1. Logic Programming & LPMLN

2. Table of Contents Logic programming Answer set programming
Markov logic network
LPMLN

3. Logic Programming

4. Definitions
Term: 𝑐 𝑥 𝑓( 𝑡 1 , …, 𝑡 𝑛 ) Atom: 𝑃 𝑡 1 ,…, 𝑡 𝑛 Literal: 𝐴 | ¬𝐴 Substitution: 𝜃={𝑥 / 𝑎, 𝑦 / 𝑧}

5. Rule & Fact
(¬𝑝∧¬𝑞∧…∧¬𝑡∧𝑢) or 𝑝∧𝑞∧…∧𝑡 →𝑢 u :- p, q, ..., t. (Rule) u :- . (Fact) head :- body.

6. Basic Datalog 𝐿 0 :− 𝐿 1 , 𝐿 2 ,…, 𝐿 𝑛
𝐿 0 :− 𝐿 1 , 𝐿 2 ,…, 𝐿 𝑛
𝐿 𝑖 is a literal of the form 𝑝 𝑖 𝑡 1 , 𝑡 2 , , 𝑡 𝑘 𝑖
𝑡 𝑗 is either a constant or a variable
Safety condition for program 𝑃
Each fact of 𝑃 is ground
Each variable which occurs in the head of a rule of 𝑃 must also occur in the body of the same rule
These conditions guarantee that the set of all facts that can be derived from a Datalog program is finite

7. Datalog Rule Example
If X is a parent of Y and if Y is a parent of Z, then X is a grandparent of Z. grandpar(Z, X) :- par(Y, X), par(Z, Y). Same Generation Cousin sgc(X, X) :- person(X). sgc(X, Y) :- par(X, X1), sgc(X1, Y1), par(Y, Y1).

8. Herbrand Model
A set of ground facts 𝐼 that satisfy a clause 𝐶 or a set of clause 𝑆 is called a Herbrand model for 𝐶 or 𝑆
Intersection property: If 𝐼 1 and 𝐼 2 are Herbrand model, then 𝐼 1 ∩ 𝐼 2 is also Herbrand model
Least Herbrand model: The intersection of all Herbrand models

9. Proof Theory Elementary Production Principle Proof Tree
𝐿 0 :- 𝐿 1 , 𝐿 2 ,…, 𝐿 𝑛
𝐹 1 , 𝐹 2 ,…, 𝐹 𝑛 are facts
If there exists a substitution 𝜃 that 𝐿 𝑖 𝜃= 𝐹 𝑖 then 𝐿 0 𝜃 can be inferred as a new fact
Proof Tree

10. Bottom-up Evaluation
r1: sgc(X, X) :- person(X). r2: sgc(X, Y) :- par(X, X1), sgc(X1, Y1), par(Y, Y1). person(a). person(b). person(c). par(a, c). par(b, c). sgc(a, a). sgc(b, b). sgc(c, c). sgc(a, b) :- par(a, c), sgc(c, c), par(b, c).

11. Top-down Evaluation
?- sgc(a, Y). person(a). -> Y = {a} par(a, X1). -> X1 = {c} sgc(c, Y1). person(c). -> Y1 = {c} par(c, X1). -> X1 = {} ... -> Y1 = {c} par(Y, Y1) -> Y = {b} -> Y = {a, b}

12. Problem Solving Example
word(d,o,g). word(r,u,n). word(t,o,p). word(f,i,v,e). word(f,o,u,r). word(l,o,s,t). word(m,e,s,s). word(u,n,i,t). word(b,a,k,e,r). word(f,o,r,u,m). word(g,r,e,e,n). word(s,u,p,e,r). word(p,r,o,l,o,g). word(v,a,n,i,s,h). word(w,o,n,d,e,r). word(y,e,l,l,o,w).

13. Problem Solving Example
solution(L1,L2,L3,L4,L5,L6,L7,L8,L 9,L10,L11,L12,L13,L14,L15,L16):- word(L1,L2,L3,L4,L5), word(L9,L10,L11,L12,L13,L14), word(L1,L6,L9,L15), word(L3,L7,L11), word(L5,L8,L13,L16).

14. Bottom-up Evaluation Optimization
Naïve approach: iteratively apply rules to all facts until no new fact can be derived
Problem
One fact is derived many times
The fact irrelevant to goal is also derived
Solution
Semi-naïve approach
Magic set rewriting

15. Answer Set Programming

16. Rules with Negation Closed world assumption
If a fact does not logically follow from a set of Datalog clauses, then we conclude that the negation of this fact is true.
Problem: multiple minimal Herbrand models
boring(chess) :- not interesting(chess)
𝐻 𝑎 = 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑖𝑛𝑔 𝑐ℎ𝑒𝑠𝑠 , 𝐻 𝑏 ={𝑏𝑜𝑟𝑖𝑛𝑔 𝑐ℎ𝑒𝑠𝑠 }

17. Stable Model Ground program
Assume a set of facts 𝑀 of program Π, the reduct of Π according to 𝑀 ( Π 𝑀 ) is defined as follows
For all literals 𝐿 𝑖 with negation and | 𝐿 𝑖 | is in 𝑀, drop it from rule body
For all rules whose body contains negation, drop the rule
Thus all the rule contains no negation
If the minimal Herbrand model coincides with 𝑀, then 𝑀 is a stable model of Π

18. Examples
p :- not q q :- not p {} -> {p, q} {p} -> {p} {q} -> {q} {p, q} -> {} Stable models are {p}, {q}
a :- not a {} -> {a} {a} -> {} No stable model

19. Constraints
a :- B, not a. If B is true, then the rule becomes a :- not a. which means there’s no stable model for the program We can write such a rule as :- B

20. Multiple Literals in Rule Head
If the minimal Herbrand models of reduct Π 𝑀 contains 𝑀, then 𝑀 is a stable model of Π
For example
a; b :- c, d. c. d.
Only {a, c, d} and {b, c, d} are stable models, {a, b, c, d} is not.

21. Example: Solving Hamiltonian Cycle
vertex(a). vertex(b). vertex(c). vertex(d). edge(a, b). edge(b, c). edge(c, d). edge(d, a). edge(b, d). in(X, Y) :- edge(X, Y), not nin(X, Y). nin(X, Y) :- edge(X, Y), not in(X, Y). :- in(X, Y), in(X1, Y), X != X1. :- in(X, Y), in(X, Y1), Y != Y1. reachable(X, X) :- vertex(X). reachable(X, Y) :- vertex(X), vertex(Y), reachable(X, X1), in(X1, Y). :- vertex(X), vertex(Y), not reachable(X, Y).
Online ASP Solver:

22. Markov Logic Network

23. Markov network Node: random variable Edge: variable dependency
Potential function 𝜙: non-negative function for clique
𝑃 𝑋=𝑥 = 1 𝑍 Π 𝑘 𝜙 𝑘 𝑥 𝑘 , 𝑍= Σ 𝑥∈𝒳 Π 𝑘 𝜙 𝑘 ( 𝑥 𝑘 )
Log-linear model
𝑃 𝑋=𝑥 = 1 𝑍 exp⁡( Σ 𝑗 𝑤 𝑗 𝑓 𝑗 𝑥 )

24. Markov Logic Network Definition
A Markov logic network 𝐿 is a set of (𝐹 𝑖 , 𝑤 𝑖 ), where 𝐹 𝑖 is a formula in first- order logic and 𝑤 𝑖 is a real number.
Together with a finite set of constants 𝐶={ 𝑐 1 , 𝑐 2 ,…, 𝑐 𝑛 } defines a Markov network 𝑀 𝐿,𝐶 .
1. 𝑀 𝐿,𝐶 contains one binary node for each possible grounding of each predicate appearing in 𝐿. The value of the node is 1 if the ground atom is true, and 0 otherwise.
2. 𝑀 𝐿,𝐶 contains one feature for each possible grounding of each formula 𝐹 𝑖 in 𝐿. The value of this feature is 1 if the ground formula is true, and 0 otherwise. The weight of the feature is the 𝑤 𝑖 associated with 𝐹 𝑖 in 𝐿.

25. Example

26. Example Given constant A and B, consider last 2 formulas in the table.
Ground atoms
Fr(A, A), Fr(A, B), Fr(B, A), Fr(B, B), Sm(A), Sm(B)
Ground formulas
¬𝑆𝑚 𝐴 ∨𝐶𝑎(𝐴)
¬𝑆𝑚 𝐵 ∨𝐶𝑎(𝐵)
¬𝐹𝑟 𝐴,𝐴 ∨𝑆𝑚 𝐴 ∨¬Sm 𝐴
¬𝐹𝑟 𝐴,𝐴 ∨¬𝑆𝑚 𝐴 ∨Sm 𝐴
¬𝐹𝑟 𝐵,𝐵 ∨𝑆𝑚 𝐵 ∨¬Sm 𝐵
¬𝐹𝑟 𝐵,𝐵 ∨¬𝑆𝑚 𝐵 ∨Sm 𝐵
¬𝐹𝑟 𝐴,𝐵 ∨𝑆𝑚 𝐴 ∨¬Sm 𝐵
¬𝐹𝑟 𝐴,𝐵 ∨¬𝑆𝑚 𝐴 ∨Sm 𝐵
¬𝐹𝑟 𝐵,𝐴 ∨𝑆𝑚 𝐵 ∨¬Sm 𝐴
¬𝐹𝑟 𝐵,𝐴 ∨¬𝑆𝑚 𝐵 ∨Sm 𝐴

27. Example

28. LPMLN
Weighted Rules under the Stable Model Semantics

29. LPMLN Rule and Definition
Rule format
𝑤 :𝑅
Hard rule
𝑤=𝛼
For a LPMLN program Π
Π : 𝑅 𝑤:𝑅∈Π}
Π 𝐼 : the set of rules 𝑤:𝑅 such that 𝐼⊨𝑅
𝑆𝑀 Π : 𝐼 𝐼 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑡𝑎𝑏𝑙𝑒 𝑚𝑜𝑑𝑒𝑙 𝑜𝑓 Π 𝐼 }

30. LPMLN Definition

31. Example

32. Example

33. Relating LPMLN to ASP & MLN

34. LPMLN Implementation

35. Reformulating LPMLN Based on Penalty

36. Turning LPMLN into ASP

37. Turning LPMLN into MLN

38. References
Ceri, Stefano, Georg Gottlob, and Letizia Tanca. "What you always wanted to know about Datalog (and never dared to ask)." IEEE transactions on knowledge and data engineering1.1 (1989):
Brewka, Gerhard, Thomas Eiter, and Mirosław Truszczyński. "Answer set programming at a glance." Communications of the ACM 54.12 (2011):
Richardson, Matthew, and Pedro Domingos. "Markov logic networks." Machine learning  (2006):
Lee, Joohyung, and Yi Wang. "Weighted Rules under the Stable Model Semantics." KR
Lee, Joohyung, Samidh Talsania, and Yi Wang. "Computing LP MLN using ASP and MLN solvers." Theory and Practice of Logic Programming  (2017):

39. Thank you