[PPT] - Probabilistic Logic Programming & Knowledge Compilation PowerPoint Presentation

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Probabilistic Logic Programming & Knowledge Compilation

Wannes Meert

DTAI, Dept. Computer Science, KU Leuven Dagstuhl, 18 September 2017 In collaboration with Jonas Vlasselaer, Guy Van den Broeck, Anton Dries, Angelika Kimmig, Hendrik Blockeel, Jesse Davis and Luc De Raedt

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StarAI

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Dealing with uncertainty:

Probability theory
Graphical models

Learning

Parameters
Structure

Reasoning with relational data

Logic
Database
Programming

?

statistical relational AI, Statistical relational learning, probabilistic logic learning, probabilistic programming, ...

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ProbLog

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Uncertainty Learning Relational data

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stress(ann). influences(ann,bob). influences(bob,carl). smokes(X) :- stress(X).  smokes(X) :- influences(Y,X), smokes(Y). 0.8::stress(ann). 0.6::influences(ann,bob). 0.2::influences(bob,carl).

→ One World → Multiple possible worlds

t(0.8)::stress(ann). t(_)::influences(ann,bob). t(_)::influences(bob,carl).

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Introduction to ProbLog

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Example

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h

toss (biased) coin & draw ball from each urn
win if (heads and a red ball) or (two balls of same color)

0.4 :: heads.  0.3 :: col(1,red); 0.7 :: col(1,blue). 0.2 :: col(2,red); 0.3 :: col(2,green); 0.5 :: col(2,blue).  win :- heads, col(_,red). win :- col(1,C), col(2,C). evidence(heads). query(win).

Probabilistic fact Annotated disjunction Logical rules / background knowledge Evidence Query

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Example

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0.4 :: heads. 0.3 :: col(1,red); 0.7 :: col(1,blue). 0.2 :: col(2,red); 0.3 :: col(2,green); 0.5 :: col(2,blue).  win :- heads, col(_,red). win :- col(1,C), col(2,C).

W

R R

H W

R R G G 0.4x0.3x0.3 (1-0.4)x0.3x0.2 (1-0.4)x0.3x0.3

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All possible worlds

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W

R R

H W

R B

H W

R G

H W

R R R G R B

H W

B B

H

G B

H W

R B R B G B

W

B B 0.024 0.036 0.060 0.036 0.054 0.090 0.056 0.084 0.084 0.126 0.140 0.210

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P(win) = ∑

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W

R R

H W

R B

H W

R G

H W

R R R G R B

H W

B B

H

G B

H W

R B R B G B

W

B B 0.024 0.036 0.060 0.036 0.054 0.090 0.056 0.084 0.084 0.126 0.140 0.210

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Alternative view: CP-logic

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throws(john). 0.5::throws(mary). 0.8 :: break :- throws(mary). 0.6 :: break :- throws(john).

probabilistic causal laws

John throws Window breaks Window breaks Window breaks doesn’t break doesn’t break doesn’t break Mary throws Mary throws doesn’t throw doesn’t throw 1.0 0.6 0.4 0.5 0.5 0.5 0.5 0.8 0.8 0.2 0.2

P(break)=0.6×0.5×0.8 + 0.6×0.5×0.2 + 0.6×0.5 + 0.4×0.5×0.8

9 [Vennekens et al 2003, Meert and Vennekens 2014]

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Sato’s Distribution semantics

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P(Q) = X

F [R| =Q

Y

f2F

p(f) Y

f62F

1 − p(f)

[Sato ICLP 95]

query subset of probabilistic facts Prolog rules sum over possible worlds where Q is true probability of possible world

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Examples from Tutorial

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Try yourself: https://dtai.cs.kuleuven.be/problog

$ pip install problog

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Tutorial: Bayes net

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Tutorial: Higher-order functions

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Tutorial: As a Python Library

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from problog.program import SimpleProgram from problog.logic import Constant,Var,Term,AnnotatedDisjunction coin,heads,tails,win = Term(‘coin'),Term('heads'),Term('tails'),Term('win') C = Var('C') p = SimpleProgram() p += coin(Constant('c1')) p += coin(Constant('c2')) p += AnnotatedDisjunction([heads(C,p=0.4), tails(C,p=0.6)], coin(C)) p += (win << heads(C)) p += query(win) lf = LogicFormula.create_from(p) # ground the program cnf = CNF.create_from(lf) # convert to CNF ddnnf = DDNNF.create_from(cnf) # compile CNF to ddnnf ddnnf.evaluate()

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Weighted Model Counting

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WMC(φ) = X

IV | =φ

Y

l∈I

w(l)

interpretations (truth value assignments) of propositional variables weight of literal propositional formula in conjunctive normal form (CNF) Given by ProbLog program and query Possible worlds for p::f, w(f) = p w(¬f) = 1-p

P(Q) = X

F [R| =Q

Y

f2F

p(f) Y

f62F

1 − p(f)

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Encodings/Compilers for WMC

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ProbLog d-DNNF SDD BDD CNF Tp-compilation Also links to MaxSAT (decisions), Bayes net inference, ... Grounding Cycle breaking Formula

usage: problog [--knowledge {sdd,bdd,nnf,ddnnf,kbest,fsdd,fbdd}] ... (+various tools)

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Impact of encoding

17 × × × × ¬y1 × ¬y0 ¬x + . . . ¬y0 ¬yn . . . smooth(Y2, . . . , Yn) y1 × × × + x smooth(Y1, . . . , Yn) y0 + WMC(x) = w(y0) + w(¬y0) · w(y1) + . . . = P

i w(yi) · Q j<i(1 − w(yj))

WMC(¬x) = Q

i w(¬yi)

= Q

i(1 − w(yi))

Since w(yi) + w(¬yi) = 1,smooth(·) = 1

y(1) ⇔ p(1,1) y(2) ⇔ p(1,2) … y(n) ⇔ p(1,n) x ⇔ (y(1) ∧ p(2,1)) ∨ … ∨ (y(n) ∧ p(2,n)) [Van den Broeck 2014, Meert 2016]

Noisy-OR

Y1 X Y2 . . . Yn

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Impact of encoding

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× . . . × × ¬yn × r × × ¬r ¬y0 ¬yn ¬y0 × ¬x . . . + x smooth(Y0, . . . , Yn) r + Since w(yi) + w(¬yi) = 1,smooth(·) = 1 WMC(¬x) = Q

i w(¬yi)

= Q

i(1 − w(yi))

WMC(x) = 1 − Q

i w(¬yi)

= 1 − Q

i(1 − w(yi))

[Van den Broeck 2014, Meert 2016] y(1) ⇔ p(1,1) y(2) ⇔ p(1,2) … y(n) ⇔ p(1,n) x ⇔ (y(1) ∧ p(2,1)) ∨ … ∨ (y(n) ∧ p(2,n))

Noisy-OR

Y1 X Y2 . . . Yn

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Is KC just a toolbox for us? No, to tackle some types of problems we need to interact while compiling or performing inference. Yes, separate concerns and conveniently use what is available (and improve timings by waiting)

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Tp-compilation

Forward inference  Incremental compilation

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Why Tp-compilation

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Domains with many cycles or long temporal chains: We encountered two problems:

1. Not always feasible to compile CNF
2. Not always feasible to create CNF

Social Genes webpages Sensor networks

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Before

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Grounding Loop breaking CNF conversion

Sampling on the CNF e.g. MC-SAT

[Poon and Domingos, NCAI ‘06]

“Approximate” Compilation e.g. via Weighted Partial MaxSAT

[Renkens et al., AAAI ‘14]

‘Exact’ Knowledge Compilation e.g. OBDD, d-DNNF, SDD

[Fierens et al., TPLP ‘15]

Horn Approximation

[Selman and Kautz, AAAI ’91]

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Tp-compilation

Generalizes the Tp operator from logic programming towards the

probabilistic setting.

Tp operator (forward reasoning):
Start with what is known.
Derive new knowledge by applying the rules.
Continue until fixpoint (interpretation unchanged)
Tp-compilation
Start with an empty formula for each probabilistic fact
Construct new formulas by applying the rules
Continue until fixpoint (formulas remains equivalent)

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Apply-operator Equivalence SDD Bounds available at every iteration

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Really a problem?

Fully connected graph with 10 nodes (90 edges)
CNF contains +25k variables and +100k clauses
Tp-compilation only requires 90 variables
Alzheimer network

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Continuous observations

Sensor measurements Circuit interacts with other representations

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Continuous sensor measurements

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“ ”

proefstand laat toe om complexere scenario’s te

WP’s in meer detail worden uitgewerkt

–
normal(0.2,0.1)::vibration(X) :- op1(X).

normal(0.6,0.2)::vibration(X) :- op2(X). normal(3.1,1.1)::vibration(X) :- fault(X). 0.2::fault(X) :- connected(X,Y), fault(Y). Restricted setting:

Sensor measurements are always available
Only used in head

Ongoing work

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Continuous values

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t(0.5)::c(ID). t(normal(1, 10))::f(ID) :- c(ID). t(normal(10,10))::f(ID) :- \+c(ID). evidence(f(1), 10). evidence(f(2), 12). evidence(f(3), 8 ). evidence(f(4), 11). evidence(f(5), 7 ). evidence(f(6), 13). evidence(f(7), 20). evidence(f(8), 21). evidence(f(9), 22). evidence(f(10), 18). evidence(f(11), 19). evidence(f(12), 19). evidence(f(13), 19). evidence(f(14), 23). evidence(f(15), 21). 0.40::c(ID). normal(10.16,2.11)::f(ID) :- c(ID). normal(20.22,1.54)::f(ID) :- \+c(ID).

Gaussian Mixture Model

9 θ₂ ¬θ₁ ¬θ₂ ⊤ 9 θ₂ ¬θ₁ ¬θ₂ θ₁ 9 ¬θ₂ θ₁ θ₂ ⊤ 9 θ₂ ¬θ₁ ¬θ₂ ⊥ 9 θ₂ θ₁ ¬θ₂ ⊥ 9 ¬θ₂ θ₁ θ₂ ⊥ 9 θ₂ θ₁ ¬θ₂ ⊤ 9 ¬θ₂¬θ₁ θ₂ ⊥ 9 ¬θ₂¬θ₁ θ₂ θ₁ 9 ¬θ₂¬θ₁ θ₂ ⊤ 7 ¬θ₃ θ₃ ⊤ 7 θ₃ ¬θ₃¬θ₂ 7 θ₃ ¬θ₃¬θ₁ 7 ¬θ₃ θ₃ 7 ¬θ₃ θ₃ θ₂ 7 ¬θ₃ θ₃ 7 θ₃ ¬θ₃ ⊥ 7 ¬θ₃ θ₃ θ₁ 5 θ₄ ¬θ₄ 5 ¬θ₄ θ₄ 5 θ₄ ¬θ₄ 5 ¬θ₄ θ₄ 3 E ¬E 3 ¬E E 1 ¬θ₆ θ₆

+ weights are functions

Ongoing work

9 E ¬F ¬E ⊥ 7 D ¬D⊥ 3 ¬B B ⊥ 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 0.05 0.1 0.15

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Resource Aware Circuits

Memory and energy Circuits that are ‘hardware-friendly’

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Why resource aware?

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spokes partner’s

f-the-art sensor fusion allows to combine sensory information streams to improve sensory info

Integrate AI and hardware to achieve dynamic attention-scalability → Adapt hardware dynamically and smart Allows to extract the maximum of information of relevant information under our limited computational bandwidth.

Resource-aware inference and fusion algorithms
Resource-scalable inference processors

Figure 24.2.4: (left) Schematic and decision tree algorithm for mixed-

“ ”

On the system’s operating mode:
μ

Previous work: Decision Trees Ongoing work

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Current challenges

We need self-reflection
To understand sensitivities to (de)activate sensors

Same Decision Probability ...

We need to steer the compilation to ‘hardware friendly’ structures
Group memory fetches, fixed-point operations, ...

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Ongoing work

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Other challenges

First-order reasoning, also for probabilities (see Guy’s talk)
First-order circuits, WFOMC
Constraints
cProbLog, MLNs
Hybrid domains (continuous variables)
Knowledge compilation + other semirings / SMT solvers
Structure learning
Incrementally updating circuits, PSDD
Link to NLP
...

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Mike has a bag of marbles with 4 white, 8 blue, and 6 red marbles. He pulls out one marble from the bag and it is red. What is the probability that the second marble he pulls out of the bag is white?

The answer is 0.235941.

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Thank you!

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