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Title: Boolean Functions I


1
Boolean Functions Ive Known and Not Known
Memories of Peter L. Hammer
Fred Roberts Rutgers University
2
Peter Hammer, RUTCOR, Boolean Functions
  • Topics of the Talk
  • Peters influence
  • A little about the history of RUTCOR
  • Some work on Boolean functions that relates to
    Peters interests

3
How I Met Peter
  • I met Peter in 1977 when I heard him give a talk
    about threshold graphs.
  • I think it was at the Southeastern conference in
    Boca Raton, Florida.

4
How I Met Peter
  • Little did I know then what influence he would
    have on my life.
  • RUTCOR didnt exist.
  • Peter was at Waterloo.
  • There was no Operations Research at Rutgers.
  • I was supervising my first Ph.D. student.
  • Endre Boros was still working on his Masters
    Thesis On Sperner Spaces.
  • Barack Obama was in high school.

5
Threshold Graphs
  • My first meeting with Peter led to a thesis topic
    for my first student, Shelly Leibowitz.
  • Let G be a graph with n vertices v1, v2, , vn.
  • G is a threshold graph iff we can associate a
    weight w(vi) with each vertex and a threshold t
    so that for all sets S of vertices
  • S is an independent set ? ?w(vi) vi ? S t

6
Threshold Graphs
  • If S is a set of vertices, associate with it a
    Boolean vector so that vi is in S iff the ith
    entry is 1.
  • Equivalently, G is a threshold graph iff there is
    a hyperplane that separates the characteristic
    vectors of independent sets of vertices from
    those of non-independent sets.
  • Chvatal and Hammer (1978) characterized threshold
    graphs.
  • That caught my attention.

7
Threshold Boolean Functions
  • Suppose f is a Boolean function assigning each
    0-1 vector (x1,x2,,xn) to 0 or 1.
  • f is a threshold Boolean function if there exist
    weights wi and a threshold t so that
  • f(x1,x2,,xn) 1 ? ?i wixi t
  • Thus, the Boolean function that assigns 1 to
    characteristic vectors of independent sets is a
    threshold Boolean function.

8
Sample Leibowitz Result Guttman Scaling
  • A famous study of soldiers physical reactions to
    battle during WW II (Suchman, reported in
    Stouffer, et al. 1950).
  • Which of the following reactions have you had?
  • Violent pounding of the heart
  • Sinking feeling of the stomach
  • Feeling of weakness or feeling faint
  • Feeling sick to the stomach
  • Cold sweat
  • Shaking or trembling all over
  • Losing bladder control

9
Sample Leibowitz Result Guttman Scaling
  • Which of the following reactions have you had?
  • Violent pounding of the heart
  • Sinking feeling of the stomach
  • Feeling of weakness or feeling faint
  • Feeling sick to the stomach
  • Cold sweat
  • Shaking or trembling all over
  • Losing bladder control
  • Suchman The reactions could be ordered so that
    if a soldier had one of them, he had all those
    below it on the list.

10
Guttman Scaling
  • This is an example of Guttman scaling
  • (after the psychologist Louis Guttman).
  • Set A of subjects and set X of items.
  • Can we order the set A?X so that a subject agrees
    with all items preceding it and disagrees with
    all items following it?
  • Useful in education students and test items
  • Useful in opinion scaling individuals and
    opinions
  • So what does this have to do with threshold
    graphs?

11
Guttman Scaling
  • Build a graph G.
  • Vertex set is A?X.
  • Edge between a in A and x in X if subject a
    agrees with item x.
  • Edge between every element of A.
  • Leibowitz observed Agreement defines a Guttman
    scale iff G is a threshold graph.
  • This led to a large number of results about
    Guttman scales, sequences, threshold graphs, etc.

12
Recruiting Peter to Rutgers
  • It was not threshold graphs that led us to
    recruit Peter to Rutgers.

13
Recruiting Peter to Rutgers
  • 1980-81, 1981-82 There was no O.R. at Rutgers,
    but there were courses in networks, graph theory,
    linear optimization, etc.
  • A group of us decided we needed a graduate
    program in O.R.
  • We decided we needed a leader to create and
    manage such a program.
  • We convinced Dean of the Graduate School, Ken
    Wolfson

14
The Recruitment
  • Search Committee (with 80 confidence)
  • Fred Roberts, Math
  • Bob Vichnevetsky, CS
  • Bill Strawderman, Stat
  • Mike Grigoriadis, CS

Bob Vichnevetsky
Bill Strawderman
Mike Grigoriadis?
15
The Recruitment
  • The recruitment encompassed numerous locations.

16
The Recruitment
  • The recruitment encompassed numerous locations.

LaGuardia Airport
17
The Recruitment
  • The recruitment encompassed numerous locations.

18
RUTCOR
  • We wanted a graduate program.
  • Peter had a much grander vision A CENTER FOR
    OPERATIONS RESEARCH
  • RUTCOR was born.

19
RUTCOR
  • How do you build a center?

20
RUTCOR Building a Center
  • Space
  • Hill Center for the Mathematical Sciences

21
RUTCOR Building a Center
  • Space A Building
  • The true story of how the RUTCOR building came to
    have a deck

22
RUTCOR Building a Center
  • Courses
  • Borrowing courses from Math, CS, Stat,
    Industrial Engineering, etc.
  • Linear Programming from CS Dept.
  • Networks and Combinatorial Optimization from CS
    Dept.
  • Design and Analysis of Data Structures and
    Algorithms from Stat Dept.
  • Etc.

23
RUTCOR Building a Center
  • Graduate Students
  • The key is to recruit bright, hard-working
    graduate students

24
RUTCOR Building a Center
  • Graduate Students
  • The key is to recruit good students
  • Peters connections in Europe and elsewhere were
    critical.

25
RUTCOR Building a Center
  • Graduate Students
  • Peters mode of operation from the beginning was
    to work closely with students and integrate them
    in the work of RUTCOR.
  • And encourage them to work hard.

26
RUTCOR Building a Center
  • Graduate Students How to Support Them
  • GA lines from others.
  • Math Dept.
  • Grad School
  • Business School
  • We started a grad program with two students in
    year 1

27
RUTCOR Building a Center
  • How are we Going to Pay for a Center?
  • Surely the publisher will support RUTCOR
  • Oh, perhaps thats not enough. I think Ill
    create a journal.

28
RUTCOR Building a Center
  • Recruiting a Faculty

29
RUTCOR Building a Center
  • How to Run a Center from Canada
  • Agree to come to Rutgers, set up the center, but
    work on RUTCOR from Waterloo.
  • Ask someone to be director during the first year.
  • Fred Roberts as Director of RUTCOR during
    1982-83.
  • Commute to Rutgers a few times each semester.
  • Work hard from a distance to
  • recruit students
  • plan for courses
  • raise funds
  • develop plans for faculty

30
RUTCOR Building a Center
  • How to Run a Center Committees
  • Universities need committees
  • Executive Committee of RUTCOR
  • Involve faculty from other departments
  • RUTCOR fellows

31
RUTCOR Building a Center
  • How to Run a Center Voting Rules
  • Many discussions to develop voting rules for
    RUTCOR.
  • Peters research also touched upon voting rules.
  • Boolean functions arise in voting.
  • I dont know if the research on Boolean functions
    influenced the voting rules of RUTCOR.

32
RUTCOR Building a Center
  • How to Run a Center Voting Rules
  • What do Boolean functions have to do with voting?

33
Power of a Voter
  • Think of a voting game
  • Every coalition (subset of the players) is either
    strong enough to win or not.
  • Represent a subset of the players as a 0-1 vector
    with the ith entry 1 iff player i is in the
    subset.
  • Then the voting game corresponds to a Boolean
    function that assigns 1 to vectors corresponding
    to winning subsets (winning coalitions) and 0
    to losing subsets.
  • If a winning coalition can never be contained in
    a losing one, we say we have a simple game.

34
Power of a Voter
  • Shapley-Shubik Power Index
  • Think of a voting game
  • Shapley-Shubik index measures the power
  • of each player in a multi-player game.
  • Consider a simple game.

Martin Shubik
Lloyd Shapley
35
Power of a Voter
  • Shapley-Shubik Power Index
  • Consider a coalition forming at random, one
    player at a time.
  • A player i is pivotal if addition of i throws
    coalition from losing to winning.
  • Shapley-Shubik index of i probability i is
    pivotal if an order of players is chosen at
    random.
  • Power measure applying to more general games than
    voting games is called Shapley Value.

36
Power of a Voter
  • Example Shareholders of Company
  • Shareholder 1 holds 3 shares.
  • Shareholders 2, 3, 4, 5, 6, 7 hold 1 share each.
  • A majority of shares are needed to make a
    decision.
  • Coalition 1,4,6 is winning.
  • Coalition 2,3,4,5,6 is winning.
  • Shareholder 1 is pivotal if he is 3rd, 4th, or
    5th.
  • So shareholder 1s Shapley value is 3/7.
  • Sum of Shapley values is 1 (since they are
    probabilities)
  • Thus, each other shareholder has Shapley value
  • (4/7)/6 2/21

37
Power of a Voter
  • Example Government of Australia
  • The previous game actually arises in the
    government of Australia.
  • There are six states and the federal government.
  • A coalition wins (can pass a law) if it consists
    of at least five states or at least two states
    and the federal government.
  • This is equivalent to the above voting game.

38
Power of a Voter
  • Example United Nations Security Council
  • 15 member nations
  • 5 permanent members
  • China, France,
  • Russia, UK, US
  • 10 non-permanent
  • Permanent members
  • have veto power
  • Coalition wins iff it has all 5 permanent members
  • and at least 4 of the 10 non-permanent members.

39
Power of a Voter
  • Example United Nations Security Council
  • What is the power of each
  • Member of the Security
  • Council?
  • Fix non-permanent member i.
  • i is pivotal in permutations in
  • which all permanent members
  • precede i and exactly 3 non-
  • permanent members do.
  • How many such permutations are there?

40
Power of a Voter
  • Example United Nations Security Council
  • Choose 3 non-permanent members preceding i.
  • Order all 8 members preceding i.
  • Order remaining 6 non-permanent members.
  • Thus the number of such permutations is
  • C(9,3) x 8! x 6! 9!/3!6! x 8! x 6! 9!8!/3!
  • The probability that i is pivotal power of
    non-permanent member
  • 9!8!/3!15! .001865
  • The power of a permanent member
  • 1 10 x .001865/5 .1963.
  • Permanent members have 100 times power of
    non-permanent members.

41
Power of a Voter
  • There are a variety of other power indices used
    in game theory and political science (Banzhaf
    index, Coleman index, )
  • Need calculate them for huge games
  • Mostly computationally intractable

42
Power of a Voter Allocation/Sharing Costs and
Revenues
  • Shapley-Shubik power index and more general
    Shapley value have been used to allocate costs to
    different users in shared projects.
  • Allocating runway fees in airports
  • Allocating highway fees to trucks of different
    sizes
  • Universities sharing library facilities
  • Fair allocation of telephone calling charges
    among users sharing complex phone systems
    (Cornells experiment)

43
Power of a Voter Allocating/Sharing Costs and
Revenues
  • Multicasting
  • Applications in multicasting.
  • Unicast routing Each packet sent from a source
    is delivered to a single receiver.
  • Sending it to multiple sites Send multiple
    copies and waste bandwidth.
  • In multicast routing Use a directed tree
  • connecting source to all receivers.
  • At branch points, a packet is duplicated as
  • necessary.

44
Multicasting
45
Power of a Voter Allocating/Sharing Costs and
Revenues
  • Multicasting
  • Multicast routing Use a directed tree connecting
    source to all receivers.
  • At branch points, a packet is duplicated as
    necessary.
  • Bandwidth is not directly attributable to a
    single receiver.
  • How to distribute costs among receivers?
  • One idea Use Shapley value.

46
Allocating/Sharing Costs and Revenues
  • Feigenbaum, Papadimitriou, Shenker (2001) no
    feasible implementation for Shapley value in
    multicasting.
  • Note Shapley value is uniquely characterized by
    four simple axioms.
  • Sometimes we state axioms as general principles
    we want a solution concept to have.
  • Jain and Vazirani (1998) polynomial time
    computable cost-sharing algorithm
  • Satisfying some important axioms
  • Calculating cost of optimum multicast tree within
    factor of two of optimal.

47
Voting Games at RUTCOR
  • Boolean functions I have not known
  • As far as I know, Peter never used the
    Shapley-Shubik index or the Banzhaf or Coleman
    indices to
  • Settle votes at RUTCOR
  • Allocate costs and revenues at RUTCOR

48
Algorithms for Container Inspection at Ports
  • My recent work has gotten me more heavily
  • into Boolean functions.
  • Im glad that RUTCOR faculty and students
  • have gotten involved.

49
Sequential Decision Making
  • Sequential decision making problems arise in many
    areas
  • Communication networks (testing connectivity,
    paging cellular customers, sequencing tasks, )
  • Manufacturing (testing machines, fault
    diagnosis, routing customer service calls, )
  • Medicine (diagnosing patients, sequencing
    treatments, )

50
Sequential Decision Making
  • These problems are especially relevant in
    homeland security inspection contexts.
  • Sheer size of modern decision problems in
    homeland security makes classical methods
    impractical quickly
  • We seek new methods that will scale to address
    modern applications

51
Container Inspection Algorithms
  • This work has gotten me and our students to
    interesting places.
  • Thanks to Capt. David Scott, US Coast Guard
    Captain of Port, Delaware Bay, we were taken on a
    tour of the port of Philadelphia

52
Container Inspection Algorithms
  • Goal Find ways to intercept illicit
  • nuclear materials and weapons
  • destined for the U.S. via the
  • maritime transportation system
  • Goal inspect all containers arriving at ports

53
Sequential Decision Making Problem
  • Stream of containers arrives at a port
  • Similar analysis for inspection prior to
    departure
  • The Decision Makers Problem
  • Which to inspect?
  • Which inspections next based on previous results?
  • Approach
  • decision logics
  • combinatorial optimization methods
  • Builds on ideas of Stroud
  • and Saeger at Los Alamos
  • Need for new models
  • and methods

54
Sequential Decision Making Problem
  • Containers arriving to be classified into
    categories.
  • Simple case 0 ok, 1 suspicious
  • Inspection scheme specifies which inspections
    are to be made based on previous observations

55
Sequential Decision Making ProblemFor Container
Inspection
  • Containers have attributes, each
  • in a number of states
  • Sample attributes
  • Levels of certain kinds of chemicals or
    biological materials
  • Levels of radiation
  • Whether or not there are items of a certain kind
    in the cargo list
  • Whether cargo was picked up in a certain
  • port

56
Sequential Decision Making Problem
  • Simplest Case Attributes are in state 0 or 1
    (absent or present)
  • Then Container is a binary string like 011001
  • So Classification is a decision function F that
    assigns each binary string to a category.

011001
F(011001)
If attributes 2, 3, and 6 are present, assign
container to category F(011001).
57
Sequential Decision Making Problem
  • If there are two categories, 0 and 1 (safe or
    suspicious), the decision function F is a
    Boolean function.
  • Example
  • F(000) F(111) 1, F(abc) 0 otherwise
  • This classifies a container as positive iff it
    has none of the attributes or all of them.

1
58
Binary Decision Tree Approach
  • Sensors (or other tests) measure
    presence/absence of attributes so 0 or 1
  • Use two outcome categories 0, 1 (safe or
    suspicious)
  • Binary Decision Tree
  • Nodes are sensors or categories
  • Two arcs exit from each sensor node, labeled left
    and right.
  • Take the right arc when sensor says the attribute
    is present, left arc otherwise

59
Binary Decision Tree Approach
  • Reach category 1 from
  • the root by
  • a0 L to a1 R a2 R 1 or
  • a0 R a2 R1
  • Container classified in category 1 iff it has
  • a1 and a2 and not a0 or
  • a0 and a2 and possibly a1.

Figure 2
60
Binary Decision Tree Approach
  • This binary decision tree corresponds to the same
    decision function
  • Container classified in category 1 iff it has
  • a1 and a2 and not a0 or
  • a0 and a2 and possibly a1
  • However, it has one less observation node ai.
  • So, it is more efficient if all observations are
    equally costly and equally likely.

Figure 3
61
Binary Decision Tree Approach
  • How do we find a low-cost or least-cost binary
    decision tree corresponding to a Boolean
    function?
  • Costs
  • Inspection costs (use of tree nodes)
  • Delay costs
  • Fixed equipment costs
  • False positive, false negative

62
Binary Decision Tree Approach
  • The problem of finding the least cost binary
    decision tree is very hard (NP-complete).
  • When the number of potential tests is small, can
    try to solve it by trying all possible binary
    decision trees.
  • But, even for n 4, not practical. (n 4 at
    Port of Long Beach-Los Angeles)

Port of Long Beach
63
Binary Decision Tree Approach
  • The number of alternative binary decision trees
    makes this brute force approach impractical.
  • Methods so far have been limited to
  • Special Boolean functions complete, monotone
  • Special binary trees
  • Limit on number of types of tests/sensors
  • Even here, the brute force methods become
    infeasible for n gt 4 types of tests/sensors

brute force
64
Binary Decision Tree Approach
  • Stroud-Saeger adopted a cost function that
    depends on
  • The expected number of tests before classifying a
    container (expected cost of utilizing the tree)
  • The cost of a false positive
  • The cost of a false negative
  • Using this cost function, they found the least
    cost binary decision trees by brute force.

65
Binary Decision Tree Approach Sensitivity
Analysis
  • We did a sensitivity analysis on the
    Stroud-Saeger results.
  • We varied three key parameters
  • A priori probability of a bad container
  • Cost of a false positive
  • Cost of a false negative

66
Sensitivity Analysis
  • Cost of false negative was varied between 25
    million and 10 billion dollars
  • (Low and high estimates of direct and indirect
    costs incurred due to a false negative.)
  • Cost of false positive was varied between 180
    and 720
  • (Cost incurred due to false positive
  • (4 men (3 -6 hrs) (15 30 /hr))
  • Probability of a bad container was varied
    between 1/10,000,000 and 1/100,000

67
Sensitivity Analysis
  • Tens of thousands of experimental runs.
  • Surprising results
  • With three types of tests, only three trees ever
    came out as least expensive.
  • Similar results with four types of tests.
  • Need an explanation of why.

68
Binary Decision Tree Approach
  • We have extended work of Stroud-Saeger in many
    ways, scaling to larger numbers of sensors
  • E.g. Defined notions of complete and monotonic
    for trees (as opposed to Boolean functions)
  • Developed heuristic search algorithms through an
    enlarged tree space of complete, monotonic
    trees
  • This has allowed us to
  • Speed up search
  • Attain at least as much accuracy
  • Allow more types of tests (sensors)

69
Binary Decision Tree Approach
  • Much more work to do
  • Explain why results so insensitive to changes in
    key parameter values
  • Understand why certain binary decision trees are
    so good.
  • Complications we will consider
  • Different cost functions
  • Role of different models for sensor errors
    thresholds (this work already begun)
  • Modeling delays due to queues
  • Simulation of port operations

70
Container Inspection
  • Collaborators on this Work
  • Saket Anand
  • David Madigan
  • Richard Mammone
  • Sushil Mittal
  • Saumitr Pathak
  • Los Alamos National Laboratory
  • Rick Picard
  • Kevin Saeger
  • Phil Stroud

71
Alternative Approaches to Container Inspection
  • Endre Boros and Paul Kantor have taken quite a
    different approach.
  • They use game theory to decide how to allocate a
    limited budget between container screening and
    actual container unpacking.
  • Their screening power index summarizes the
    cost-effectiveness of different screening tests.
  • The approach can yield increases of as much as
    100 in inspection with virtually no increase in
    cost.

72
Container Inspection
  • I can only think that Peter would have liked this
    problem.
  • It uses Boolean functions in a serious way.
  • It is a blend of practical Operations Research
    and theoretical O.R.
  • The work has engaged both students and faculty.

73
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