towards querying probabilistic knowledge basesguyvdb/slides/akbc19.pdfdesired query answer ernst...
TRANSCRIPT
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Towards Querying
Probabilistic Knowledge Bases
Guy Van den Broeck
First Conference on Automated Knowledge Base Construction
May 20, 2019
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Cartoon Motivation
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Cartoon Motivation:
Relational Embedding Models
∃x Coauthor(Einstein,x)
∧Coauthor(Erdos,x)
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?
Cartoon Motivation 2:
Relational Embedding Models
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?
Goal 1:
Probabilistic Query Evaluation
∃x Coauthor(Einstein,x)
∧Coauthor(Erdos,x)
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?
Goal 2: Querying
Relational Embedding Models
∃x Coauthor(Einstein,x)
∧Coauthor(Erdos,x)
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Probabilistic Query Evaluation
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What we’d like to do…
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What we’d like to do…
∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
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Einstein is in the Knowledge Graph
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Erdős is in the Knowledge Graph
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This guy is in the Knowledge Graph
… and he published with both Einstein and Erdos!
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Desired Query Answer
Ernst Straus
Barack Obama, …
Justin Bieber, …
1. Fuse uncertain
information from web
⇒ Embrace probability!
2. Cannot come from
labeled data
⇒ Embrace query eval!
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• Tuple-independent probabilistic database
• Learned from the web, large text corpora, ontologies,
etc., using statistical machine learning.
Co
au
tho
r
Probabilistic Databases
x y P
Erdos Renyi 0.6
Einstein Pauli 0.7
Obama Erdos 0.1
Scie
nti
st x P
Erdos 0.9
Einstein 0.8
Pauli 0.6
[VdB&Suciu’17]
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x y
A B
A C
B C
Tuple-Independent Probabilistic DB
x y P
A B p1
A C p2
B C p3
Possible worlds semantics:
p1p2p3
(1-p1)p2p3
(1-p1)(1-p2)(1-p3)
Probabilistic database D:
x y
A C
B C
x y
A B
A C
x y
A B
B C
x y
A B x y
A C x y
B C x y
Co
auth
or
[VdB&Suciu’17]
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x y P
A D q1 Y1
A E q2 Y2
B F q3 Y3
B G q4 Y4
B H q5 Y5
x P
A p1 X1
B p2 X2
C p3 X3
P(Q) = 1-(1-q1)*(1-q2) p1*[ ]
1-(1-q3)*(1-q4)*(1-q5) p2*[ ]
1- {1- } *
{1- }
Probabilistic Query Evaluation
Q = ∃x∃y Scientist(x) ∧ Coauthor(x,y)
Scie
nti
st
Co
auth
or
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Lifted Inference Rules
P(Q1 ∧ Q2) = P(Q1) P(Q2) P(Q1 ∨ Q2) =1 – (1– P(Q1)) (1–P(Q2))
P(∀z Q) = ΠA ∈Domain P(Q[A/z]) P(∃z Q) = 1 – ΠA ∈Domain (1 – P(Q[A/z]))
P(Q1 ∧ Q2) = P(Q1) + P(Q2) - P(Q1 ∨ Q2) P(Q1 ∨ Q2) = P(Q1) + P(Q2) - P(Q1 ∧ Q2)
Preprocess Q (omitted), Then apply rules (some have preconditions)
Decomposable ∧,∨
Decomposable ∃,∀
Inclusion/ exclusion
P(¬Q) = 1 – P(Q) Negation
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Example Query Evaluation
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
Decomposable ∃-Rule
Check independence:
Scientist(A) ∧ ∃y Coauthor(A,y)
Scientist(B) ∧ ∃y Coauthor(B,y)
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
Complexity PTIME
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Limitations
H0 = ∀x∀y Smoker(x) ∨ Friend(x,y) ∨ Jogger(y)
The decomposable ∀-rule:
… does not apply:
H0[Alice/x] and H0[Bob/x] are dependent:
∀y (Smoker(Alice) ∨ Friend(Alice,y) ∨ Jogger(y))
∀y (Smoker(Bob) ∨ Friend(Bob,y) ∨ Jogger(y))
Dependent
Lifted inference sometimes fails.
P(∀z Q) = ΠA ∈Domain P(Q[A/z])
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Are the Lifted Rules Complete?
Dichotomy Theorem for Unions of
Conjunction Queries / Monotone CNF
• If lifted rules succeed, then PTIME query
• If lifted rules fail, then query is #P-hard
Lifted rules are complete for UCQ!
[Dalvi and Suciu;JACM’11]
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Commercial Break
• Survey book http://www.nowpublishers.com/article/Details/DBS-052
• IJCAI 2016 tutorial http://web.cs.ucla.edu/~guyvdb/talks/IJCAI16-tutorial/
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Throwing Relational
Embedding Models
Over the Wall
• Notion of distance 𝑑 ℎ, 𝑟, 𝑡 in vector space
(Euclidian, 1-cosine, …)
• Probabilistic semantics:
– Distance 𝑑 ℎ, 𝑟, 𝑡 = 0 is certainty
– Distance 𝑑 ℎ, 𝑟, 𝑡 > 0 is uncertainty
P r(h,t) ≈ e−𝛼⋅𝑑 ℎ,𝑟,𝑡
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What About Tuple-Independence?
• Deterministic databases = tuple-independent
• Relational embedding models = tuple-independent
At no point do we model joint uncertainty between tuples
• We can capture correlations, but query evaluation becomes much harder!
– See probabilistic database literature
– See statistical relational learning literature
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So everything is solved?
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What we’d like to do…
∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Ernst Straus
Kristian Kersting, …
Justin Bieber, …
![Page 26: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/26.jpg)
Open World DB
• What if fact missing?
• Probability 0 for:
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q2 = ∃x Coauthor(Bieber,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
Q5 = Coauthor(Einstein,Bieber) ∧ ¬Coauthor(Einstein,Bieber)
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Intuition
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
We know for sure that P(Q1) ≥ P(Q3), P(Q1) ≥ P(Q4)
and P(Q3) ≥ P(Q5), P(Q4) ≥ P(Q5) because P(Q5) = 0.
We have strong evidence that P(Q1) ≥ P(Q2).
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q2 = ∃x Coauthor(Bieber,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
Q5 = Coauthor(Einstein,Bieber) ∧ ¬Coauthor(Einstein,Bieber)
[Ceylan, Darwiche, Van den Broeck; KR’16]
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Problem: Curse of Superlinearity
Reality is worse: tuples
intentionally missing!
x y P
… … …
Sibling
Facebook scale
All Google storage is 2 exabytes…
⇒ 200 Exabytes of data
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 29: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/29.jpg)
Bayesian Learning Loop
Bayesian view on learning:
1. Prior belief:
P(Coauthor(Straus,Pauli)) = 0.01
2. Observe page
P(Coauthor(Straus,Pauli| ) = 0.2
3. Observe page
P(Coauthor(Straus,Pauli)| , ) = 0.3
Principled and sound reasoning!
![Page 30: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/30.jpg)
Problem: Broken Learning Loop
Bayesian view on learning:
1. Prior belief:
P(Coauthor(Straus,Pauli)) = 0
2. Observe page
P(Coauthor(Straus,Pauli| ) = 0.2
3. Observe page
P(Coauthor(Straus,Pauli)| , ) = 0.3
[Ceylan, Darwiche, Van den Broeck; KR’16]
This is mathematical nonsense!
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Problem: Model Evaluation
Given:
Learn:
0.8::Coauthor(x,y) :- Coauthor(z,x) ∧ Coauthor(z,y).
x y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
… … …
Co
au
tho
r
0.6::Coauthor(x,y) :- Affiliation(x,z) ∧ Affiliation(y,z).
OR
What is the likelihood, precision, accuracy, …?
[De Raedt et al; IJCAI’15]
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Open-World Prob. Databases
Intuition: tuples can be added with P < λ
Q2 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
X Y P
Einstein Straus 0.7
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
X Y P
Einstein Straus 0.7
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Erdos Straus λ
Coauthor
0.7 * λ ≥ P(Q2) ≥ 0
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Open-world query evaluation
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UCQ / Monotone CNF
• Lower bound = closed-world probability
• Upper bound = probability after adding all
tuples with probability λ
• Polynomial time☺
• Quadratic blow-up
• 200 exabytes … again
![Page 35: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/35.jpg)
Closed-World Lifted Query Eval
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
Decomposable ∃-Rule
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
Complexity PTIME
Check independence:
Scientist(A) ∧ ∃y Coauthor(A,y)
Scientist(B) ∧ ∃y Coauthor(B,y)
![Page 36: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/36.jpg)
Closed-World Lifted Query Eval
No supporting facts
in database!
Complexity linear time!
Probability 0 in closed world
Ignore these sub-queries!
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
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Open-World Lifted Query Eval
No supporting facts
in database!
Complexity PTIME!
Probability λ in open world
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
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Open-World Lifted Query Eval
No supporting facts
in database!
Probability p in closed world
All together, probability (1-p)k
Exploit symmetry
Lifted inference Complexity linear time!
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 39: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/39.jpg)
[Ceylan’16]
Complexity Results
![Page 40: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/40.jpg)
Implement PDB Query in SQL
– Convert to nested SQL recursively
– Open-world existential quantification
– Conjunction
– Run as single query!
SELECT (1.0-(1.0-pUse)*power(1.0-0.0001,(4-ct))) AS pUse
FROM
(SELECT ior(COALESCE(pUse,0)) AS pUse,
count(*) AS ct
FROM SQL(conjunction)
0.0001 = open-world probability; 4 = # open-world query instances
ior = Independent OR aggregate function
Q = ∃x P(x) ∧ Q(x)
SELECT q9.c5,
COALESCE(q9.pUse,λ)*COALESCE(q10.pUse,λ) AS pUse
FROM
SQL(Q(X)) OUTER JOIN SQL(P(X)) SELECT Q.v0 AS c5,
p AS pUse
FROM Q
[Tal Friedman, Eric Gribkoff]
![Page 41: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/41.jpg)
0
100
200
300
400
500
600
0 10 20 30 40 50 60 70
Size of Domain
OpenPDB vs Problog Running Times (s)
PDB Problog Linear (PDB)
Out of memory trying to run the ProbLog query with 70 constants in domain
[Tal Friedman]
![Page 42: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/42.jpg)
0
100
200
300
400
500
600
0 500 1000 1500 2000 2500 3000
Size of Domain
OpenPDB vs Problog Running Times (s)
PDB Problog Linear (PDB)12.5 million
random variables!
[Tal Friedman]
![Page 43: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/43.jpg)
Querying
Relational Embedding Models
![Page 44: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/44.jpg)
Ongoing Work
• Run query evaluation in vector space?
– DistMult model on Wordnet18RR
– Query most likely answers to:
– Solver: gradient descent on query probability
• Fast approximate query answering
Q(h) = R(h,’c’) Q(t) = R(’c’,t)
[Tal Friedman, Pasquale Minervini]
![Page 45: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/45.jpg)
Ongoing Work
• Join algorithms in vector space?
– DistMult model on Wordnet18RR
– Query most likely answers to:
– Answer: x = ‘SpiceTree’
• Projection (∃x)?
– Skolemization in vector space
• More to come!
Q(x) = Hypernym (‘OliveTree’, x) ∧ Hypernym(x, ‘FloweringTree’)
[Tal Friedman, Pasquale Minervini]
![Page 46: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/46.jpg)
Conclusions
You can do much more with knowledge
bases/graphs than just completing missing tuples
Let’s tear down the wall(s) between
statistical models for knowledge base completion and
query evaluation systems
Relational probabilistic reasoning is frontier and
integration of AI, KR, ML, DB, TH, etc.
Forward pointers @AKBC: Arcchit Jain, Tal Friedman, Ondrej Kuzelka, Guy Van den Broeck and Luc
De Raedt. Scalable Rule Learning in Probabilistic Knowledge Bases
Tal Friedman and Guy Van den Broeck.
On Constrained Open-World Probabilistic Databases
![Page 47: Towards Querying Probabilistic Knowledge Basesguyvdb/slides/AKBC19.pdfDesired Query Answer Ernst Straus Barack Obama, … Justin Bieber, … 1. Fuse uncertain information from web](https://reader033.vdocuments.mx/reader033/viewer/2022042915/5f51a45d3c42b87fa516fcd5/html5/thumbnails/47.jpg)