1. Montague meets Markov: Combining Logical and Distributional Semantics
Raymond J. Mooney
Katrin Erk
Islam Beltagy
University of Texas at Austin 1 1 1 1

2.  Unable to handle uncertain knowledge and probabilistic reasoning.
Logical AI Paradigm
Represents knowledge and data in a binary symbolic logic such as FOPC.
+ Rich representation that handles arbitrary sets of objects, with properties, relations, quantifiers, etc.
 Unable to handle uncertain knowledge and probabilistic reasoning.

3. Probabilistic AI Paradigm
Represents knowledge and data as a fixed set of random variables with a joint probability distribution.
+ Handles uncertain knowledge and probabilistic reasoning.
 Unable to handle arbitrary sets of objects, with properties, relations, quantifiers, etc.

4. Statistical Relational Learning (SRL)
SRL methods attempt to integrate methods from predicate logic (or relational databases) and probabilistic graphical models to handle structured, multi-relational data.

5. SRL Approaches (A Taste of the “Alphabet Soup”)
Stochastic Logic Programs (SLPs)
(Muggleton, 1996)
Probabilistic Relational Models (PRMs)
(Koller, 1999)
Bayesian Logic Programs (BLPs)
(Kersting & De Raedt, 2001)
Markov Logic Networks (MLNs) (Richardson & Domingos, 2006)
Probabilistic Soft Logic (PSL) (Kimmig et al., 2012)

6. SRL Methods Based on Probabilistic Graphical Models
BLPs use definite-clause logic (Prolog programs) to define abstract templates for large, complex Bayesian networks (i.e. directed graphical models).
MLNs use full first order logic to define abstract templates for large, complex Markov networks (i.e. undirected graphical models).
PSL uses logical rules to define templates for Markov nets with real-valued propositions to support efficient inference.
McCallum’s FACTORIE uses an object-oriented programming language to define large, complex factor graphs.
Goodman & Tanenbaum’s CHURCH uses a functional programming language to define, large complex generative models.

7. Markov Logic Networks [Richardson & Domingos, 2006]
Set of weighted clauses in first-order predicate logic.
Larger weight indicates stronger belief that the clause should hold.
MLNs are templates for constructing Markov networks for a given set of constants
MLN Example: Friends & Smokers 7

8. Example: Friends & Smokers
Two constants: Anna (A) and Bob (B) 8

9. Example: Friends & Smokers
Two constants: Anna (A) and Bob (B)
Friends(A,B)
Friends(A,A)
Smokes(A)
Smokes(B)
Friends(B,B)
Cancer(A)
Cancer(B)
Friends(B,A) 9

10. Example: Friends & Smokers
Two constants: Anna (A) and Bob (B)
Friends(A,B)
Friends(A,A)
Smokes(A)
Smokes(B)
Friends(B,B)
Cancer(A)
Cancer(B)
Friends(B,A) 10

11. Example: Friends & Smokers
Two constants: Anna (A) and Bob (B)
Friends(A,B)
Friends(A,A)
Smokes(A)
Smokes(B)
Friends(B,B)
Cancer(A)
Cancer(B)
Friends(B,A) 11

12. Probability of a possible world
Weight of formula i
No. of true groundings of formula i in x
A possible world becomes exponentially less likely as the total weight of all the grounded clauses it violates increases. 12

13. MLN Inference
Infer probability of a particular query given a set of evidence facts.
P(Cancer(Anna) | Friends(Anna,Bob), Smokes(Bob))
Use standard algorithms for inference in graphical models such as Gibbs Sampling or belief propagation.

14. MLN Learning Learning weights for an existing set of clausesEM
Max-margin
On-line
Learning logical clauses (a.k.a. structure learning)
Inductive Logic Programming methods
Top-down and bottom-up MLN clause learning
On-line MLN clause learning

15. Strengths of MLNs Fully subsumes first-order predicate logic
Just give  weight to all clauses
Fully subsumes probabilistic graphical models.
Can represent any joint distribution over an arbitrary set of discrete random variables.
Can utilize prior knowledge in both symbolic and probabilistic forms.
Large existing base of open-source software (Alchemy)

16. Weaknesses of MLNs
Inherits computational intractability of general methods for both logical and probabilistic inference and learning.
Inference in FOPC is semi-decidable
Inference in general graphical models is P-space complete
Just producing the “ground” Markov Net can produce a combinatorial explosion.
Current “lifted” inference methods do not help reasoning with many kinds of nested quantifiers.

17. PSL: Probabilistic Soft Logic [Kimmig & Bach & Broecheler & Huang & Getoor, NIPS 2012]
Probabilistic logic framework designed with efficient inference in mind.
Input: set of weighted First Order Logic rules and a set of evidence, just as in BLP or MLN
MPE inference is a linear-programming problem that can efficiently draw probabilistic conclusions. 17

18. PSL vs. MLN PSL MLN Atoms have boolean truth values {0, 1}.
Atoms have continuous truth values in the interval [0,1].
Inference finds truth value of all atoms that best satisfy the rules and evidence.
MPE inference: Most Probable Explanation.
Linear optimization problem.
MLN
Atoms have boolean truth values {0, 1}.
Inference finds probability of atoms given the rules and evidence.
Calculates conditional probability of a query atom given evidence.
Combinatorial counting problem. 18

19. PSL Example First Order Logic weighted rules Evidence Inference
I(friend(John,Alex)) = 1 I(spouse(John,Mary)) = 1
I(votesFor(Alex,Romney)) = 1 I(votesFor(Mary,Obama)) = 1
Inference
I(votesFor(John, Obama)) = 1
I(votesFor(John, Romney)) = 0 19

20. PSL’s Interpretation of Logical Connectives
Łukasiewicz relaxation of AND, OR, NOT
I(ℓ1 ∧ ℓ2) = max {0, I(ℓ1) + I(ℓ2) – 1}
I(ℓ1 ∨ ℓ2) = min {1, I(ℓ1) + I(ℓ2) }
I(¬ ℓ1) = 1 – I(ℓ1)
Distance to satisfaction
Implication: ℓ1 → ℓ2 is Satisfied iff I(ℓ1) ≤ I(ℓ2)
d = max {0, I(ℓ1) - I(ℓ2) }
Example
I(ℓ1) = 0.3, I(ℓ2) = ⇒ d = 0
I(ℓ1) = 0.9, I(ℓ2) = ⇒ d = 0.6 20

21. PSL Probability Distribution
PDF:
Distance to satisfaction of rule r
a possible continuous truth assignment
Normalization constant
For all rules
Weight of formula r 21

22. PSL Inference MPE Inference: (Most probable explanation)
Find interpretation that maximizes PDF
Find interpretation that minimizes summation
Distance to satisfaction is a linear function
Linear optimization problem 22

23. Semantic Representations
Formal Semantics
Uses first-order logic
Deep
Brittle
Distributional Semantics
Statistical method
Robust
Shallow
Combining both logical and distributional semantics
Represent meaning using a probabilistic logic
Markov Logic Network (MLN)
Probabilistic Soft Logic (PSL)
Generate soft inference rules
From distributional semantics 23

24. System Architecture [Garrette et al. 2011, 2012; Beltagy et al., 2013]
Sent1
LF1
BOXER
Dist. Rule
Constructor
Rule Base
Sent2
LF2
Vector Space
MLN/PSL Inference
BOXER [Bos, et al. 2004]: maps sentences to logical form
Distributional Rule constructor: generates relevant soft inference rules based on distributional similarity
MLN/PSL: probabilistic inference
Result: degree of entailment or semantic similarity score (depending on the task)
result 24

25. Markov Logic Networks [Richardson & Domingos, 2006]
Two constants: Anna (A) and Bob (B)
P(Cancer(Anna) | Friends(Anna,Bob), Smokes(Bob))
Friends(A,B)
Friends(A,A)
Smokes(A)
Smokes(B)
Friends(B,B)
Cancer(A)
Cancer(B)
Friends(B,A) 25

26. Recognizing Textual Entailment (RTE)
Premise: “A man is cutting pickles”
x,y,z. man(x) ∧ cut(y) ∧ agent(y, x) ∧ pickles(z) ∧ patient(y, z)
Hypothesis: “A guy is slicing cucumber”
x,y,z. guy(x) ∧ slice(y) ∧ agent(y, x) ∧ cucumber(z) ∧ patient(y, z)
Inference: Pr(Hypothesis | Premise)
Degree of entailment 26

27. Distributional Lexical Rules
For all pairs of words (a, b) where a is in S1 and b is in S2 add a soft rule relating the two
x a(x) → b(x) | wt(a, b)
wt(a, b) = f( cos(a, b) )
Premise: “A man is cutting pickles”
Hypothesis: “A guy is slicing cucumber”
x man(x) → guy(x) | wt(man, guy)
x cut(x) → slice(x) | wt(cut, slice)
x pickle(x) → cucumber(x) | wt(pickle, cucumber)
x man(x) → cucumber(x) | wt(man, cucumber)
x pickle(x) → guy(x) | wt(pickle, guy)
→ → 27

28. Distributional Phrase Rules
Premise: “A boy is playing”
Hypothesis: “A little kid is playing”
Need rules for phrases
x boy(x) → little(x) ∧ kid(x) | wt(boy, "little kid")
Compute vectors for phrases using vector addition [Mitchell & Lapata, 2010]
"little kid" = little + kid 28

29. Paraphrase Rules [by: Cuong Chau]
Generate inference rules from pre-compiled paraphrase collections like Berant et al. [2012]
e.g,
“X solves Y” => “X finds a solution to Y ” | w 29 29 29

30. Evaluation (RTE using MLNs)
Dataset
RTE-1, RTE-2, RTE-3
Each dataset is 800 training pairs and 800 testing pairs
Use multiple parses to reduce impact of misparses 30

31. Evaluation (RTE using MLNs) [by: Cuong Chau]
Logic-only baseline
KB is wordnet
RTE-1 RTE-2 RTE-3
Bos & Markert[2005] – –
MLN
MLN-multi-parse
MLN-paraphrases 31

32. Semantic Textual Similarity (STS)
Rate the semantic similarity of two sentences on a 0 to 5 scale
Gold standards are averaged over multiple human judgments
Evaluate by measuring correlation to human rating
S S2 score
A man is slicing a cucumber A guy is cutting a cucumber 5
A man is slicing a cucumber A guy is cutting a zucchini 4
A man is slicing a cucumber A woman is cooking a zucchini 3
A man is slicing a cucumber A monkey is riding a bicycle 1 32

33. Softening Conjunction for STS
Premise: “A man is driving”
x,y. man(x) ∧ drive(y) ∧ agent(y, x)
Hypothesis: “A man is driving a bus”
x,y,z. man(x) ∧ drive(y) ∧ agent(y, x) ∧ bus(z) ∧ patient(y, z)
Break the sentence into “mini-clauses” then combine their evidences using an “averaging combiner” [Natarajan et al., 2010]
Becomes
x,y,z. man(x) ∧ agent(y, x)→ result()
x,y,z. drive(y) ∧ agent(y, x)→ result()
x,y,z. drive(y) ∧ patient(y, z) → result()
x,y,z. bus(z) ∧ patient(y, z) → result() 33

34. Evaluation (STS using MLN)
Microsoft video description corpus (SemEval 2012)
Short video descriptions
System Pearson r
Our System with no distributional rules [Logic only] 0.52
Our System with lexical rules
Our System with lexical and phrase rules 34

35. PSL: Probabilistic Soft Logic [Kimmig & Bach & Broecheler & Huang & Getoor, NIPS 2012]
MLN's inference is very slow
PSL is a probabilistic logic framework designed with efficient inference in mind
Inference is a linear program 35

36. STS using PSL - Conjunction
Łukasiewicz relaxation of AND is very restrictive
I(ℓ1 ∧ ℓ2) = max {0, I(ℓ1) + I(ℓ2) – 1}
Replace AND with weighted average
I(ℓ1 ∧ … ∧ ℓn) = w_avg( I(ℓ1), …, I(ℓn))
Learning weights (future work)
For now, they are equal
Inference
“weighted average” is a linear function
no changes in the optimization problem 36

37. Evaluation (STS using PSL)
msr-vid: Microsoft video description corpus (SemEval 2012)
Short video description sentences
msr-par: Microsoft paraphrase corpus (SemEval 2012)
Long news sentences
SICK: (SemEval 2014)
msr-vid msr-par SICK
vec-add (dist. only)
vec-mul (dist. only)
MLN (logic + dist.)
PSL-no-DIR (logic only)
PSL (logic + dist.)
PSL+vec-add (ensemble) 37

38. Evaluation (STS using PSL)
msr-vid msr-par SICK
PSL time/pair s 30s 10s
MLN time/pair 1m 31s 11m 49s 4m 24s
MLN timeouts(10 min) 9% 97% 36% 38