The Relational Database Engine: An Efficient Validator of T emporal - - PowerPoint PPT Presentation

Sylvain Hallé

Jason Vallet, Aouatef Mrad, Sylvain Hallé*, Éric Beaudet

CRSNG NSERC

The Relational Database Engine:

An Efficient Validator of T emporal Properties

• n Event T

races

Université du Québec à Chicoutimi CANADA Novum Solutions CANADA

Sylvain Hallé

How high is this building? How high is this building?

?

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How high is this building?

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h = 4.9 t 2

How high is this building?

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T = 2π 9.8 h

How high is this building?

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2h = T c

How high is this building?

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p

p = p 1 - L h T0

( (

g M R L

How high is this building?

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How high is this building?

Answer derived from indirect measurements No single "best" solution

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How high is this building?

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τ ⊧ φ ?

A sequence of events Some assertion

• n that sequence

satisfies Web server log Execution trace . . . FSM LTL . . .

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T race validation

Algos Tools

I t e r a t

• r

< T >

I t e r a t

• r

< T >

hasNext next

I t e r a t

• r

< T >

hasNext next A call to next must be preceded by a call to hasNext

B A

B A

No CartCreate request can occur before a LoginResponse message

Login

Login

Three successive login attempts should trigger an alarm

Receive order

Receive order Ready?

Receive order Ready? Yes

Receive order Ready? Yes File order No Ship

Receive order Ready? Yes File order No Ship

A received order must eventually be shipped

A

1 2 3 4 . . .

a a b c b

ℕ A trace m is a mapping from ℕ to the set of events :

A

Let be a set of event symbols.

A

Ground terms

→ ¬ ∧

→ ¬ ∧ Boolean connectives Temporal

• perators

X G F U next globally eventually until

+ + = Linear Temporal Logic

A

1 2 3 4 . . .

a a b c b

ℕ

Φ

Let be the set of all possible LTL formulas. The function ℒ : Φ → 2 labels each state with a set of LTL formulas

ℕ

a ∧ b a ∧ b

G (a→b)

b ∨ c b∨c a ∧ b

G (a→b)

ℒ(a∧b) = {0,1,4,...} Example: ℒ

A

1 2 3 4 . . .

a a b c b

ℕ

Φ

Let be the set of all possible LTL formulas. The function ℒ : Φ → 2 labels each state with a set of LTL formulas

ℕ

i ∈ ℒ(φ∨ψ) ⇔ i ∈ ℒ(φ) or i ∈ ℒ(ψ) i ∈ ℒ(φ∧ψ) ⇔ i ∈ ℒ(φ) and i ∈ ℒ(ψ) i ∈ ℒ(¬φ) ⇔ i ∉ ℒ(φ) i ∈ ℒ(G φ) ⇔ j ∈ ℒ(φ) for all j ≥ i i ∈ ℒ(X φ) ⇔ i+1 ∈ ℒ(φ) i ∈ ℒ(F φ) ⇔ j ∈ ℒ(φ) for some j ≥ i i ∈ ℒ(φ U ψ) ⇔ j ∈ ℒ(ψ) for some j ≥ i and k ∈ ℒ(φ) for all j ≥ k ≥ i i ∈ ℒ(a) ⇔ m(i) = a

i ∈ ℒ(φ) exactly when the trace m(i), m(i+1), ... satisfies φ Theorem

ψ φ σ

1 2 3 4 . . .

ψ φ σ

i ∈ ℒ(φ) exactly when the trace m(i), m(i+1), ... satisfies φ Theorem

ψ φ σ

1 2 3 4 . . .

ψ φ σ

0 ∈ ℒ(φ) ⇔ m ⊧ φ Therefore...

A call to next must be followed by a call to hasNext No CartCreate request can occur before a LoginResponse message A received order must eventually be shipped Three successive login attempts should trigger an alarm

A call to next must be followed by a call to hasNext No CartCreate request can occur before a LoginResponse message A received order must eventually be shipped Three successive login attempts should trigger an alarm G (next → X hasNext)

A call to next must be followed by a call to hasNext No CartCreate request can occur before a LoginResponse message A received order must eventually be shipped Three successive login attempts should trigger an alarm G (next → X hasNext) ¬ CartCreate U hasNext

A call to next must be followed by a call to hasNext No CartCreate request can occur before a LoginResponse message A received order must eventually be shipped Three successive login attempts should trigger an alarm G (next → X hasNext) ¬ CartCreate U hasNext G (receive → F ship)

A call to next must be followed by a call to hasNext No CartCreate request can occur before a LoginResponse message A received order must eventually be shipped Three successive login attempts should trigger an alarm G (next → X hasNext) ¬ CartCreate U hasNext G (receive → F ship) G ¬(fail ∧ (X (fail ∧ X fail)))

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Four types of traces

τ : a mapping from ℕ to events in E = {e₀, e₁, ...}

Multi-valued eᵢ : Pᵢ → 2V

Parameters Values <event> <p>13</p> <q>8</q> <q>6</q> </event>

1

Single-valued eᵢ : Pᵢ → V

<event> <p>13</p> <q>8</q> <r>6</r> </event>

2 3

Fixed schema eᵢ : P → V

p 13 12 9 q 8 4 2 r 6 10 8

4

Atomic eᵢ = a

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Specification languages

φ : an expression in some specification language

Linear Temporal Logic (LTL) G p₀ ≠ 0 First-order Linear Temporal Logic (LTL-FO+) F (∀x ∈ /event/p₀ : x = 0) Regular expressions

^a+b.*b?c$

SQL

SELECT * FROM events WHERE p0=0...

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Software and algorithms

⊧ : an algorithm to evaluate satisfiability

for inputs in format T against assertions in language L BeepBeep, Monid, Logscope, MySQL, ProM, NuSMV , Spin, SEQ.OPEN, . . . 〈T,L〉

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Islands?

〈LTL-FO+, multi-valued〉 〈SQL, fixed schema〉 〈XQuery, multi-valued〉 〈MFOTL, single-valued〉 Saxon BeepBeep NuSMV Spin MySQL Maude MonPoly 〈LTL, atomic〉 〈LTL, single-valued〉

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BabelT race

XML CSV XES

Reader T ranslator

τ File

φ

Launcher Application

T/F

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T ransduction

T ransducer

(τ,φ) ∈ 〈T,L〉 (τ',φ') ∈ 〈T ',L'〉

⊧〈T,L〉

τ φ

⊧〈T ',L'〉

τ' φ'

⇔

Transduction preserves logical equivalence Chaining transducers allows an application to verify a property on a trace expressed in a different format/language pair

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Islands?

〈LTL-FO+, multi-valued〉 〈SQL, fixed schema〉 〈XQuery, multi-valued〉 〈MFOTL, single-valued〉 Saxon BeepBeep NuSMV Spin MySQL Maude MonPoly ⊆

*

〈LTL, atomic〉 〈LTL, single-valued〉 〈LTL-FO+, single-valued〉 ⊆ ⊆

*

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Islands?

〈LTL-FO+, multi-valued〉 〈SQL, fixed schema〉 〈XQuery, multi-valued〉 〈MFOTL, single-valued〉 Saxon BeepBeep NuSMV Spin MySQL Maude MonPoly ⊆

*

〈LTL, atomic〉 〈LTL, single-valued〉 〈LTL-FO+, single-valued〉 ⊆ ⊆

*

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Islands?

〈LTL-FO+, multi-valued〉 〈SQL, fixed schema〉 〈XQuery, multi-valued〉 〈MFOTL, single-valued〉 Saxon BeepBeep NuSMV Spin MySQL Maude MonPoly ⊆

*

〈LTL, atomic〉 〈LTL, single-valued〉 〈LTL-FO+, single-valued〉 ⊆ ⊆

*

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Islands?

〈LTL-FO+, multi-valued〉 〈SQL, fixed schema〉 〈XQuery, multi-valued〉 〈MFOTL, single-valued〉 Saxon BeepBeep NuSMV Spin MySQL Maude MonPoly ⊆

*

〈LTL, atomic〉 〈LTL, single-valued〉 〈LTL-FO+, single-valued〉 ⊆ ⊆

*

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Islands?

〈LTL-FO+, multi-valued〉 〈SQL, fixed schema〉 〈XQuery, multi-valued〉 〈MFOTL, single-valued〉 Saxon BeepBeep NuSMV Spin MySQL Maude MonPoly ⊆

*

〈LTL, atomic〉 〈LTL, single-valued〉 〈LTL-FO+, single-valued〉 ⊆ ⊆

*

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Islands?

〈LTL-FO+, multi-valued〉 〈SQL, fixed schema〉 〈XQuery, multi-valued〉 〈MFOTL, single-valued〉 Saxon BeepBeep NuSMV Spin MySQL Maude MonPoly ⊆

*

〈LTL, atomic〉 〈LTL, single-valued〉 〈LTL-FO+, single-valued〉 ⊆ ⊆

*

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Format of an event trace

n Action SessionId CartId ItemId CartCreate 1234 null null 1 CartCreateResponse 1234 45603 null 2 CartAdd 1234 45603 005-40958 3 CartClear 1234 45603 null

Event attributes Event Sequential number

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From LTL to SQL

Recursive function

ω

LTL formula SQL query

n

2 ...

φ

i ∈ T

Evaluate

T : if and only if τ ⊧ φ

i

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From LTL to SQL

ω (x = y) ≡

SELECT n FROM T WHERE x = y

ℓ

Base case: assertions of the form x=y Create a table containing event numbers n that satisfy the assertion

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From LTL to SQL

ω (ϕ ∨ψ) ≡ ω +1(ϕ) UNION ω +1(ψ) ω (ϕ ∧ψ) ≡ ω +1(ϕ) INTERSECT ω +1(ψ) ω (¬ ϕ) ≡ T MINUS ω +1(ϕ)

ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ

Boolean connectives: combine tables computed at a previous step

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From LTL to SQL

ω (X ϕ) ≡

SELECT n−1 FROM ω +1(ϕ)

ℓ ℓ

X φ: put n-1 in the output table if n is in the table computed for φ

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From LTL to SQL

ω (F ϕ) ≡

SELECT DISTINCT T

A, .n

FROM T AS T

A, JOIN ω +1(ϕ) AS T B,

WHERE T

A, .n ≤ T B, .n ℓ ℓ ℓ ℓ ℓ ℓ

F φ: put in the output table all values of T that are smaller than some value n is in the table computed for φ (can be optimized)

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From LTL to SQL

T ≡

SELECT MAX(T

A, .n) AS n1, T B, .n AS n2

FROM ω +1(¬

ϕ) AS T

A, JOIN ω +1(ψ) AS T B,

WHERE n1 < n GROUP BY T

B, .n

ω (ϕ U ψ) ≡

SELECT T.n FROM T JOIN T WHERE T.n > T .n1 AND T.n ≤ T .n2

ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ'

'

ℓ

'

.

ℓ

'

ℓ

φ U ψ: done in two steps Most expensive translation (two joins)

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Experimental evaluation

Using BabelTrace, compared SQL translation with 7 other tools using the same inputs

Tool Input language Available? BeepBeep [12] LTL-FO+ Yes Logscope [7] Logscope — Maude [20] LTL Yes Monid [19] EAGLE — MonPoly [4] MFOTL Yes NuSMV [5] LTL Yes ProM [24] LTL Yes RuleR [3] RuleR — Saxon [11] XQuery Yes SEQ.OPEN [6] µ-calculus2 Yes SPIN [14] LTL Yes

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Experimental evaluation

2 4 6 8 10 12 14 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (s) Trace length Scenario Random property P9 BeepBeep Filter Monpoly MySQL MySQL-Opt ProMLTL Saxon

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Experimental evaluation

Surprise! When used as a trace validator, a relational database engine outperforms most software designed for that purpose ⇒ Trace validation capabilities are available through commodity software

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Future work

Event data stored across multiple tables (n:1 or n:n relations) Optimize translation Handle past operators Row vs. column-oriented storage

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The end

Thank you! Questions?