cProbLog: Restricting the Possible Worlds of Probabilistic Logic - - PowerPoint PPT Presentation

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cProbLog: Restricting the Possible Worlds of Probabilistic Logic - - PowerPoint PPT Presentation

Jan 11, 2023 41 likes •675 views

cProbLog: Restricting the Possible Worlds of Probabilistic Logic Programs Dimitar Shterionov Prof. Gerda Janssens 1 Weight: 3 Weight: 4 Weight: 8 Weight: 6 2 Weight: 3 Weight: 4 0.33 Weight: 8 Weight: 6 0.25 0.125 0.16 3 Weight:

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cProbLog: Restricting the Possible Worlds of Probabilistic Logic Programs

Dimitar Shterionov

  • Prof. Gerda Janssens
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Weight: 6 Weight: 4 Weight: 8 Weight: 3

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Weight: 6 Weight: 4 Weight: 8 Weight: 3 0.16 0.25 0.33 0.125

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Luggage weight 10 kg. Weight: 6 Weight: 4 Weight: 8 Weight: 3 0.16 0.25 0.33 0.125

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Luggage weight 10 kg. Weight: 6 Weight: 4 Weight: 8 Weight: 3 0.16 0.25 0.33 0.125 Query: what is the probability of carrying items without exceeding the limit?

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Luggage weight 10 kg. Weight: 6 Weight: 4 Weight: 8 Weight: 3 0.16 0.25 0.33 0.125 Query: what is the probability of carrying items without exceeding the limit?

ProbLog

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Luggage weight 10 kg. Weight: 6 Weight: 4 Weight: 8 Weight: 3 0.16 0.25 0.33 0.125 Query: what is the probability of carrying items without exceeding the limit? Additional knowledge: if skis are packed then boots also need to be packed.

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Luggage weight 10 kg. Weight: 6 Weight: 4 Weight: 8 Weight: 3 0.16 0.25 0.33 0.125 Query: what is the probability of carrying items without exceeding the limit? Additional knowledge: if skis are packed then boots also need to be packed.

cProbLog

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Outline

  • ProbLog and Possible Worlds
  • cProbLog – FOL sentences expressing constraints
  • Implementation – constraint-evidence approach
  • Examples
  • Comparison to other systems
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Outline

  • ProbLog and Possible Worlds
  • cProbLog – FOL sentences expressing constraints
  • Implementation – constraint-evidence approach
  • Examples
  • Comparison to other systems
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Logic

Uncertainties

ML

ProbLog

  • Compute the marginal

probability given evidence

  • Learn from Interpretations
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weight(skis,6). weight(board, 8). weight(boots,4). weight(helmet,3). 0.160::pack(skis). 0.125::pack(board). 0.250::pack(boots). 0.330::pack(helmet). inlimit(Limit):- inlimit([skis,boots,board,helmet],Limit). inlimit([],Limit):- Limit>=0. inlimit([I|R],Limit):- pack(I), weight(I,W), L is Limit-W, inlimit(R,L). inlimit([I|R],Limit):- \+pack(I), inlimit(R,Limit).

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weight(skis,6). weight(board, 8). weight(boots,4). weight(helmet,3). 0.160::pack(skis). 0.125::pack(board). 0.250::pack(boots). 0.330::pack(helmet). inlimit(Limit):- inlimit([skis,boots,board,helmet],Limit). inlimit([],Limit):- Limit>=0. inlimit([I|R],Limit):- pack(I), weight(I,W), L is Limit-W, inlimit(R,L). inlimit([I|R],Limit):- \+pack(I), inlimit(R,Limit).

Probabilistic Facts

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weight(skis,6). weight(board, 8). weight(boots,4). weight(helmet,3). 0.160::pack(skis). 0.125::pack(board). 0.250::pack(boots). 0.330::pack(helmet). inlimit(Limit):- inlimit([skis,boots,board,helmet],Limit). inlimit([],Limit):- Limit>=0. inlimit([I|R],Limit):- pack(I), weight(I,W), L is Limit-W, inlimit(R,L). inlimit([I|R],Limit):- \+pack(I), inlimit(R,Limit).

Probabilistic Facts pack(ski) pack(board) pack(boots) pack(helmet) 0.16 0.84 0.125 0.875 0.25 0.75 0.33 0.67

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pack(ski) pack(board) pack(boots) pack(helmet) 0.84 0.875 0.25 0.33

A possible world

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pack(ski) pack(board) pack(boots) pack(helmet) 0.84 0.875 0.25 0.33

* * * = 0.061 A possible world

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pack(ski) pack(board) pack(boots) pack(helmet) 0.84 0.875 0.25 0.33

* * * = 0.061 A possible world

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A possible world corresponds to a model of the ProbLog program A query atom is true in some possible worlds. (the models of the ProbLog program and the query)

pack(ski) pack(board) pack(boots) pack(helmet) 0.84 0.875 0.25 0.33

* * * = 0.061 A possible world

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A possible world corresponds to a model of the ProbLog program A query atom is true in some possible worlds. (the models of the ProbLog program and the query)

(the success probability of q) pack(ski) pack(board) pack(boots) pack(helmet) 0.84 0.875 0.25 0.33

* * * = 0.061 A possible world

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query(inlimit(10)).

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P(inlimit(10)) = 0.9162 query(inlimit(10)).

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Outline

  • ProbLog and Possible Worlds
  • cProbLog – FOL sentences expressing constraints
  • Implementation – constraint-evidence approach
  • Examples
  • Comparison to other systems
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The MARG task

Evidence – a set of atoms (E) with observed truth values (e)

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The MARG task

boots are already packed

Evidence – a set of atoms (E) with observed truth values (e)

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The MARG task

boots are already packed

Evidence – a set of atoms (E) with observed truth values (e)

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P(inlimit(10)|pack(boots)=true)

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P(inlimit(10)|pack(boots)=true)

What about more complex additional knowledge? Eg.,“if the skis are packed then also the boots need to be packed” FOL:

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cProbLog: Generalization of Evidence to FOL Constraints

What about more complex additional knowledge? Eg.,“if the skis are packed then also the boots need to be packed” FOL:

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cProbLog Syntax

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cProbLog Syntax

pack(skis) implies pack(boots) for_all X in {1, 2, 3}: exists Y in {a, b}: match(X, Y)

for_all X of init_node(X): exists Y of end_node(Y): edge(X, Y) implies not edge(X, 1)

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cProbLog Syntax

pack(skis) implies pack(boots) for_all X in {1, 2, 3}: exists Y in {a, b}: match(X, Y)

for_all X of init_node(X): exists Y of end_node(Y): edge(X, Y) implies not edge(Y, 1)

determine the relevant grounding ensure finite grounding

  • f the constraints

domains

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cProbLog Semantics

Constraints are true in a set of possible worlds (subset of the possible worlds of the initial ProbLog program)

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cProbLog Semantics

Constraints are true in a set of possible worlds (subset of the possible worlds of the initial ProbLog program)

Probability of a Query

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cProbLog Semantics

Constraints are true in a set of possible worlds (subset of the possible worlds of the initial ProbLog program)

Probability of a Query

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query(inlimit(10)). constraint(pack(skis) implies pack(boots)).

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P(inlimit(10)|pack(skis) implies pack(boots)) = 0.9218.

query(inlimit(10)). constraint(pack(skis) implies pack(boots)).

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Outline

  • ProbLog and Possible Worlds
  • cProbLog – FOL sentences expressing constraints
  • Implementation – constraint-evidence approach
  • Examples
  • Comparison to other systems
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Additional rules + Evidence

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Constraints Additional rules + Evidence

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Constraints Additional rules + Evidence

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Constraints

Constraint – Evidence Approach

Additional rules + Evidence

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Constraint – Evidence Approach

A transformation of a single constraint into prolog rules and setting evidence.

1.Convert to Prenex Normal Form 2.Generate Prolog rules + Instantiate every variable ▪ Conjunction ▪ Disjunction 3.Add evidence 4.Do ProbLog MARG

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Constraint – Evidence Approach

A transformation of a single constraint into prolog rules and setting evidence.

Rewriting rules 1.Convert to Prenex Normal Form 2.Generate Prolog rules + Instantiate every variable ▪ Conjunction ▪ Disjunction 3.Add evidence 4.Do ProbLog MARG

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Constraint – Evidence Approach

A transformation of a single constraint into prolog rules and setting evidence.

for_all X in {1, 2, 3}: exists Y in {a, b} : match(X, Y) Rewriting rules 1.Convert to Prenex Normal Form 2.Generate Prolog rules + Instantiate every variable ▪ Conjunction ▪ Disjunction 3.Add evidence 4.Do ProbLog MARG

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Constraint – Evidence Approach

A transformation of a single constraint into prolog rules and setting evidence.

for_all X in {1, 2, 3}: exists Y in {a, b} : match(X, Y) Rewriting rules 1.Convert to Prenex Normal Form 2.Generate Prolog rules + Instantiate every variable ▪ Conjunction ▪ Disjunction 3.Add evidence 4.Do ProbLog MARG

constraint_aux(1):-match(1, a); match(1, b). constraint_aux(2):-match(2, a); match(2, b). constraint_aux(3):-match(3, a); match(3, b). constraint_aux(0):-constraint_aux(1), constraint_aux(2), constraint_aux(3).

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Constraint – Evidence Approach

A transformation of a single constraint into prolog rules and setting evidence.

for_all X in {1, 2, 3}: exists Y in {a, b} : match(X, Y) Rewriting rules 1.Convert to Prenex Normal Form 2.Generate Prolog rules + Instantiate every variable ▪ Conjunction ▪ Disjunction 3.Add evidence 4.Do ProbLog MARG evidence(constraint_aux(0), true).

constraint_aux(1):-match(1, a); match(1, b). constraint_aux(2):-match(2, a); match(2, b). constraint_aux(3):-match(3, a); match(3, b). constraint_aux(0):-constraint_aux(1), constraint_aux(2), constraint_aux(3).

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Constraints Additional rules + Evidence ProbLog MARG inference

Constraint – Evidence Approach

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Outline

  • ProbLog and Possible Worlds
  • cProbLog – FOL sentences expressing constraints
  • Implementation – constraint-evidence approach
  • Examples
  • Comparison to other systems
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Burglary – Earthquake – Alarm

burglary earthquake alarm John calls Merry calls

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Burglary – Earthquake – Alarm

burglary earthquake alarm John calls Merry calls

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Burglary – Earthquake – Alarm

person(john). person(mary). 0.1::burglary. 0.2::earthquake. 0.7::hears_alarm(X):- person(X). alarm:- burglary. alarm:- earthquake. calls(X):- alarm, hears_alarm(X). query(burglary). query(earthquake). constraint((for_all X of person(X): for_all Y of person(Y)): (calls(X) and calls(Y)) implies X == Y).

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Burglary – Earthquake – Alarm

person(john). person(mary). 0.1::burglary. 0.2::earthquake. 0.7::hears_alarm(X):- person(X). alarm:- burglary. alarm:- earthquake. calls(X):- alarm, hears_alarm(X). query(burglary). query(earthquake). constraint((for_all X of person(X): for_all Y of person(Y)): (calls(X) and calls(Y)) implies X == Y). constraint_aux(0):- ((\+ calls(mary); \+ calls(mary)); mary==mary), (\+ calls(mary); \+ calls(john)); mary==john)), ((\+ calls(john); \+ calls(mary)); john==mary), (\+ calls(john); \+ calls(john)); john==john)). evidence(constraint_aux(0), true).

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Burglary – Earthquake – Alarm

person(john). person(mary). 0.1::burglary. 0.2::earthquake. 0.7::hears_alarm(X):- person(X). alarm:- burglary. alarm:- earthquake. calls(X):- alarm, hears_alarm(X). query(burglary). query(earthquake). constraint((for_all X of person(X): for_all Y of person(Y)): (calls(X) and calls(Y)) implies X == Y). constraint_aux(0):- ((\+ calls(mary); \+ calls(mary)); mary==mary), (\+ calls(mary); \+ calls(john)); mary==john)), ((\+ calls(john); \+ calls(mary)); john==mary), (\+ calls(john); \+ calls(john)); john==john)). evidence(constraint_aux(0), true). ev1:- \+ calls(mary). ev1:- \+ calls(john). evidence(ev1, true).

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Burglary – Earthquake – Alarm

person(john). person(mary). 0.1::burglary. 0.2::earthquake. 0.7::hears_alarm(X):- person(X). alarm:- burglary. alarm:- earthquake. calls(X):- alarm, hears_alarm(X). query(burglary). query(earthquake). constraint((for_all X of person(X): for_all Y of person(Y)): (calls(X) and calls(Y)) implies X == Y). constraint_aux(0):- ((\+ calls(mary); \+ calls(mary)); mary==mary), (\+ calls(mary); \+ calls(john)); mary==john)), ((\+ calls(john); \+ calls(mary)); john==mary), (\+ calls(john); \+ calls(john)); john==john)). evidence(constraint_aux(0), true). ev1:- \+ calls(mary). ev1:- \+ calls(john). evidence(ev1, true).

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Student's Exams

0.8::pass_exam(Exam, studied_enough). 0.4::pass_exam(Exam, luck). 0.7::pass_exam(Exam, cheating). P::pass_exam(Exam, experience, Years):- P is Years/7.

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Student's Exams

0.8::pass_exam(Exam, studied_enough). 0.4::pass_exam(Exam, luck). 0.7::pass_exam(Exam, cheating). P::pass_exam(Exam, experience, Years):- P is Years/7. student(john, [prolog, cog_sci, anns, comp_vis, m_learn], 3). succeed(Student):- student(Student, Exams, Experience), pass_all(Exams, Experience). pass_all([], _). pass_all([F|Rest], Exp):- pass_one(F, Exp), pass_all(Rest, Exp). pass_one(Ex, _):-pass_exam(Ex, _). pass_one(Ex, Years):-pass_exam(Ex, experience, Years).

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Student's Exams

0.8::pass_exam(Exam, studied_enough). 0.4::pass_exam(Exam, luck). 0.7::pass_exam(Exam, cheating). P::pass_exam(Exam, experience, Years):- P is Years/7. student(john, [prolog, cog_sci, anns, comp_vis, m_learn], 3). succeed(Student):- student(Student, Exams, Experience), pass_all(Exams, Experience). pass_all([], _). pass_all([F|Rest], Exp):- pass_one(F, Exp), pass_all(Rest, Exp). pass_one(Ex, _):-pass_exam(Ex, _). pass_one(Ex, Years):-pass_exam(Ex, experience, Years). exam(E):- member(E, [prolog, cog_sci, anns, comp_vis, m_learn]). constraint(pass_exam(m_learn, luck) implies not pass_exam(anns, luck)). constraint((for_all X of exam(X), for_all Y in {0,1,2,3}) : not pass_exam(X, experience, Y)). query(succeed(john)).

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Outline

  • ProbLog and Possible Worlds
  • cProbLog – FOL sentences expressing constraints
  • Implementation – constraint-evidence approach
  • Examples
  • Comparison to other systems
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cProbLog and ...

  • CLP(BN)
  • PCLP
  • CHRiSM
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cProbLog and ...

  • CLP(BN)
  • PCLP
  • CHRiSM

Constraints define values of Random Variables: generative

cProbLog can be employed by other systems.

cProbLog constraints limit the set of possible worlds: restrictive

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Summary

  • ProbLog defines a distribution over possible worlds
  • Constraints, similar to evidence, restrict the set of

possible worlds

  • Constraint-Evidence approach:

– Transforming – Setting evidence – ProbLog MARG inference

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Take-away message

  • cProbLog: a system which combines ProbLog and

FOL constraints, to increase the expressivity of the language

  • cProbLog implementation is not dependent on the

host and can easily be employed by other PLP systems.

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Thank you for your attention!