Applications of Probabilistic Logic to Materials Discovery: Solving problems with hard and soft...

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 ScienceInstitute of Mathematics and Statistics

University of Sao Paulo, Brazil

Nov 2013

Marcelo Finger

HardSoft & PSAT


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


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6 Conclusions

Marcelo Finger

HardSoft & PSAT


My Anguish

How to Lead My Career?

Basic Research

Application Oriented Research

Marcelo Finger

HardSoft & PSAT


My Anguish


Basic Research

Advantages: Does not require a lot of resources; there is acommunity to talk to; can be beautiful


Marcelo Finger

HardSoft & PSAT


My Anguish


Basic Research

Advantages: Does not require a lot of resources; there is acommunity to talk to; can be beautifulProblems: Beauty is hard to achieve; boring, not interestingto most; attracts little money


Marcelo Finger

HardSoft & PSAT


My Anguish


Basic Research



Advantages: Sexy, attracts visibility and money

Marcelo Finger

HardSoft & PSAT


My Anguish


Basic Research



Advantages: Sexy, attracts visibility and moneyProblems: Very hard to obtain data (and money is a must!),easily obsolescent

Marcelo Finger

HardSoft & PSAT


My Anguish


Basic Research



Advantages: Sexy, attracts visibility and moneyProblems: Very hard to obtain data (and money is a must!),easily obsolescent

No answer, but keep this in mind

Marcelo Finger

HardSoft & PSAT


What you most certainly don’t know

Combinatorial Chemistry: discovery of new products

Lots of experimentsEach has a different settingsSearch for the right setting that produces an interestingproductVery useful to find new medicine, paint, adhesives, etc.

Marcelo Finger

HardSoft & PSAT


Combinatorial Materials Science

Search for new materials.

Aim: new catalysts for the purification of water and air

Buzz word: Sustainability

Marcelo Finger

HardSoft & PSAT


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


The Materials Discovery Problem

In a reaction, where are the materials formed?

Marcelo Finger

HardSoft & PSAT


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


The Crazy Idea


Consider only peaks





Different models, different graphs, different defects

Marcelo Finger

HardSoft & PSAT


The Crazy Idea


Consider only peaks





Different models, different graphs, different defects

Limit the probability of the occurrence of defects

Marcelo Finger

HardSoft & PSAT


Challenges: Only for the tough

Logic

Probability

Marcelo Finger

HardSoft & PSAT


Challenges: Only for the tough

Logic

Probability

Linear Algebra

Linear Programming

Optimization

Algorithms

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HardSoft & PSAT


Student Profile

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HardSoft & PSAT


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6 Conclusions

Marcelo Finger

HardSoft & PSAT


Pragmatic Motivation

Practical problems combine real-world hard constraints withsoft constraints

Soft constraints: preferences, uncertainties, flexiblerequirements

We explore probabilistic logic as a mean of dealing withcombined soft and hard constraints

Marcelo Finger

HardSoft & PSAT


Goals

Aim: Combine Logic and Probabilistic reasoning to dealwith Hard (L) and Soft (P) constraints

Method: develop optimized Probabilistic Satisfiability(oPSAT)

Application: Demonstrate effectiveness on a real-worldreasoning task in the domain of Materials Discovery.

Marcelo Finger

HardSoft & PSAT


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 mostthree courses.There is a limit of ℓ students per course.

Soft Avoid having students with no teammate.

Marcelo Finger

HardSoft & PSAT


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 mostthree 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” orbounded

Marcelo Finger

HardSoft & PSAT


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





Our choice: Probabilistic Satisfiability (PSAT)

Natural extension of Boolean LogicDesirable properties, e.g. respects Kolmogorov axiomsProbabilistic reasoning free of independence presuppositions

Marcelo Finger

HardSoft & PSAT





Our choice: Probabilistic Satisfiability (PSAT)

Natural extension of Boolean LogicDesirable properties, e.g. respects Kolmogorov axiomsProbabilistic reasoning free of independence presuppositions

What is PSAT?

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HardSoft & PSAT


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6 Conclusions

Marcelo Finger

HardSoft & PSAT


Is PSAT a Zombie Idea?

An idea that refuses to die!

Marcelo Finger

HardSoft & PSAT


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 AIPSAT is NP-complete [Georgakopoulos et. al 1988]Nilsson [1993]: “complete impracticability” of PSATcomputationMany other works; see Hansen & Jaumard [2000]

Marcelo Finger

HardSoft & PSAT


Or a Wild Amazonian Flower?

Awaits special conditions to bloom!

(Linear programming + SAT-based techniques)

Marcelo Finger

HardSoft & PSAT


The Setting

Formulas α1, . . . , αℓ over logical variables P = {x1, . . . , xn}

Propositional valuation v : P → {0, 1}

Marcelo Finger

HardSoft & PSAT


The Setting



A probability distribution over propositional valuations

π : V → [0, 1]

2n∑

i=1

π(vi ) = 1

Marcelo Finger

HardSoft & PSAT


The Setting



A probability distribution over propositional valuations

π : V → [0, 1]

2n∑

i=1

π(vi ) = 1

Probability of a formula α according to π

Pπ(α) =∑

{π(vi )|vi (α) = 1}

Marcelo Finger

HardSoft & PSAT


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 ) S pi |1 ≤ i ≤ ℓ}

Probabilistic Satisfiability: is there a π that satisfies Σ?

Marcelo Finger

HardSoft & PSAT


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 ) S 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


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) ≤ 0.25P(y ∧ ¬pxy ) ≤ 0.60P(z ∧ ¬pxz) ≤ 0.60

Marcelo Finger

HardSoft & PSAT


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) ≤ 0.25P(y ∧ ¬pxy ) ≤ 0.60P(z ∧ ¬pxz) ≤ 0.60

(Small) solution: distribution π

π(x , y , z ,¬pxy ,¬pxz) = 0.1 π(x , y , z , pxy ,¬pxz) = 0.4π(x , y , z ,¬pxy , pxz) = 0.5 π(v) = 0 for other 29 valuations

Marcelo Finger

HardSoft & PSAT


Solving PSAT

Linear algebraic formulation

Optimization problem with exponential columns

Columns are not explicitly represented, but provided by acolumn generation process

Goal: minimize the probability of inconsistent columns

A SAT solver is employed for column generation and to forcea decrease in cost

Interface between logic and algebra is an algebraic constraintthat can be seen as a logic formula

Marcelo Finger

HardSoft & PSAT


Example of PSAT solution

Add variables for each soft violation: sx , sy , sz .

Γ =

{

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 }

Iteration 0:

sxsysz

1 1 1 10 0 0 10 0 1 10 1 1 1

·

0.40

0.350.25

=

10.250.600.60

cost(0) = 0.4

b(0) = [1 0 1 0]′ : col 3

Marcelo Finger

HardSoft & PSAT




Γ =

{


}

Ψ = { P(sx) = 0.25, P(sy ) = 0.6, P(sz) = 0.6 }

Iteration 1:

sxsysz

1 1 1 10 0 0 10 0 1 10 1 0 1

·

0.050.350.350.25

=

10.250.600.60

cost(1) = 0.05

b(1) = [1 1 0 1]′ : col 1

Marcelo Finger

HardSoft & PSAT




Γ =

{


}

Ψ = { P(sx) = 0.25, P(sy ) = 0.6, P(sz) = 0.6 }

Iteration 2:

sxsysz

1 1 1 11 0 0 10 0 1 11 1 0 1

·

0.050.350.400.20

=

10.250.600.60

cost(2) = 0

Marcelo Finger

HardSoft & PSAT


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6 Conclusions

Marcelo Finger

HardSoft & PSAT


Optimizing PSAT solutions

Solutions to PSAT are not unique

First optimization phase: determines if constraints are solvable

Second optimization phase to obtain a distribution withdesirable properties.

A different objective (cost) function

Marcelo Finger

HardSoft & PSAT


Minimizing Expected Violations

Idea: minimize the expected number of soft constraintsviolated by each valuation, S(v)

E (S) =∑

vi |vi (Γ)=1

S(vi )π(vi )

Theorem: Every linear function of a model (valuation) hasconstant 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 forminimization

Marcelo Finger

HardSoft & PSAT


Minimizing Variance

Idea: penalize high S , minimize E (S2)

Lemma: The distribution that minimizes E (S2) alsominimizes variance, Var(S) = E ((S − E (S))2)

oPSAT is a second phase minimization whose objectivefunction is E (S2)

Problem: computing a SAT formula that decreases cost isharder than in PSAT

oPSAT needs a more elaborate interface logic/linear algebra

Marcelo Finger

HardSoft & PSAT


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6 Conclusions

Marcelo Finger

HardSoft & PSAT


Problem Definition

Marcelo Finger

HardSoft & PSAT


Problem Modeling

Find an association of peaks to phases, respecting structuralconstraints, such that the probability of defect is limited.

Marcelo Finger

HardSoft & PSAT


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 betweenvariables

Identify the possible defects

Limit defects with “acceptable” upper probabilities. Mayrequire iteration and/or learning

Marcelo Finger

HardSoft & PSAT


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


Experimental Results

Dataset SMT oPSAT

System P L∗ K #Peaks Time(s) Time(s) Accuracy

Al/Li/Fe 28 6 6 170 346 5.3 84.7%Al/Li/Fe 28 8 6 424 10076 8.8 90.5%Al/Li/Fe 28 10 6 530 28170 12.6 83.0%Al/Li/Fe 45 7 6 651 18882 121.1 82.0%Al/Li/Fe 45 8 6 744 46816 128.0 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

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HardSoft & PSAT


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6 Conclusions

Marcelo Finger

HardSoft & PSAT


Conclusions and the Future

oPSAT can be effectively implemented to deal with hard andsoft constraints

Can be successfully applied to non-trivial problems ofmaterials discovery with acceptable precision and superior runtimes than existing methods

Other forms of logic-probabilistic inference are underinvestigation

Marcelo Finger

HardSoft & PSAT

Applications of Probabilistic Logic to Materials Discovery: Solving problems with hard and soft...

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