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CS-INFO 372:Explorations in Artificial Intelligence
Prof. Carla P. [email protected]
Module 2Examples of Different Modeling Formalisms
http://www.cs.cornell.edu/courses/cs372/2008sp
Example of a reasoning formalism:
Constraint Satisfaction Problems
Escher:Waterfall, 1961
Escher:Belvedere, May 1958
Escher:Ascending and Descending, 1960
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:
Convex – from the viewer’s perspective
represented with
+
Concave – from the viewer’s perspective
represented with
-
Huffman-ClowesLabeling
Types of Junctions
Type of junction: L Fork T Arrow
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;
Constraint Satisfaction Problem (CSP)
• 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 - CubeCSP Model - Cube
A B
C D
E F
G
How to label the cube?
Constraint Satisfaction Problem (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(upper right corner)
The Impossible Objects is Escher’s Worlds
Penrose & Penrose Stairs
Penrose Triangle
Impossible Objects:No labeling!
Other examples using a Constraint Satisfaction formalism
SudokuSudoku
9 55 ~ 3x 10 52 possible completionsConstraint Satisfaction Problem (CSP) (but also 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.
Nodesconnect point to point
fiber optic links
Each fiber optic link supports alarge number of wavelengths
Wavelength
Division
Multiplexing (WDM)
the most promising
technology for the
next generation of
wide-area
backbone networks.
Routing in Fiber Optic Networks
Routing Node
How can we achieve conflict-free routing in each node of the network?
Dynamic wavelength routing is an 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
Spatially Balanced Latin Squares
Really hard to build balanced LS’s
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
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
Graph Coloring
Constraint Satisfaction Problem (CSP) and Satisfiability and Integer Programming
Graph coloring formulations can be used to solve different problems.
Can we color agraph such that no two nodesconnected by an edge have the same color?
CSP:Variables Nodes Domains ColorsConstraints Edges
Scheduling of Final Exams
How can the final exams at Cornell be scheduled so that no student has
two exams at the same time? (Note not obvious this has anything to do
with graphs or graph coloring.)
Graph:A vertex correspond to a course.An edge between two vertices denotes that there is at least one common student in the courses they represent.Each time slot for a final exam is represented by a different color.
A coloring of the graph corresponds to a valid schedule of the exams.
1
7 2
36
5 4
Scheduling of Final Exams
1
7 2
36
5 4
What are the constraints between courses?Find a valid coloring
1
7 2
36
5 4
TimePeriod
IIIIIIIV
Courses
1,62
3,54,7
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)
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.
A “real world” example
i.e. ((not x1) or x7) and ((not x1) or x6)
and … etc.
Bounded Model Checking instance:
(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:
…
4000 pages later:
…
!!!!!!a 59-cnf clause…
Finally, 15,000 pages later:
The Chaff SAT solver solves this instance in less than one minute.
Note that: … !!!
Another example of a reasoning formalism
Integer Programming
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
Combinatorial Problems
• 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!
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
• “Nice” combinatorial problem (Shortest Path) – 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
Nice Problems !
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;