info 372: explorations in artificial intelligence prof. carla p. gomes [email protected] module 2...
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INFO 372:Explorations in Artificial Intelligence
Prof. Carla P. [email protected]
Module 2Example Applications
Previous Lecture:What's involved in Intelligence?
Previous Lecture:What's involved in Intelligence?
A) Ability to interact with the real world to perceive, understand, and act
speech recognition and understanding
image understanding (computer vision)
B) Reasoning and Planningmodelling the external world
problem solving, planning, and decision making
ability to deal with unexpected problems, uncertainties
C) Learning and Adaptation We are continuously learning and adapting.
We want systems that adapt to us!
INFO 372
Previous Lecture
• AI goals:– understand “intelligent” behavior– build “intelligent” agents
• Intelligence not necessarily human like
Issue: The HardwareIssue: The Hardware
• The brain– a neuron, or nerve cell, is the basic information
– processing unit (10^11 )
– many more synapses (10^14) connect the neurons
– cycle time: 10^(-3) seconds (1 millisecond)
• How complex can we make computers?– 10^8 or more transistors per CPU
– supercomputer: hundreds of CPUs, 10^10 bits of RAM
– cycle times: order of 10^(-9) seconds (1 nanosecond)
• Conclusion– In near future we can have computers with as many
processing elements as our brain, but:
far fewer interconnections (wires or synapses)
much faster updates.
Fundamentally different hardware may require fundamentally different algorithms!– Very much an open question.
Previous Lecture
• AI goals:– understand “intelligent” behavior– build “intelligent” agents
• Intelligence not necessarily human like
Focus of INFO372 (most recent progress).
Developing methods to match or exceed human performance in certain domains, possibly by very different
means e.g., Deep Blue;
1997:Deep Blue beats the World Chess Champion
I could feel human-level intelligence across the room -Gary Kasparov, World Chess Champion (human…)
vs.
How Intelligent is Deep Blue?How Intelligent is Deep Blue?
• Saying Deep Blue doesn't really think about chess is like saying an airplane doesn't really fly because it doesn't flap its wings.
- Drew McDermott
On Game 2On Game 2
(Game 2 - Deep Blue took an early lead.
Kasparov resigned, but it turned out he could
have forced a draw by perpetual check.)
This was real chess. This was a game any human
grandmaster would have been proud of.
Joel Benjamin grandmaster, member Deep Blue team
Kasparov on Deep BlueKasparov on Deep Blue
• 1996: Kasparov Beats Deep Blue
“I could feel --- I could smell --- a new kind
of intelligence across the table.”
• 1997: Deep Blue Beats Kasparov
“Deep Blue hasn't proven anything.”
Game Tree SearchGame Tree Search
• How to search a game tree was independently invented by Shannon (1950) and Turing (1951).
• Technique called: MiniMax search.
• Evaluation function combines material & position.
History of Search InnovationsHistory of Search Innovations
•Shannon, Turing Minimax search 1950•Kotok/McCarthy Alpha-beta pruning1966•MacHack Transposition tables1967•Chess 3.0+ Iterative-deepening1975•Belle Special hardware 1978•Cray Blitz Parallel search 1983•Hitech Parallel evaluation 1985•Deep Blue All of the above 1997
Transposition TablesTransposition Tables
• Introduced by Greenblat's Mac Hack (1966)
• Basic idea: caching– once a board is evaluated, save it in a hash table, avoid re-
evaluating.
– called “transposition” tables, because different orderings (transpositions) of the same set of moves can lead to the same board.
– Form of root learning (memorization)
– Don’t repeat blunders can’t beat the computer twice in a row using same moves
Deep Blue --- huge transposition tables (100,000,000+),
must be carefully managed.
Positions with Smart PruningPositions with Smart Pruning
Search Depth Positions
2 604 2,0006 60,0008 2,000,00010 (<1 second DB) 60,000,00012 2,000,000,00014 (5 minutes DB) 60,000,000,00016 2,000,000,000,000
How many lines of play does a grand master consider?
Around 5 to 7
Special-Purpose and Parallel HardwareSpecial-Purpose and Parallel Hardware
• Belle (Thompson 1978)• Cray Blitz (1993)• Hitech (1985)• Deep Blue (1987-1996)
– Parallel evaluation: allows more complicated evaluation functions
– Hardest part: coordinating parallel search– Deep Blue never quite plays the same game, because
of “noise” in its hardware!
Deep BlueDeep Blue
• Hardware– 32 general processors– 220 VSLI chess chips
• Overall: 200,000,000 positions per second– 5 minutes = depth 14
• Selective extensions - search deeper at unstable positions– down to depth 25 !
Tactics into StrategyTactics into Strategy
• As Deep Blue goes deeper and deeper into a position, it displays elements of strategic understanding. Somewhere out there mere tactics translate into strategy. This is the closet thing I've ever seen to computer intelligence. It's a very weird form of intelligence, but you can feel it. It feels like thinking.– Frederick Friedel (grandmaster), Newsday, May 9, 1997
Goals of INFO 372
Introduce the students to a range of computational modeling approaches and solution strategies using examples from AI and Information Science.
Formalisms:Logical representations;Constraint-based languages, Mathematical programming – Linear and Integer programming;Multi-agent formalisms (including adversarial games);
Solution strategies: Logical inference;General complete backtrack search; (e.g., Iterative Deepening)Local search;Dynamic Programming;Game tree search (e.g., alpha-beta pruning)
Goals of INFO 372
Special models: Satisfiability (SAT); Maximum SAT; Horn
Constraint Satisfaction; Binary Constraint Satisfaction;
Mixed Integer Programming, Linear Programming and
Network Flow Models;Themes:
Expressiveness and efficiency tradeoffs of the various representation formalisms
Students learn about the tradeoffs in modeling choices.;Concrete examples to move from one representation modeling formalism to another formalism;
How do we Interpret the Scenes in Escher’s Worlds?
Analysis of Polyhedral Scenesorigins of Constraint Reasoning
researchers in computer vision in the 60s-70s were
interested in developing a procedure to assign 3-
dimensional interpretations to scenes;
They identified
Three types of edgesFour types of junctions
Edge Types
Hidden – if one of its planes cannot be seen
represented with arrows:
or Convex – from the viewer’s perspective
represented with
+
Concave – from the viewer’s perspective
represented with
-
Huffman-ClowesLabeling
Scene InterpretationConstraint Reasoning Problem:
Variables Edges;
Domains {+,-,,}
Constraints:
1- The different type junctions define constraints:
L, Fork, T, Arrow;
L = {(, ) , ( , ), (+, ), (,+), (-, ), (,-)}
Fork = { (+,+,+), (-,-,-), (,,-), (,-,),(-,,)}
L(A,B) the pair of values assigned to variables A,B
has to belong in the set L;
Fork(A,B,C) the trio of values assigned to variables A,B,C
has to belong in the set Fork;
CSP Model
• T = {(, , ) , ( ,,), (,,+), (,-)}
• Arrow = { (,,+), (+,+,-), (-,-,+)}
T(A,B,C) the trio of values assigned to variables A,B,C
has to belong in the set T;
Arrow(A,B,C) the trio of values assigned to variables A,B,C
has to belong in the set Arrow;
2- For each edge XY its reverse YX has a compatible value
Edge = { +,+), (-,-), (,),(,)}
Edge(A,B) the pair of values assigned to variables A,B
has to belong in the set Edge;
CSP Model
Variables: Edges: AB, BA,AC,CA,AE,EA,CD,
DC,BD,DB,DG,GD,GF,FG,EF,FE,AE,EA;
Domains {+,-,,}
Constraints:
L(AC,CD); L(AE,EF); L(DG,GF);
Arrow(AC,AE,AB); Arrow(EF,FG,BF); Arrow(CD,DG,DB);
Fork(AB,BF,BD);
Edge(AB,BA); Edge(AC,CA); Edge(AE,EA);
Edge(EF,FE); Edge(BF,FB); Edge(FG,GF);
Edge(CD,DC); Edge(BD,DB); Edge(DG,GD);
A B
C D
E F
G
CSP Model
Variables: Edges: AB, BA,AC,CA,AE,EA,CD,
DC,BD,DB,DG,GD,GF,FG,EF,FE,AE,EA;
Domains {+,-,,}
Constraints:
L(AC,CD); L(AE,EF); L(DG,GF);
Arrow(AC,AE,AB); Arrow(EF,FG,BF); Arrow(CD,DG,DB);
Fork(AB,BF,BD);
Edge(AB,BA); Edge(AC,CA); Edge(AE,EA);
Edge(EF,FE); Edge(BF,FB); Edge(FG,GF);
Edge(CD,DC); Edge(BD,DB); Edge(DG,GD);
A B
C D
E F
G+
+
+
One (out of four) possible labelings
SudokuSudoku
9 55 ~ 3x 10 52 possible completionsConstraint Satisfaction Problem (CSP) and Satisfiability and Integer Programming
Given an N X N matrix, and given N colors, a Latin Square of order N is a a colored matrix, such that:
-all cells are colored.
- each color occurs exactly once in each row.
- each color occurs exactly once in each column.
Quasigroup or Latin Square(Order 4)
Constraint Satisfaction Problem (CSP) and Satisfiability and Integer Programming
Latin Squares
Given an N X N matrix, and given N colors, a Latin Square of order N is a a colored matrix, such that:
-all cells are colored
-a color is not repeated in a row
-a color is not repeated in a column
Quasigroup or Latin Square
(Order 4)Constraint Satisfaction Problem (CSP) and Satisfiability and Integer Programming
Latin Squares
Latin Square Completion ProblemLatin Square Completion Problem
Given a partial assignment of colors (10 colors in this case), can the partial latin square be completed so we obtain a full Latin square?
Example:
32% preassignment 10 68 possible completions
Fiber Optic Networks
Nodes are capable of photonic switching --dynamic wavelength routing --
which involves the setting of the wavelengths.
Wavelength
Division
Multiplexing (WDM)
the most promising
technology for the
next generation of
wide-area
backbone networks.
Nodesconnect point to point
fiber optic links
Each fiber optic link supports alarge number of wavelengths
Routing in Fiber Optic Networks
Routing Node
How can we achieve conflict-free routing in each node of the network?
Dynamic wavelength routing is a NP-hard problem.
Input Ports Output Ports1
2
3
4
1
2
3
4
preassigned channels
LSCP Application Example: Routers in Fiber Optic Networks
LSCP Application Example: Routers in Fiber Optic Networks
Dynamic wavelength routing in Fiber Optic Networks can be directly mapped into the Latin Square Completion Problem.
•each channel cannot be repeated in the same input port (row constraints);
• each channel cannot be repeated in the same output port (column constraints);
CONFLICT FREELATIN ROUTER
Inp
ut
po
rts
Output ports
3
1
2
4
Input Port Output Port
1
2
43
Design of Statistical ExperimentsDesign of Statistical Experiments
We have 5 treatments for growing beans. We want to know what treatments are effective in increasing yield, and by how much.
The objective is to eliminate bias and distribute the treatments somewhat evenly over the test plot.
Latin Square Analysis of Variance
A D E BB C
C B A E D
D C BB A E
E A C D B
B E D C A
(*) Already in use (*) Already in use in this sub-plotin this sub-plot
Timetabling: Constraint Satisfaction Problem (CSP) and
Integer Programming
An 8 Team Round Robin Timetable
Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7Period 1 0 vs 1 0 vs 2 4 vs 7 3 vs 6 3 vs 7 1 vs 5 2 vs 4
Period 2 2 vs 3 1 vs 7 0 vs 3 5 vs 7 1 vs 4 0 vs 6 5 vs 6
Period 3 4 vs 5 3 vs 5 1 vs 6 0 vs 4 2 vs 6 2 vs 7 0 vs 7
Period 4 6 vs 7 4 vs 6 2 vs 5 1 vs 2 0 vs 5 3 vs 4 1 vs 3
The problem of generating schedules with complex constraints (in this case for sports teams).
28 28 ~ 3.3 x 1040 possibilities
Why Sports Scheduling???
Big Business!US National TV pays $500 million / year for baseballCollege basketball conferences get up to $30 millionManchester United has (had) a market cap of £400 million
No rights holder wants to pay those sums and then get a “bad” schedule Difficult to automate: Huge variety of problem typesSmall instances are difficult
Strong break between easy/hard (for all algorithms)Significant theoretical backgroundCP and IP differ in modeling
CP has clean models with [1..n] variables IP uses 0-1 variables reasonably naturally
Practical interest in instances at the easy/hard interfaceSource:Mike Trick
Graph Coloring
Coloring the nodes of the graph:What’s the minimum number of colors such that any two nodes
connected by an edge have different colors?
nn ~ possible colorings for n nodes
Another example of a reasoning formalism
A restricted form of Constraint Satisfaction:
Satisfiability
Propositional Satisfiability problem
Satifiability (SAT): Given a formula in propositional calculus, is there
an assignment to its variables making it true?
We consider clausal form, e.g.:
( a OR NOT b OR NOT c ) AND ( b OR NOT c) AND ( a OR c)n2
possible assignments
SAT: prototypical hard combinatorial search and reasoning problem. Problem is NP-Complete. (Cook 1971)
Surprising “power” of SAT for encoding computational problems.
Significant progress in Satisfiability Methods
Software and hardware verification – complete methods are critical - e.g. for verifying the correctness of chip design, using SAT encodings
Current methods can verify automatically the correctness of > 1/7 of a Pentium IV.
Going from 50 variable, 200 constraints to 1,000,000 variables and 5,000,000 constraints in the last 10 years
Applications: Hardware and
Software Verification Planning,
Protocol Design, etc.
(x177 or x169 or x161 or x153 … or x17 or x9 or x1 or (not x185))
clauses / constraints are getting more interesting…
10 pages later:
…
Finally, 15,000 pages later:
The Chaff SAT solver solves this instance in less than one minute.
Note that: … !!!
Knapsack Problem (one resource)
A hiker trying to fill her knapsack to maximum total value. Each item she considers taking with
her has a certain value and a certain weight. Goal – maximize the value of the contents of the
knapsack considering the overall weight constraint.
• This problem is an abstraction with many practical applications:
Project selection and capital budgeting allocation problems
Storing a warehouse to maximum value given the indivisibility of goods and space limitations
Sub-problem of other problems e.g., generation of columns for a given model in the course of optimization – cutting stock problem (beyond the scope of this course)
Investment 1 2 3 4 5 6
Cash Required (1000s)
$5
$7
$4
$3
$4
$6
NPV added (1000s)
$16
$22
$12
$8
$11
$19
Capital Budgeting Example
Investment budget = $14,000
maximize 16x1 + 22x2 + 12x3 + 8x4 +11x5 + 19x6
subject to 5x1 + 7x2 + 4x3 + 3x4 +4x5 + 6x6 14
xj binary for j = 1 to 6
Binary Optimization: Applications in Regional
Planning
I
Mxw
xc
I
iii
I
iii
,...1i and }1,0{x
Subject to
Maximize
i
1
1
Stream FootagePhosphorousPathogenParcel SizeParcel ValueBudget Constraint
Riparian Buffer in the Skaneateles Lake Watershed
Town of Skaneateles:-1834 parcels-12341 acres52 land use class.
Preservation in NY State
2,345 barns registered in year 2000464 barns in Finger Lakes Region only.
Contribution to a scenic landscape or agricultural
setting Historic significance
Budget: $2 million; Max of $25,000 grant per barn
Office of Parks, Recreation and Historic Preservation Unique Natural Areas in
Tompkins County
Important Natural CommunityGeological ImportanceAesthetic/Cultural QualitiesBudget Constraint
ModelsKnapsack and VariantsZevi Azzaino
Jon ConradCarla Gomes
Objective: Identify the best collection of parcels to include
in a riparian buffer subject to a budget constraint
Southwestern Airways Crew Scheduling
• Southwestern Airways needs to assign crews to cover all its upcoming flights.
• Simple example assigning 3 crews based in San Francisco (SFO) to 11 flights.
Question: How should the 3 crews be assigned 3 sequences of flights so that every one of the 11 flights is covered?
Southwestern Airways FlightsSeat tl e (SEA)
San Francisco (SFO)
Los Angel es (LAX)
Denver (DEN)
Chicago ORD)
Data for the Southwestern Airways Problem
Feasible Sequence of Flights (pairings)
Flights 1 2 3 4 5 6 7 8 9 10 11 12
1. SFO–LAX 1 1 1 1
2. SFO–DEN 1 1 1 1
3. SFO–SEA 1 1 1 1
4. LAX–ORD 2 2 3 2 3
5. LAX–SFO 2 3 5 5
6. ORD–DEN 3 3 4
7. ORD–SEA 3 3 3 3 4
8. DEN–SFO 2 4 4 5
9. DEN–ORD 2 2 2
10. SEA–SFO 2 4 4 5
11. SEA–LAX 2 2 4 4 2
Cost, $1,000s 2 3 4 6 7 5 7 8 9 9 8 9
Algebraic Formulation
Let xj = 1 if flight sequence (paring) j is assigned to a crew; 0 otherwise. (j = 1, 2, … , 12).
Minimize Cost = 2x1 + 3x2 + 4x3 + 6x4 + 7x5 + 5x6 + 7x7 + 8x8 + 9x9 + 9x10 + 8x11 + 9x12
(in $thousands)
subject to
Flight 1 covered: x1 + x4 + x7 + x10 ≥ 1
Flight 2 covered: x2 + x5 + x8 + x11 ≥ 1
: :
Flight 11 covered: x6 + x9 + x10 + x11 + x12 ≥ 1
Three Crews: x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 ≤ 3
and
xj are binary (j = 1, 2, … , 12).
pairings
• Many computational tasks, such as planning or scheduling, can in principle be reduced to an exploration of a large set of all possible scenarios.
• Try all possible schedules, try all possible plans, pick the best.
Combinatorial Problems
Problem: combinatorial explosion!
AI PLANNING
In AI, planning involves the generation of an actionplan (i.e. a sequence of actions) for an agent, such as a robot ora software system or a living artefact, that can alter its surroundings.
Planning implies the notion of synthesis: synthesis of actions, to go from an initial state to a goal state.Examples:
•plan to perform astronomical observations for the Hubble space telescope;
•plan for a robot to assemble pieces in a factory
Planning Example: Blocks world
• objects: blocks and a table• actions: move blocks ‘on’ one object to ‘on’
another object• goals: configurations of blocks• plan: sequence of actions to achieve goals
TA B C
D
Initial State
A
B
C
D
Goal State
Blocks world: propositional and first order logic representation
Knowledge Base:On(A,T)^On(B,T)^On(C,T)^On(D,C)^Block(A)^Block(B)^Block(C)^Block(D) )^Table(T)^Clear(A)^Clear(B)^Clear(D)
TA B C
D
KB:
On(A,D)^On(B,T)^On(C,T)^On(D,C)
^Block(A)^Block(B)^Block(C)^Block(D)^Table(T)
^Clear(A) ^Clear(B)T
A
B C
D
Move(A,T,D)
Planning ComplexityPlanning (single-agent): find the right sequence of actions
HARD: 10 actions, 10! = 3 x 106 possible plans
REALLY HARD: 10 x 92 x 84 x 78 x … x 2256 = 10224 possible contingency plans!
Contingency planning (multi-agent): actions may or may not produce the desired effect!
…1 outof 10
2 outof 9
4 outof 8
100 ! = 9.33262154 × 10157
Networks are Everywhere
Physical Networks
Road Networks
Railway Networks
Airline traffic Networks
Electrical networks, e.g., the power grid
Computer networks
New areas of application
Social networks - e.g. relationships networks (6 degrees of Kevin Bacon ); communities such as researchers, CEO’s etc
Economic/ technological networks – e.g. patents:
ApplicationsPhysical analog
of nodes Physical analog
of arcsFlow
Communicationsystems
phone exchanges, computers, transmission
facilities, satellites
Cables, fiber optic links, microwave
relay links
Voice messages, Data,
Video transmissions
Hydraulic systemsPumping stationsReservoirs, Lakes
PipelinesWater, Gas, Oil,Hydraulic fluids
Integrated computer circuits
Gates, registers,processors
Wires Electrical current
Mechanical systems JointsRods, Beams,
SpringsHeat, Energy
Transportationsystems
Intersections, Airports,
Rail yards
Highways,Airline routes
Railbeds
Passengers, freight,
vehicles, operators
Applications of Network Optimization
• “Nice” combinatorial problem (Min Cost Flow) – exception to combinatorial explosition polynomial scaling !
• General formulation for special problems:– shortest paths
– transportation problem
– assignment problem
– plus more
• Important subproblem of many optimization problems, including multicommodity flows
Network Flow Algorithms:
EXPONENTIAL FUNCTION
POLYNOMIAL FUNCTIONHard Computational
ProblemsScale Exponentially
EXPONENTIAL-TIMEALGORITHMS
EXPLOSIVECOMBINATORICS
ExperimentDesignGoal
Start
Software & HardwareVerification
Satisfiability
(A or B) (D or E or not A)
Data Analysis& Data Mining
Fiber optics routing
Capital BudgetingAnd Financial Appl. Information
Retrieval
Protein Folding
And Medical ApplicationsCombinatorial
Auctions
Planning and SchedulingAnd Supply Chain Management
Many more applications!!!
Require powerful computational and
mathematical tools!
NP-Complete andNP-Hard Problems
But most interesting real-world problems are:
Goals of INFO 372
Introduce the students to a range of computational modeling approaches and solution strategies using examples from AI and Information Science.
Formalisms:Logical representations;Constraint-based languages, Mathematical programming – Linear and Integer programming;Multi-agent formalisms (including adversarial games);
Solution strategies: Logical inference;General complete backtrack search; (e.g., Iterative Deepening)Local search;Dynamic Programming;Game tree search (e.g., alpha-beta pruning)
Goals of INFO 372
Special models: Satisfiability (SAT); Maximum SAT; Horn
Constraint Satisfaction; Binary Constraint Satisfaction;
Mixed Integer Programming, Linear Programming and
Network Flow Models;Themes:
Expressiveness and efficiency tradeoffs of the various representation formalisms
Students learn about the tradeoffs in modeling choices.;Concrete examples to move from one representation modeling formalism to another formalism;