Admissible?
• Delete all preconditions from actions, solve
  easier relaxed problem (why easier?), use length
• Admissible?
• Delete negative effects from actions, solve
  easier relaxed problem, use length
• Admissible?

64
Planning as A Search

• HSP (Geffner Bonet 1999), introduced admissible
  ignore negative effects heuristic
• FF (Hoffman Nebel 2000), used a modified
  non-admissible heuristic
• Often dramatically faster, but usually
  non-optimal solutions found
• Best overall performance AIPS 2000 planning
  competition

65
Search Algorithms

• Backtrack Search
• DFS
• BFS / Dijkstras Algorithm
• Iterative Deepening
• Best-first search
• A
• Constraint Propagation
• Forward Checking
• k-Consistency
• DPLL Resolution
• Local Search
• Hillclimbing
• Simulated annealing
• Walksat

66
Guessing versus Inference

• All the search algorithms weve seen so far are
  variations of guessing and backtracking
• But we can reduce the amount of guesswork by
  doing more reasoning about the consequences of
  past choices
• Example planning a trip
• Idea
• Problem solving as constraint satisfaction
• As choices (guesses) are made, propagate
  constraints

67
Map Coloring
68
CSP

• V is a set of variables v1, v2, , vn
• D is a set of finite domains D1, D2, , Dn
• C is a set of constraints C1, C2, , Cm
• Each constraint specifies a restriction over
  joint values of a subset of the variables
• E.g.
• v1 is Spain, v2 is France, v3 is Germany,
• Di Red, Blue, Green for all i
• For each adjacent vi, vj there is a constraint
  Ck
• (vi,vj) ? (R,G), (R,B), (G,R), (G,B), (B,R),
  (B,G)

69
Variations

• Find a solution that satisfies all constraints
• Find all solutions
• Find a tightest form for each constraint
• (v1,v2) ? (R,G), (R,B), (G,R), (G,B), (B,R),
  (B,G)
• ?
• (v1,v2) ? (R,G), (R,B), (B,G)
• Find a solution that minimizes some additional
  objective function

70
Chinese Dinner Constraint Network
Must be HotSour
Soup
No Peanuts
Chicken Dish
Appetizer
Total Cost lt 30
No Peanuts
Pork Dish
Vegetable
Rice
Seafood
Not Both Spicy
Not Chow Mein
71
Exploiting CSP Structure

• Interleave inference and guessing
• At each internal node
• Select unassigned variable
• Select a value in domain
• Backtracking try another value
• Branching factor?
• At each node
• Propagate Constraints

72
Running Example 4 Queens
Constraints

• Variables
• Q1 ? 1,2,3,4
• Q2 ? 1,2,3,4
• Q3 ? 1,2,3,4
• Q3 ? 1,2,3,4

Q1 Q2
1 3
1 4
2 4
3 1
4 1
4 2
Q
Q
Q
Q
73
Constraint Checking



Q

Q
x
Q x
x
Q x
x x
Q x x
Q
x
x
Q x
Q
x
x Q
Q x x
Q x
x x
x Q x
Q x x x
Takes 5 guesses to determine first guess was wrong
74
Forward Checking
x
x
x
Q x x x
x x
Q x x
x x x
Q x x x
When variable is set, immediately remove
inconsistent values from domains of other
variables
Q x x
x x
x x
Q x x x
Q x x
x x x
x Q x
Q x x x
Takes 3 guesses to determine first guess was wrong
75
Arc Consistency
x
x
x
Q x x x
x
x
x
Q x x x
x
x
x x
Q x x
x x
x x
x x
Q x x

• Iterate forward checking
• Propagations
• Q33 inconsistent with Q4 ? 2,3,4
• Q21 and Q22 inconsistent with Q3 ? 1

Inference alone determines first guess was wrong!
76
Huffman-ClowesLabeling


-




77
Waltzs Filtering Arc-Consistency

• Lines variables
• Conjunctions constraints
• Initially Di ,-, ?, ? )
• Repeat until no changes
• Choose edge (variable)
• Delete labels on edge not consistent with both
  endpoints

78
No labeling!
79
Path Consistency

• Path consistency (3-consistency)
• Check every triple of variables
• More expensive!
• k-consistency
• n-consistency backtrack-free search

80
Variable and Value Selection

• Select variable with smallest domain
• Minimize branching factor
• Most likely to propagate most constrained
  variable heuristic
• Which values to try first?
• Most likely value for solution
• Least propagation! Least constrained value
• Why different?
• Every constraint must be eventually satisfied
• Not every value must be assigned to a variable!
• Tie breaking?
• In general randomized tie breaking best less
  likely to get stuck on same bad pattern of choices

81
CSPs in the real world

• Scheduling Space Shuttle Repair
• Transportation Planning
• Computer Configuration
• ATT CLASSIC Configurator
• 5ESS Switching System
• Configuring new orders 2 months ? 2 hours

82
Quasigroup Completion Problem (QCP)
Given a partial assignment of colors (10 colors
in this case), can the partial quasigroup (latin
square) be completed so we obtain a full
quasigroup? Example
32 preassignment
(Gomes Selman 97)
83
QCP Example Use Routers in Fiber Optic Networks
Dynamic wavelength routing in Fiber Optic
Networks can be directly mapped into the
Quasigroup Completion Problem.
(Barry and Humblet 93, Cheung et al. 90, Green
92, Kumar et al. 99)
84
QCP as a CSP

• Variables -
• Constraints -

row
column
85
Hill Climbing

• Idea
• Always choose best child,no backtracking
• Evaluation
• Complete?
• Space Complexity?
• Complexity of random restart hillclimbing, with
  success probability P

86
Simulated Annealing / Random Walk

• Objective avoid local minima
• Technique
• For the most part use hill climbing
• Occasionally take non-optimal step
• Annealing Reduce probability (non-optimal) over
  time
• Comparison to Hill Climbing
• Completeness?
• Speed?
• Space Complexity?

Temperature Objective
Time ?
87
Backtracking with Randomized Restarts

• Idea
• If backtracking algorithm does not find solution
  quickly, it is like to be stuck in the wrong part
  of the search space
• Early decisions were bad!
• So kill the run after T seconds, and restart
• Requires randomized heuristic, so choices not
  always the same
• Why does it often work?
• Many problems have a small set of backdoor
  variables guess them on a restart, and your are
  done! (Andrew, Selman, Gomes 2003)
• Completeness?

88
Demos!

• N-Queens
• Backtracking vs. Local Search
• Quasigroup Completion
• Randomized Restarts
• Travelling Salesman
• Simulated Annealing

89
Exercise

• Peer interviews Real-world constraint
  satisfaction problems
• Break into pairs
• 7 minute interview example of needing to solve
  a CSP type problem (work or life). Interviewer
  takes notes
• Describe problem
• What techniques actually used
• Any techniques from class that could have been
  used?
• Switch roles
• A few teams present now
• Hand in notes (MSR have someone collect and
  mail to me at dept)

90
Planning as CSP

• Phase 1 - Convert planning problem in a CSP
• Choose a fixed plan length
• Boolean variables
• Action executed at a specific time point
• Proposition holds at a specific time point
• Constraints
• Initial conditions true in first state, goals
  true in final state
• Actions do not interfere
• Relation between action, preconditions, effects
• Phase 2 - Solution Extraction
• Solve the CSP

91
Planning Graph Representation of CSP
Precondition constraints
Effect constraints
Proposition Init State
Action Time 1
Proposition Time 1
Action Time 2
92
Constructing the planning graph

• Initial proposition layer
• Just the initial conditions
• Action layer i
• If all of an actions preconditionss are in i-1
• Then add action to layer I
• Proposition layer i1
• For each action at layer i
• Add all its effects at layer i1

93
Mutual Exclusion

• Actions A,B exclusive (at a level) if
• A deletes Bs precondition, or
• B deletes As precondition, or
• A B have inconsistent preconditions
• Propositions P,Q inconsistent (at a level) if
• All ways to achieve P exclude all ways to achieve
  Q
• Constraint propagation (arc consistency)
• Can force variables to become true or false
• Can create new mutexes

94
Solution Extraction

• For each goal G at last time slice N
• Solve( G, N )
• Solve( G, t )
• CHOOSE action A making G true _at_t that is not
  mutex with a previously chosen action
• If no such action, backtrack to last choice point
• For each precondition P of A
• Solve(P, t-1)

95
Graphplan

• Create level 0 in planning graph
• Loop
• If goal ? contents of highest level (nonmutex)
• Then search graph for solution
• If find a solution then return and terminate
• Else Extend graph one more level

A kind of double search forward direction checks
necessary (but insufficient) conditions for a
solution, ... Backward search verifies...
96
Dinner Date
Initial Conditions (and (cleanHands)
(quiet)) Goal (and (noGarbage)
(dinner) (present)) Actions (operator carry
precondition effect (and (noGarbage)
(not (cleanHands))) (operator fire
precondition effect (and (noGarbage)
(not (paper))) (operator cook precondition
(cleanHands) effect (dinner)) (operator
wrap precondition (paper) effect
(present))
97
Planning Graph
noGarb cleanH paper dinner
present
cleanH paper
carry fire cook wrap
98
Are there any exclusions?
noGarb cleanH paper dinner
present
cleanH paper
carry noop fire noop cook wrap
99
Do we have a solution?
noGarb cleanH paper dinner
present
cleanH paper
carry noop fire noop cook wrap
100
Extend the Planning Graph
noGarb cleanH paper dinner
present
cleanH paper
noGarb cleanH paper dinner
present
noop carry noop fire noop cook noop wrap noop
carry noop fire noop cook wrap
101
One (of 4) Solutions
noGarb cleanH paper dinner
present
cleanH paper
noGarb cleanH paper dinner
present
noop carry noop fire noop cook noop wrap noop
carry noop fire noop cook wrap
102
Search Algorithms

• Backtrack Search
• DFS
• BFS / Dijkstras Algorithm
• Iterative Deepening
• Best-first search
• A
• Constraint Propagation
• Forward Checking
• k-Consistency
• DPLL Resolution
• Local Search
• Hillclimbing
• Simulated annealing
• Walksat

103
Representing Knowledge in Propositional Logic

• RN Chapter 7

104
Basic Idea of Logic

• By starting with true assumptions, you can deduce
  true conclusions.

105
Truth

• Francis Bacon (1561-1626)
• No pleasure is comparable to the standing upon
  the vantage-ground of truth.
• Thomas Henry Huxley (1825-1895)
• Irrationally held truths may be more harmful than
  reasoned errors.
• John Keats (1795-1821)
• Beauty is truth, truth beauty that is all
• Ye know on earth, and all ye need to know.

• Blaise Pascal (1623-1662)
• We know the truth, not only by the reason, but
  also by the heart.
• François Rabelais (c. 1490-1553)
• Speak the truth and shame the Devil.
• Daniel Webster (1782-1852)
• There is nothing so powerful as truth, and often
  nothing so strange.

106
Propositional Logic

• Ingredients of a sentence
• Propositions (variables)
• Logical Connectives ?, ?, ?, ?
• literal a variable or a negated variable
• A possible world assigns every proposition the
  value true or false
• A truth value for a sentence can be derived from
  the truth value of its propositions by using the
  truth tables of the connectives
• The meaning of a sentence is the set of possible
  worlds in which it is true

107
Truth Tables for Connectives
?
?
0 0
0 1
1 0
1 1
0
1
?
?
0 0
0 1
1 0
1 1
0 0
0 1
1 0
1 1
108
Special Syntactic Forms

• General PL
• ((q?? r) ? s)) ? ? (s ? t)
• Conjunction Normal Form (CNF)
• (? q ? r ? s ) ? (? s ? ? t)
• Set notation (? q, r, s ), (? s, ? t)
• empty clause () false
• Binary clauses 1 or 2 literals per clause
• (? q ? r) (? s ? ? t)
• Horn clauses 0 or 1 positive literal per clause
• (? q ? ? r ? s ) (? s ? ? t)
• (q?r) ? s (s?t) ? false

109
Satisfiability, Validity, Entailment

• S is satisfiable if it is true in some world
• Example
• S is unsatisfiable if it is false all worlds
• S is valid if it is true in all worlds
• S1 entails S2 if wherever S1 is true S2 is true

110
Reasoning Tasks

• Model finding
• KB background knowledge
• S description of problem
• Show (KB ? S) is satisfiable
• A kind of constraint satisfaction
• Deduction
• S question
• Prove that KB ? S
• Two approaches
• Rules to derive new formulas from old (inference)
• Show (KB ? ? S) is unsatisfiable

111
Inference

• Mechanical process for computing new sentences
• Resolution
• (p ? ?), (? p ? ?) ?R (? ? ?)
• Correctness
• If S1 ?R S2 then S1 ? S2
• Refutation Completeness
• If S is unsatisfiable then S ?R ()

112
Resolution
If the unicorn is mythical, then it is immortal,
but if it is not mythical, it is a mammal. If
the unicorn is either immortal or a mammal, then
it is horned. Prove the unicorn is horned. M
mythical I immortal A mammal H horned
113
New Variable Trick

• Putting a formula in clausal form may increase
  its size exponentially
• But can avoid this by introducing dummy variables
• (a?b?c)?(d?e?f) ? (a?d),(a?e),(a?f),
• (b?d),(b?e),(b?f),
• (c?d),(c?e),(c?f)
• (a?b?c)?(d?e?f) ? (g?h),
• (?a??b??c?g),(?g?a),(?g?b),(?g?c),
• (?d??e??f?h),(?h?d),(?h?e),(?h?f)
• Dummy variables dont change satisfiability!

114
DPLLDavis Putnam Loveland Logmann

• Model finding Backtrack search over space of
  partial truth assignments

DPLL( wff ) Simplify wff for each unit
clause (Y) Remove clauses containing Y
if no clause left then return true
(satisfiable) Shorten clauses contain
? Y if empty clause then return false
Choose a variable Choose a value (0/1)
yields literal X if DPLL( wff, X ) return
true (satisfiable) else return DPLL(wff, ? X)
115
DPLLDavis Putnam Loveland Logemann

• Backtrack search over space of partial truth
  assignments

DPLL( wff ) Simplify wff for each unit
clause (Y) Remove clauses containing Y
if no clause left then return true
(satisfiable) Shorten clauses contain
? Y if empty clause then return false
Choose a variable Choose a value (0/1)
yields literal X if DPLL( wff, X ) return
true (satisfiable) else return DPLL(wff, ? X)
unit propagation arc consistency
116
Horn Theories

• Recall the special case of Horn clauses
• (? q ? ? r ? s ), (? s ? ? t)
• ((q?r) ? s ), ((s?t) ? false)
• Many problems naturally take the form of such
  if/then rules
• If (fever) AND (vomiting) then FLU
• Unit propagation is refutation complete for Horn
  theories
• Good implementation linear time!

117
DPLL

• Developed 1962 still the best complete
  algorithm for propositional reasoning
• State of the art solvers use
• Smart variable choice heuristics
• Clause learning at backtrack points,
  determine minimum set of choices that caused
  inconsistency, add new clause
• Limited resolution (Agarwal, Kautz, Beame 2002)
• Randomized tie breaking restarts
• Chaff fastest complete SAT solver
• Created by 2 Princeton undergrads, for a summer
  project!
• Superscaler processor verification
• AI planning - Blackbox

118
Exercise

• How could we represent the Quasigroup Completion
  Problem as a Boolean formula in CNF form?
• (take 10 minutes to sketch solution)

119
WalkSat

• Local search over space of complete truth
  assignments
• With probability P flip any variable in any
  unsat clause
• With probability (1-P) flip best variable in any
  unsat clause
• Like fixed-temperature simulated annealing
• SAT encodings of QCP, N-Queens, scheduling
• Best algorithm for random K-SAT
• Best DPLL 700 variables
• Walksat 100,000 variables

120
Random 3-SAT

• Random 3-SAT
• sample uniformly from space of all possible
  3-clauses
• n variables, l clauses
• Which are the hard instances?
• around l/n 4.3

121
Random 3-SAT

• Varying problem size, n
• Complexity peak appears to be largely invariant
  of algorithm
• backtracking algorithms like Davis-Putnam
• local search procedures like GSAT
• Whats so special about 4.3?

122
Random 3-SAT

• Complexity peak coincides with solubility
  transition
• l/n lt 4.3 problems under-constrained and SAT
• l/n gt 4.3 problems over-constrained and UNSAT
• l/n4.3, problems on knife-edge between SAT and
  UNSAT

123
Real-World Phase Transition Phenomena

• Many NP-hard problem distributions show phase
  transitions -
• job shop scheduling problems
• TSP instances from TSPLib
• exam timetables _at_ Edinburgh
• Boolean circuit synthesis
• Latin squares (alias sports scheduling)
• Hot research topic predicting hardness of a
  given instance, using hardness to control
  search strategy (Horvitz, Kautz, Ruan 2001-3)

124

• Logical Reasoning about Time Change
• AKA
• Planning as Satisfiability

Salvidor Dali, Persistance of Memory
125
Actions

• We want to relate changes in the world over time
  to actions associated with those changes
• How are actions represented?
• As functions from one state to another
• As predicates that are true in the state in which
  they (begin to) occur

126
Actions as Functions Situation Calculus

• On(cat, mat, S0)
• Happy(cat, S0)
• On(cat, mat, kick(S0))
• Happy(cat, kick(S0))
• Happy(cat, pet(kick(S0)))
• Branching time

kick
kick
pet
S0
kick
pet
pet
127
Actions as PredicatesAction Calculus

• On(cat, mat, S0) ? Happy(S0)
• Kick(cat, S0)
• On(cat, mat, S1) ? ?Happy(CAT99, S1)
• Pet(CAT99, S1)
• On(CAT99, MAT37, S2) ? Happy(CAT99, S2)
• Linear time

S0
S1
S2
kick
pet
128
Relating Actions to Preconditions Effects

• Strips notation
• Action Fly( plane, start, dest )
• Precondition Airplane(plane), City(start),
  City(dest), At(plane, start)
• Effect At(plane, dest), ? At(plane, start)
• Pure strips no negative preconditions!
• Need to represent logically
• An action requires its predications
• An action causes its effects
• Interfering actions do not co-occur
• Changes in the world are the result of actions.

129
Preconditions Effects

• ? plane, start, dest, s . Fly(plane, start, dest,
  s) ? At(plane, start, s) ?
  Airplane(plane,s) ? City(start) ? City(dest)
• Note state indexes on predicates that never
  change not necessary.
• ? plane, start, dest, s . Fly(plane, start, dest,
  s) ? At(plane, dest, s1)
• In action calculus, the logical representation of
  requires and causes is the same!
• Not a full blown theory of causation, but good
  enough

130
Interfering Actions

• Want to rule out
• Fly( PLANE32, NYC, DETROIT, S4) ? Fly( PLANE32,
  NYC, DETROIT, S4)
• Actions interfere if one changes a precondition
  or effect of the other
• They are mutually exclusive mutex
• ? p, c1, c2, c3, c4, s . Fly(p, c1, c2, s) ?
  (c1 ? c3 ? c2 ? c4) ? ? Fly(p, c3, c4, s)
• (Similar for any other actions Fly is mutex
  with)

131
Explanatory Axioms

• Dont want world to change by magic only
  actions change things
• If a proposition changes from true to false (or
  vice-versa), then some action that can change it
  must have occurred
• plane, start, s . Airplane(plane) ? City(start)
  At(plane,start,s) ? ?At(plane,city,s1) ?
  ? dest . City(dest) ? Fly(plane, start, dest,
  s)
• plane, dest, s . Airplane(plane) ? City(start)
  ?At(plane,dest,s) ? At(plane,dest,s1) ?
  ? start . City(start) ? Fly(plane, start, dest,
  s)

132
The Frame Problem

• General form of explanatory axioms
• p(s) ? ?p(s1) ? A1(s) ? A2(s) ? ? An(s)
  
• As a logical consequence, if none of these
  actions occurs, the proposition does not change
• ?A1(s) ? ?A2(s) ? ? ?An(s) ? p(s) ? p(s1)
  
• This solves the frame problem being able to
  deduce what does not change when an action occurs

133
Frame Problem in AI

• Frame problem identified by McCarthy in his first
  paper on the situation calculus (1969)
• 667 papers in researchindex !
• Lead to a (misguided?) 20 year effort to develop
  non-standard logics where no frame axioms are
  required (non-monotonic)
• 7039 papers!
• 1990 - Haas and Schubert independently pointed
  out that explanatory axioms are pretty easy to
  write down

134
Planning as Satisfiability

• Idea in action calculus assert that initial
  state holds at time 0 and goal holds at some time
  (in the future)
• Axioms ? Initial ? ? s . Goal(s)
• Any model that satisfies these assertions and the
  axioms for actions corresponds to a plan
• Bounded model finding, i.e. satisfiability
  testing
• Assert goal holds at a particular time K
• Ground out (instantiate) the theory up to time K
• Try to find a model if so, done!
• Otherwise, increment K and repeat

135
Reachability Analysis

• Problem many irrelevant propositions, large
  formulas
• Reachability analysis what propositions actually
  connect to initial state or goal in K steps?
• Graphplans plan graph computes reachable set!
• Blackbox (Kautz Selman 1999)
• Run graphplan to generate plan graph
• Translate plan graph to CNF formula
• Run any SAT solver

136
Translation of Plan Graph
Act1
Pre1
Fact
Pre2
Act2
Fact ? Act1 ? Act2 Act1 ? Pre1 ? Pre2 Act1 ?
Act2
137
Improved Encodings

• Translations of Logistics.a
• STRIPS ? Axiom Schemas ? SAT
• 3,510 variables, 16,168 clauses
• 24 hours to solve
• STRIPS ? Plan Graph ? SAT
• (Blackbox)
• 2,709 variables, 27,522 clauses
• 5 seconds to solve!

138
Model-Based Diagnosis

• Idea
• Create a logical model of the correct functioning
  of a device
• When device is broken, observations model is
  inconsistent
• Create diagnosis by restoring consistency

139
Simplified KB

• Knowledge Base
• SignalValueA ? ValveAok ? ValveAopen
• SignalValueB ? ValveBok ? ValveBopen
• SignalValueC ? ValveCok ? ValveCopen
• ValveAopen? ? EngineHasFuel
• ValveBopen? ? EngineHasFuel
• ValveCopen ? EngineHasOxy
• EngineHasFuel ? EngineHasOxy ? EngineFires
• Normal Assumptions
• ValveAok, ValveBok, ValveCok
• Direct Actions (cannot fail)
• SignalValveA, SignalValveB, SignalValveC
• Observed
• ? EngineFires

140
Diagnosis 1

• Knowledge Base
• SignalValueA ? ValveAok ? ValveAopen
• SignalValueB ? ValveBok ? ValveBopen
• SignalValueC ? ValveCok ? ValveCopen
• ValveAopen? ? EngineHasFuel
• ValveBopen? ? EngineHasFuel
• ValveCopen ? EngineHasOxy
• EngineHasFuel ? EngineHasOxy ? EngineFires
• Normal Assumptions
• ValveAok, ValveBok, ValveCok
• Direct Actions (cannot fail)
• SignalValveA, SignalValveB, SignalValveC
• Observed
• ? EngineFires

Inconsistent by Unit Propagation
141
Diagnosis 2

• Knowledge Base
• SignalValueA ? ValveAok ? ValveAopen
• SignalValueB ? ValveBok ? ValveBopen
• SignalValueC ? ValveCok ? ValveCopen
• ValveAopen? ? EngineHasFuel
• ValveBopen? ? EngineHasFuel
• ValveCopen ? EngineHasOxy
• EngineHasFuel ? EngineHasOxy ? EngineFires
• Normal Assumptions
• ValveAok, ValveBok, ValveCok
• Direct Actions (cannot fail)
• SignalValveA, SignalValveB, SignalValveC
• Observed
• ? EngineFires

Still Inconsistent
142
Diagnosis 3

• Knowledge Base
• SignalValueA ? ValveAok ? ValveAopen
• SignalValueB ? ValveBok ? ValveBopen
• SignalValueC ? ValveCok ? ValveCopen
• ValveAopen? ? EngineHasFuel
• ValveBopen? ? EngineHasFuel
• ValveCopen ? EngineHasOxy
• EngineHasFuel ? EngineHasOxy ? EngineFires
• Normal Assumptions
• ValveAok, ValveBok, ValveCok
• Direct Actions (cannot fail)
• SignalValveA, SignalValveB, SignalValveC
• Observed
• ? EngineFires

Consistency Restored! Diagnosis Valve A and
Valve B broken (double fault)
143
Diagnosis 4

• Knowledge Base
• SignalValueA ? ValveAok ? ValveAopen
• SignalValueB ? ValveBok ? ValveBopen
• SignalValueC ? ValveCok ? ValveCopen
• ValveAopen? ? EngineHasFuel
• ValveBopen? ? EngineHasFuel
• ValveCopen ? EngineHasOxy
• EngineHasFuel ? EngineHasOxy ? EngineFires
• Normal Assumptions
• ValveAok, ValveBok, ValveCok
• Direct Actions (cannot fail)
• SignalValveA, SignalValveB, SignalValveC
• Observed
• ? EngineFires

A different way to restore consistency Diagnosis
Valve C broken (single fault)
144

• Diagnostic Engine for NASAs Deep Space One
• 2,000 variable CNF formula
• Real-time planning and diagnosis

145
Beyond Logic

• Often you want most likely diagnosis rather than
  all possible diagnoses
• Can assign probabilities to sets of fault, and
  search for most likely way to restore consistency
• But suppose observations and model of the device
  are also uncertain?
• Next Probabilistic Reasoning in Bayesian Networks