1. Probabilistic Inference Modulo Theories
Rodrigo de Salvo - Braz Ciaran O’Reilly - Vibhav Gogate - Rina Dechter
Presented by Shaked Or
Special thanks to Ian Gent for watched literals explanation

2. Lecture Objective Introduce Probabilistic Inference
Explain why current approaches are lacking
Introduce SGDPLL
Seeing SGDPLL applications in probabilistic calculations

3. Lecture layout: What is Probabilistic Inference? Definitions
Real life example
Boolean Satisfiability problem (SAT) – reminder
Satisfiability Modulo Theories (SMT)– SAT generalization
SGDPLL(T) – one step further
Generalize the problem further
The SGDPL algorithm
Applications in probability
Experiments and Optimizations

4. Lecture layout: What is Probabilistic Inference? Definitions
Real life example
Boolean Satisfiability problem (SAT) – reminder
Satisfiability Modulo Theories (SMT)– SAT generalization
SGDPLL(T) – one step further
Generalize the problem further
The SGDPL algorithm
Applications in probability
Experiments and Optimizations

5. Probabilistic Inference
a=P(Retake Course)
As we find more evidence, the probability Changes

6. Probabilistic Inference - Formally
P - probability distribution
a - event
{Ei |1 ≤ i ≤ n} – evidence
Given new incoming facts (evidence), we derive the posterior probability of a:

7. Probabilistic Inference
Lets look at an example of a “real” problem.
Given N voters, the Amount of voters that like an incumbent Senator
is influenced by:
The number of terror attacks
The Dow Jones index
The number of newly created jobs
The number of people who like the challenging candidate.

8. Probabilistic Inference
Let us take: Attacks ~ Uniform[20] newJobs ~ Uniform[10000] Dow ~ Uniform[11000,18000] likeChallenger ~ Uniform [N] And take the influence on P(likeIncumbant) to be:

9. Probabilistic Inference
We want to calculate examples like:
For a constant value of new jobs
While treating new jobs as a parameter

10. Problems with current approaches
Regular approach: building a database of the problem
In interesting problems, database will be too large
Requires many calculations to get parametric relationships
Logical approach:
Can help reduce redundancies while inferring data from logical relationships
Has really efficient solvers for private cases
Not expressive enough for real problems
We are going to solve this last problem.

11. Lecture layout: What is Probabilistic Inference? Definitions
Real life example
Boolean Satisfiability problem (SAT) – reminder
Satisfiability Modulo Theories (SMT)– SAT generalization
SGDPLL(T) – one step further
Generalize the problem further
The SGDPL algorithm
Applications in probability
Experiments and Optimizations

12. Reminder – SAT problem
Given a CNF: xi are Boolean variables X=(x1,x2,x3)

13. Reminder – SAT problem
Given a CNF: xi are Boolean variables X=(x1,x2,x3) We ask: Is f satisfiable? Meaning:

14. Reminder – SAT problem
Given a CNF: xi are Boolean variables X=(x1,x2,x3) We ask: Is f satisfiable? Meaning: In this case, for We get:

15. Lecture layout: What is Probabilistic Inference? Definitions
Real life example
Boolean Satisfiability problem (SAT) – reminder
Satisfiability Modulo Theories (SMT)– SAT generalization
SGDPLL(T) – one step further
Generalize the problem further
The SGDPL algorithm
Applications in probability
Experiments and Optimizations

16. Generalization - SMT
We add literals from a first order theory T, but still connect them with Boolean operators. Examples: T is the theory of difference arithmetic, meaning {≤,≥,<,>,=,+,x}

17. SMT -formally Given: a first order logic theory T
an expression containing literals of T and Boolean operators.
We ask if that expression is satisfiable.
In SAT, assignment is independent
In SMT, not always the case

18. SMT -formally
Example:

19. SMT -formally
Example: We want g(x)=T and h(x)=F But

20. SMT -formally
Variables are not only True and False:

21. SMT - Currently There are very efficient SMT solvers
However SMT is not expressive enough
Can’t use it for probability

22. So far
SMT
Probabilistic
Inference
SAT

23. And now
SGDPLL
SMT
Probabilistic
Inference
SAT

24. Lecture layout: What is Probabilistic Inference? Definitions
Real life example
Boolean Satisfiability problem (SAT) – reminder
Satisfiability Modulo Theories (SMT)– SAT generalization
SGDPLL(T) – one step further
Generalize the problem further
The SGDPL algorithm
Applications in probability
Experiments and Optimizations

25. Taking it one step further – SGDPLL(T)
We can be:
symbolic parametric
quantifier parametric
input theory parametric.
Example:

26. Taking it one step further – SGDPLL(T)
We can be:
symbolic parametric
quantifier parametric
input theory parametric.
Our answer can be in terms of y , not all terms are implicitly quantified
Possible Answer:

27. Taking it one step further – SGDPLL(T)
We can be:
symbolic parametric
quantifier parametric
input theory parametric.
Our answer can be in terms of y , not all terms are implicitly quantified
Gives as bigger problem solving capabilities.

28. In SAT,SMT

29. Taking it one step further – SGDPLL(T)
We can be:
symbolic parametric
quantifier parametric
input theory parametric.
We don’t only ask for exists, we can ask for summation etc

30. Taking it one step further – SGDPLL(T)
We can be:
symbolic parametric
quantifier parametric
input theory parametric.
Caveat: We can only use associative quantifiers.
Examples:

31. Taking it one step further – SGDPLL(T)
We can be:
symbolic parametric
quantifier parametric
input theory parametric.
Not only Boolean operators.
Can use if, then, else, x, etc.

32. SGDPLL(T) – formally
SGDPLL(T) takes a theory T=(TC ,TL) TC = Constraint theory TL = Input Theory

33. SGDPLL(T) – formally
SGDPLL(T) Recieves a T Problem = Associative Quantifier x = indexed variable y = free variables ( y=(y1 ,y2 ,…) ) F(x,y) = Conjunction of literals in TC E(x,y) = Expression in TL connecting literals in TC

34. SGDPLL(T) – formally Output of SGDPLL(T) is a T solution. T Solution:
A quantifier free expression in TL
Equivalent to T problem

35. SGDPLL(T) – formally
Example: TC = Difference Arithmetic TL = F(x,y,w)= E(x,y,w)=

36. SGDPLL(T) – formally
T problem: T Solution:

37. SGDPLL algorithm – formally
Base Cases and non base Cases: Base Case: E(x,y) contains no Literals in TC. Non base case: The general case.

38. SGDPLL algorithm - illustration

39. SGDPLL algorithm - illustration
Base Case:
Now its time for the base case solver

40. SGDPLL algorithm - illustration

41. SGDPLL algorithm – illustration
The problem: Becomes:

42. SGDPLL algorithm – formally
Requirements:
Base quantifiers solver, quantifiers must be associative.
Base case solvers.
Consistency checkers
Note that in the following steps we do theory independent operations. This is the power of SGDPLL, it gives you the power to do complex calculation when supplying only the basic tools.

43. SGDPLL algorithm – Algorithm
SGDPLL Psuedo Code:
Choose a splitter literal L
If L contains x, do Quantifier Splitting
Else do if splitting

44. SGDPLL algorithm – Algorithm
Quantifier splitting:
If x is in L, Create 2 sub-problems.
First with L
Second with not L
Combine them using Quantifier

45. SGDPLL algorithm – Algorithm
IF splitting:
If x is not in L, Create 2 sub-problems.
First with implication of L
Second with implication of not L
Combine them using the if clause

46. SGDPLL algorithm – Algorithm

47. Lecture layout: What is Probabilistic Inference? Definitions
Real life example
Boolean Satisfiability problem (SAT) – reminder
Satisfiability Modulo Theories – SAT generalization
SGDPLL(T) – one step further
Generalize the problem further
The SGDPL algorithm
Applications in probability
Experiments and Optimizations

48. SGDPLL algorithm – Applications in probability
Discreet probabilistic calculations can be reduced to SGDLL summation problems.
Example: Calculating marginal probability
P= joint probability distribution
X=X1,X2X3…..

49. SGDPLL algorithm – Applications in probability
Example: calculating posterior probability, useful in probabilistic inference. P and X like before We solve this with 2 SGDPLL(T)s One to get a summation free expression in the numerator One for the denominator

50. SGDPLL algorithm – Back to our problem
Given N voters, the Amount of voters that like an incumbent Senator
is influenced by:
The number of terror attacks
The Dow Jones index
The number of newly created jobs
The number of people who like the challenging candidate.

51. SGDPLL algorithm – Applications in probability
Let us take: Attacks ~ Uniform[20] newJobs ~ Uniform[10000] Dow ~ Uniform[11000,18000] likeChallenger ~ Uniform[N] And take the influence on P(likeIncumbant) to be:

52. SGDPLL algorithm – Applications in probability
Let us take: Attacks ~ Uniform[20] newJobs ~ Uniform[10000] Dow ~ Uniform[11000,18000] likeChallenger ~ Uniform[N] And take the influence on P(likeIncumbant) to be:

53. SGDPLL algorithm – Applications in probability
Since our Constraints are expressed in a language receivable by SGDPLL We don’t have to generate a database We can just transform the rules to SGDPLL format. For example, Attacks ~ Uniform[20] become: If attacks ≥ 0 and attacks ≤ 20 then 1/21 else 0

54. SGDPLL algorithm – Applications in probability
P(likeIncumbant | dow, newjobs, attacks, likeChallenger) After converting, we can easily calculate complex queries on this system.

55. Probabilistic Inference
We want to calculate examples like: Now, this is much simpler: For N=108 we get the anwer:

56. Probabilistic Inference
This is a T Solution:
Expression in TL
No quantifier
Equivalent to problem =

57. Probabilistic Inference
This answer is parametric, newJob is a free variable here
We did not have to iterate over all values of newJob
We did not have to iterate over all values of attacks etc for posterior probability =

58. Lecture layout: What is Probabilistic Inference? Definitions
Real life example
Boolean Satisfiability problem (SAT) – reminder
Satisfiability Modulo Theories (SMT)– SAT generalization
SGDPLL(T) – one step further
Generalize the problem further
The SGDPL algorithm
Applications in probability
Experiments and Optimizations

59. Experimental results
SGDPLL was compared against a state of the art probabilistic inference solver, VEC. Given the aforementioned system and the query: SGDPLL took 2 seconds to complete the task, Constant in N. For N=108, VEC could not solve the problem exactly because the database it needed to create was too big to instantiate. For N = 500, newJobs ~ uniform[100], dow~[110,180]: Its took VEC 51 seconds to complete the task.

60. SGDPLL algorithm – Possible optimization
Implication trimming: Sometimes we get redundancies For example: Solution: We can keep a conjunction of all chosen literals If we find contradiction, we prince the search Notice that this can be used by any SMT solver (and there are many) so we can always use a state of the art SMT solver.

61. SGDPLL algorithm – Possible optimization
Unit propagation: In SAT, if we have a clause with one unassigned literal, we assume it to be true. For example: Assume a=T

62. SGDPLL algorithm – Possible optimization
Watched literals:
There are smart data structures that help us decide when to attempt unit propagation.
Idea:
Unit propagation only fires when all except 1 are false
So, if 2 vars are unassigned or true, we do nothing.

63. SGDPLL algorithm – Possible optimization
Watched literals: Data structure: Array with 2 triggers per clause. Clause: D C B A
T/F

64. SGDPLL algorithm – Possible optimization
Watched literals: If a is assigned false, update pointer/trigger D C B A
T/F F

65. SGDPLL algorithm – Possible optimization
Watched literals: If a is assigned false, update pointer/trigger D C B A
T/F F

66. SGDPLL algorithm – Possible optimization
Watched literals: When back tracking, don’t move back D C B A
T/F

67. SGDPLL algorithm – Possible optimization
Watched literals: If other variables are assigned, do nothing D C B A F
T/F

68. SGDPLL algorithm – Possible optimization
Watched literals: If we cant find something new and unassigned for the trigger… D C B A F
T/F

69. SGDPLL algorithm – Possible optimization
Unit
Propa
gation
Watched literals: D C B A F T

70. SGDPLL algorithm – Possible optimization
Watched literals:
Not useful for small problems (3-SAT)
Only have 2 watch 2 literals per or-clause
Rest of the time no work
Useful for big problems

71. References
Original Paper Watched literal presentation: