Author: Marcelo Finger (http://www.ime.usp.br/~mfinger/)

1. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Applications of Probabilistic Logic to
Materials Discovery
Solving problems with hard and soft constraints
Marcelo Finger
Department of Computer Science
Institute of Mathematics and Statistics
University of Sao Paulo, Brazil
Nov 2013
Marcelo Finger
HardSoft & PSAT

2. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Topics
1
What You Probably Don’t Know
2
Pragmatic Motivation
3
Probabilistic Satisfiability
4
Optimizing Probability Distributions with oPSAT
5
oPSAT and Combinatorial Materials Discovery
6
Conclusions
Marcelo Finger
HardSoft & PSAT

3. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Next Topic
1
What You Probably Don’t Know
2
Pragmatic Motivation
3
Probabilistic Satisfiability
4
Optimizing Probability Distributions with oPSAT
5
oPSAT and Combinatorial Materials Discovery
6
Conclusions
Marcelo Finger
HardSoft & PSAT

4. Ignorance
Motivation
PSAT
oPSAT
My Anguish
How to Lead My Career?
Basic Research
Application Oriented Research
Marcelo Finger
HardSoft & PSAT
Application
Conclusion

5. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
My Anguish
How to Lead My Career?
Basic Research
Advantages: Does not require a lot of resources; there is a
community to talk to; can be beautiful
Application Oriented Research
Marcelo Finger
HardSoft & PSAT

6. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
My Anguish
How to Lead My Career?
Basic Research
Advantages: Does not require a lot of resources; there is a
community to talk to; can be beautiful
Problems: Beauty is hard to achieve; boring, not interesting
to most; attracts little money
Application Oriented Research
Marcelo Finger
HardSoft & PSAT

7. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
My Anguish
How to Lead My Career?
Basic Research
Advantages: Does not require a lot of resources; there is a
community to talk to; can be beautiful
Problems: Beauty is hard to achieve; boring, not interesting
to most; attracts little money
Application Oriented Research
Advantages: Sexy, attracts visibility and money
Marcelo Finger
HardSoft & PSAT

8. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
My Anguish
How to Lead My Career?
Basic Research
Advantages: Does not require a lot of resources; there is a
community to talk to; can be beautiful
Problems: Beauty is hard to achieve; boring, not interesting
to most; attracts little money
Application Oriented Research
Advantages: Sexy, attracts visibility and money
Problems: Very hard to obtain data (and money is a must!),
easily obsolescent
Marcelo Finger
HardSoft & PSAT

9. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
My Anguish
How to Lead My Career?
Basic Research
Advantages: Does not require a lot of resources; there is a
community to talk to; can be beautiful
Problems: Beauty is hard to achieve; boring, not interesting
to most; attracts little money
Application Oriented Research
Advantages: Sexy, attracts visibility and money
Problems: Very hard to obtain data (and money is a must!),
easily obsolescent
No answer, but keep this in mind
Marcelo Finger
HardSoft & PSAT

10. Ignorance
Motivation
PSAT
oPSAT
Application
What you most certainly don’t know
Combinatorial Chemistry: discovery of new products
Lots of experiments
Each has a different settings
Search for the right setting that produces an interesting
product
Very useful to find new medicine, paint, adhesives, etc.
Marcelo Finger
HardSoft & PSAT
Conclusion

11. Ignorance
Motivation
PSAT
oPSAT
Application
Combinatorial Materials Science
Search for new materials.
Aim: new catalysts for the purification of water and air
Buzz word: Sustainability
Marcelo Finger
HardSoft & PSAT
Conclusion

12. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Combinatorial Materials Science
Search for new materials.
Aim: new catalysts for the purification of water and air
Buzz word: Sustainability
This work was developed within the
Institute of Computational Sustainability
at the Department of Computer Science, Cornell University
Marcelo Finger
HardSoft & PSAT

13. Ignorance
Motivation
PSAT
oPSAT
Application
The Materials Discovery Problem
In a reaction, where are the materials formed?
Marcelo Finger
HardSoft & PSAT
Conclusion

14. Ignorance
Motivation
PSAT
oPSAT
Application
The Crazy Idea
Use Probabilistic Logic to solve this problem
Consider only peaks
Each peak is in a layer
Connect all peaks in the same layer with a graph
Model graphs with propositional logic
Each disconnected peak is a defect
Marcelo Finger
HardSoft & PSAT
Conclusion

15. Ignorance
Motivation
PSAT
oPSAT
Application
The Crazy Idea
Use Probabilistic Logic to solve this problem
Consider only peaks
Each peak is in a layer
Connect all peaks in the same layer with a graph
Model graphs with propositional logic
Each disconnected peak is a defect
Different models, different graphs, different defects
Marcelo Finger
HardSoft & PSAT
Conclusion

16. Ignorance
Motivation
PSAT
oPSAT
Application
The Crazy Idea
Use Probabilistic Logic to solve this problem
Consider only peaks
Each peak is in a layer
Connect all peaks in the same layer with a graph
Model graphs with propositional logic
Each disconnected peak is a defect
Different models, different graphs, different defects
Limit the probability of the occurrence of defects
Marcelo Finger
HardSoft & PSAT
Conclusion

17. Ignorance
Motivation
PSAT
oPSAT
Application
Challenges: Only for the tough
Logic
Probability
Marcelo Finger
HardSoft & PSAT
Conclusion

18. Ignorance
Motivation
PSAT
oPSAT
Application
Challenges: Only for the tough
Logic
Probability
Linear Algebra
Linear Programming
Optimization
Algorithms
Marcelo Finger
HardSoft & PSAT
Conclusion

19. Ignorance
Motivation
Student Profile
Marcelo Finger
HardSoft & PSAT
PSAT
oPSAT
Application
Conclusion

20. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Next Topic
1
What You Probably Don’t Know
2
Pragmatic Motivation
3
Probabilistic Satisfiability
4
Optimizing Probability Distributions with oPSAT
5
oPSAT and Combinatorial Materials Discovery
6
Conclusions
Marcelo Finger
HardSoft & PSAT

21. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Pragmatic Motivation
Practical problems combine real-world hard constraints with
soft constraints
Soft constraints: preferences, uncertainties, flexible
requirements
We explore probabilistic logic as a mean of dealing with
combined soft and hard constraints
Marcelo Finger
HardSoft & PSAT

22. Ignorance
Motivation
PSAT
oPSAT
Application
Goals
Aim: Combine Logic and Probabilistic reasoning to deal
with Hard (L) and Soft (P) constraints
Method: develop optimized Probabilistic Satisfiability
(oPSAT)
Application: Demonstrate effectiveness on a real-world
reasoning task in the domain of Materials Discovery.
Marcelo Finger
HardSoft & PSAT
Conclusion

23. Ignorance
Motivation
PSAT
oPSAT
Application
An Example
Summer course enrollment
m students and k summer courses.
Potential team mates, to develop coursework. Constraints:
Hard Coursework to be done alone or in pairs.
Students must enroll in at least one and at most
three courses.
There is a limit of ℓ students per course.
Soft Avoid having students with no teammate.
Marcelo Finger
HardSoft & PSAT
Conclusion

24. Ignorance
Motivation
PSAT
oPSAT
Application
An Example
Summer course enrollment
m students and k summer courses.
Potential team mates, to develop coursework. Constraints:
Hard Coursework to be done alone or in pairs.
Students must enroll in at least one and at most
three courses.
There is a limit of ℓ students per course.
Soft Avoid having students with no teammate.
In our framework: P(student with no team mate) “minimal” or
bounded
Marcelo Finger
HardSoft & PSAT
Conclusion

25. Ignorance
Motivation
PSAT
oPSAT
Application
Combining Logic and Probability
Many proposals in the literature
Markov Logic Networks [Richardson & Domingos 2006]
Probabilistic Inductive Logic Prog [De Raedt et. al 2008]
Relational Models [Friedman et al 1999], etc
Marcelo Finger
HardSoft & PSAT
Conclusion

26. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Combining Logic and Probability
Many proposals in the literature
Markov Logic Networks [Richardson & Domingos 2006]
Probabilistic Inductive Logic Prog [De Raedt et. al 2008]
Relational Models [Friedman et al 1999], etc
Our choice: Probabilistic Satisfiability (PSAT)
Natural extension of Boolean Logic
Desirable properties, e.g. respects Kolmogorov axioms
Probabilistic reasoning free of independence presuppositions
Marcelo Finger
HardSoft & PSAT

27. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Combining Logic and Probability
Many proposals in the literature
Markov Logic Networks [Richardson & Domingos 2006]
Probabilistic Inductive Logic Prog [De Raedt et. al 2008]
Relational Models [Friedman et al 1999], etc
Our choice: Probabilistic Satisfiability (PSAT)
Natural extension of Boolean Logic
Desirable properties, e.g. respects Kolmogorov axioms
Probabilistic reasoning free of independence presuppositions
What is PSAT?
Marcelo Finger
HardSoft & PSAT

28. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Next Topic
1
What You Probably Don’t Know
2
Pragmatic Motivation
3
Probabilistic Satisfiability
4
Optimizing Probability Distributions with oPSAT
5
oPSAT and Combinatorial Materials Discovery
6
Conclusions
Marcelo Finger
HardSoft & PSAT

29. Ignorance
Motivation
PSAT
oPSAT
Is PSAT a Zombie Idea?
An idea that refuses to die!
Marcelo Finger
HardSoft & PSAT
Application
Conclusion

30. Ignorance
Motivation
PSAT
oPSAT
Application
A Brief History of PSAT
Proposed by [Boole 1854],
On the Laws of Thought
Rediscovered several times since Boole
De Finetti [1937, 1974], Good [1950], Smith [1961]
Studied by Hailperin [1965]
Nilsson [1986] (re)introduces PSAT to AI
PSAT is NP-complete [Georgakopoulos et. al 1988]
Nilsson [1993]: “complete impracticability” of PSAT
computation
Many other works; see Hansen & Jaumard [2000]
Marcelo Finger
HardSoft & PSAT
Conclusion

31. Ignorance
Motivation
PSAT
oPSAT
Application
Or a Wild Amazonian Flower?
Awaits special conditions to bloom!
(Linear programming + SAT-based techniques)
Marcelo Finger
HardSoft & PSAT
Conclusion

32. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
The Setting
Formulas α1 , . . . , αℓ over logical variables P = {x1 , . . . , xn }
Propositional valuation v : P → {0, 1}
Marcelo Finger
HardSoft & PSAT

33. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
The Setting
Formulas α1 , . . . , αℓ over logical variables P = {x1 , . . . , xn }
Propositional valuation v : P → {0, 1}
A probability distribution over propositional valuations
π : V → [0, 1]
2n
π(vi ) = 1
i=1
Marcelo Finger
HardSoft & PSAT

34. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
The Setting
Formulas α1 , . . . , αℓ over logical variables P = {x1 , . . . , xn }
Propositional valuation v : P → {0, 1}
A probability distribution over propositional valuations
π : V → [0, 1]
2n
π(vi ) = 1
i=1
Probability of a formula α according to π
Pπ (α) =
Marcelo Finger
HardSoft & PSAT
{π(vi )|vi (α) = 1}

35. Ignorance
Motivation
PSAT
oPSAT
Application
The PSAT Problem
Consider ℓ formulas α1 , . . . , αℓ defined on n atoms
{x1 , . . . , xn }
A PSAT problem Σ is a set of ℓ restrictions
Σ = {P(αi )
pi |1 ≤ i ≤ ℓ}
Probabilistic Satisfiability: is there a π that satisfies Σ?
Marcelo Finger
HardSoft & PSAT
Conclusion

36. Ignorance
Motivation
PSAT
oPSAT
Application
The PSAT Problem
Consider ℓ formulas α1 , . . . , αℓ defined on n atoms
{x1 , . . . , xn }
A PSAT problem Σ is a set of ℓ restrictions
Σ = {P(αi )
pi |1 ≤ i ≤ ℓ}
Probabilistic Satisfiability: is there a π that satisfies Σ?
In our framework, ℓ = m + k, Σ = Γ ∪ Ψ:
Hard Γ = {α1 , . . . , αm }, P(αi ) = 1 (clauses)
Soft Ψ = {P(si ) ≤ pi |1 ≤ i ≤ k}
si atomic; pi given, learned or minimized
Marcelo Finger
HardSoft & PSAT
Conclusion

37. Ignorance
Motivation
PSAT
oPSAT
Application
Example continued
Only one course, three student enrollments: x, y and z
Potential partnerships: pxy and pxz , mutually exclusive.
Hard constraint
P(x ∧ y ∧ z ∧ ¬(pxy ∧ pxz )) = 1
Soft constraints
P(x ∧ ¬pxy ∧ ¬pxz )
P(y ∧ ¬pxy )
P(z ∧ ¬pxz )
Marcelo Finger
HardSoft & PSAT
≤
≤
≤
0.25
0.60
0.60
Conclusion

38. Ignorance
Motivation
PSAT
oPSAT
Application
Example continued
Only one course, three student enrollments: x, y and z
Potential partnerships: pxy and pxz , mutually exclusive.
Hard constraint
P(x ∧ y ∧ z ∧ ¬(pxy ∧ pxz )) = 1
Soft constraints
P(x ∧ ¬pxy ∧ ¬pxz )
P(y ∧ ¬pxy )
P(z ∧ ¬pxz )
≤
≤
≤
0.25
0.60
0.60
(Small) solution: distribution π
π(x, y , z, ¬pxy , ¬pxz ) = 0.1
π(x, y , z, ¬pxy , pxz ) = 0.5
Marcelo Finger
HardSoft & PSAT
π(x, y , z, pxy , ¬pxz ) = 0.4
π(v ) = 0 for other 29 valuations
Conclusion

39. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Solving PSAT
Linear algebraic formulation
Optimization problem with exponential columns
Columns are not explicitly represented, but provided by a
column generation process
Goal: minimize the probability of inconsistent columns
A SAT solver is employed for column generation and to force
a decrease in cost
Interface between logic and algebra is an algebraic constraint
that can be seen as a logic formula
Marcelo Finger
HardSoft & PSAT

40. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Example of PSAT solution
Add variables for each soft violation: sx , sy , sz .
x, y , z, ¬pxy ∨ ¬pxz ,
(x ∧ ¬pxy ∧ ¬pxz ) → sx , (y ∧ ¬pxy ) → sy , (z ∧ ¬pxz ) → sz
Ψ = { P(sx ) = 0.25, P(sy ) = 0.6, P(sz ) = 0.6 }
Γ=
Iteration 0:
sx
sy
sz

1
 0

 0
0
1
0
0
1
1
0
1
1
 
0.4
1
1   0
·
1   0.35
0.25
1

HardSoft & PSAT
=
cost(0)
b (0)
Marcelo Finger



=
=


1
 0.25 


 0.60 
0.60
0.4
[1 0 1 0]′ : col 3

41. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Example of PSAT solution
Add variables for each soft violation: sx , sy , sz .
x, y , z, ¬pxy ∨ ¬pxz ,
(x ∧ ¬pxy ∧ ¬pxz ) → sx , (y ∧ ¬pxy ) → sy , (z ∧ ¬pxz ) → sz
Ψ = { P(sx ) = 0.25, P(sy ) = 0.6, P(sz ) = 0.6 }
Γ=
Iteration 1:
sx
sy
sz

1
 0

 0
0
1
0
0
1
1
0
1
0
 
0.05
1
1   0.35
·
1   0.35
0.25
1

HardSoft & PSAT
=
cost(1)
b (1)
Marcelo Finger



=
=


1
 0.25 


 0.60 
0.60
0.05
[1 1 0 1]′ : col 1

42. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Example of PSAT solution
Add variables for each soft violation: sx , sy , sz .
x, y , z, ¬pxy ∨ ¬pxz ,
(x ∧ ¬pxy ∧ ¬pxz ) → sx , (y ∧ ¬pxy ) → sy , (z ∧ ¬pxz ) → sz
Ψ = { P(sx ) = 0.25, P(sy ) = 0.6, P(sz ) = 0.6 }
Γ=
Iteration 2:
sx
sy
sz

1
 1

 0
1
1
0
0
1
1
0
1
0
 
0.05
1
1   0.35
·
1   0.40
1
0.20

HardSoft & PSAT
=
cost(2)
Marcelo Finger



=


1
 0.25 


 0.60 
0.60
0

43. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Next Topic
1
What You Probably Don’t Know
2
Pragmatic Motivation
3
Probabilistic Satisfiability
4
Optimizing Probability Distributions with oPSAT
5
oPSAT and Combinatorial Materials Discovery
6
Conclusions
Marcelo Finger
HardSoft & PSAT

44. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Optimizing PSAT solutions
Solutions to PSAT are not unique
First optimization phase: determines if constraints are solvable
Second optimization phase to obtain a distribution with
desirable properties.
A different objective (cost) function
Marcelo Finger
HardSoft & PSAT

45. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Minimizing Expected Violations
Idea: minimize the expected number of soft constraints
violated by each valuation, S(v )
S(vi )π(vi )
E (S) =
vi |vi (Γ)=1
Theorem: Every linear function of a model (valuation) has
constant expected value for any PSAT solution
In particular, E (S) is constant, no point in minimizing it
Any other linear expectation function not a candidate for
minimization
Marcelo Finger
HardSoft & PSAT

46. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Minimizing Variance
Idea: penalize high S, minimize E (S 2 )
Lemma: The distribution that minimizes E (S 2 ) also
minimizes variance, Var(S) = E ((S − E (S))2 )
oPSAT is a second phase minimization whose objective
function is E (S 2 )
Problem: computing a SAT formula that decreases cost is
harder than in PSAT
oPSAT needs a more elaborate interface logic/linear algebra
Marcelo Finger
HardSoft & PSAT

47. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Next Topic
1
What You Probably Don’t Know
2
Pragmatic Motivation
3
Probabilistic Satisfiability
4
Optimizing Probability Distributions with oPSAT
5
oPSAT and Combinatorial Materials Discovery
6
Conclusions
Marcelo Finger
HardSoft & PSAT

48. Ignorance
Motivation
PSAT
Problem Definition
Marcelo Finger
HardSoft & PSAT
oPSAT
Application
Conclusion

49. Ignorance
Motivation
PSAT
oPSAT
Application
Problem Modeling
Find an association of peaks to phases, respecting structural
constraints, such that the probability of defect is limited.
Marcelo Finger
HardSoft & PSAT
Conclusion

50. Ignorance
Motivation
PSAT
oPSAT
Application
Modeling Process
Logic modeling
Identify the structural elements of the problem
Identify the {0,1}-variables of the problem
Formulas encode the constraints on relationships between
variables
Identify the possible defects
Limit defects with “acceptable” upper probabilities. May
require iteration and/or learning
Marcelo Finger
HardSoft & PSAT
Conclusion

51. Ignorance
Motivation
PSAT
oPSAT
Application
Implementation
Implementated in C++
Linear Solver uses blas and lapack
SAT-solver: minisat
PSAT formula (DIMACS extension) generated by C++
formula generator
P(dp ) ≤ 2% =⇒ P(di ) ≤ 1 − (1 − P(dp ))Li
Input: Peaks at sample point
Output: oPSAT most probable model
Compare with SMT implementation in SAT 2012
psat.sourceforge.net
Marcelo Finger
HardSoft & PSAT
Conclusion

52. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Experimental Results
System
Al/Li/Fe
Al/Li/Fe
Al/Li/Fe
Al/Li/Fe
Al/Li/Fe
Dataset
P L∗ K
28
28
28
45
45
6
8
10
7
8
#Peaks
SMT
Time(s)
170
424
530
651
744
346
10076
28170
18882
46816
6
6
6
6
6
oPSAT
Time(s) Accuracy
5.3
8.8
12.6
121.1
128.0
84.7%
90.5%
83.0%
82.0%
80.3%
The accuracy of SMT is 100%
P: n. of sample points; L∗ : the average n, of peaks per phase
K : n. of basis patterns; #Peaks: overall n. of peaks
#Aux variables > 10 000
Marcelo Finger
HardSoft & PSAT

53. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Next Topic
1
What You Probably Don’t Know
2
Pragmatic Motivation
3
Probabilistic Satisfiability
4
Optimizing Probability Distributions with oPSAT
5
oPSAT and Combinatorial Materials Discovery
6
Conclusions
Marcelo Finger
HardSoft & PSAT

54. Ignorance
Motivation
PSAT
oPSAT
Application
Conclusion
Conclusions and the Future
oPSAT can be effectively implemented to deal with hard and
soft constraints
Can be successfully applied to non-trivial problems of
materials discovery with acceptable precision and superior run
times than existing methods
Other forms of logic-probabilistic inference are under
investigation
Marcelo Finger
HardSoft & PSAT