analysis of defaults in real world variability models

Defaults Analyses in non-Boolean Feature Models

Leonardo Passos([email protected])

February 2012

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Outline

Proposed Analysis

Initial Definitions

Analysis Encoding

Case Study: CDL

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Proposed Analysis

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Analysis

Given a model with defaults, is it the case that for every correctpartial configuration, its completion using defaults result in aconflict-free completion?

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Initial Definitions

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Encoding of Boolean FMs with no Defaults

I Given a feature model (FM) M, let F = {f1, . . . , fn} be theset of its features.

I Each analysis over F is modelled as a SAT problem (given toSMT Solvers).

I Encoding scheme for Boolean FMs with no defaults:I E = {e1, . . . , en}: set of variables, each denoting a feature’s

enabled state.

I Φ = ΦFM ∧ ΦCTC: satisfiability constraint

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Encoding of Boolean FMs with Defaults

I E and Φ as before.

I DE = {de1 , . . . , den}: set of variables, each denoting afeature’s default enable state.

I OE = {oe1 , . . . , oen}: set of variables, each controlling if afeature’s default enable state has been overridden by the user.

I UE = {ue1 , . . . , uen}: set of variables, each denoting afeature’s enable state as set by the user.

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I New satisfiability constraint:

Φ ∧ ΦSE ∧ ΦDE ∧ ΦOE

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I ΦSE : source of a feature’s enabled value.

n∧i=1

ei = (oei ?uei : dei )

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I ΦDE : defaults setting.

n∧i=1

dei = default enabled expression for fi

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I ΦOE : controls which features cannot have the enabled valueoverridden. ∧

i∈{k | fk∈F , ¬overridableE(fk )}

oei = false

where

overridableE(fi ) =

{false, fi is mandatory

true, otherwise

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Encoding of Non-Boolean Feature Models with Defaults

I E , DE , OE , UE , ΦSE , ΦDE , ΦOE as before

I Create set of variables and constraints for the data value part:V, DV , OV , UV , ΦSV , ΦDV , ΦOV .

I Φ = ΦFM ∧ ΦCTC is now defined over E and V.

I Note that:I |V = {v1, . . . , vm}| ≤ |E|

I |DV | = |V| ≤ |DE |

I |OV | = |V| ≤ |OE |

I |UV | = |V| ≤ |UE |

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Analysis Encoding

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Analysis Statement

Given a model with defaults, is it the case that for every correctpartial configuration, its completion using defaults result in a

conflict-free completion?

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Abstract Idea

Partial configuration+

completion(SAT)

Partial configuration+

default completion with conflicts(SAT)

User provided value Correct completion value Default completion value

Coun

ter e

xample

copied

copied

copied

M1 M2

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1) Build M1I Model M1 has no defaults.

I Solution: use completion variables instead.

I Sets of variables:

I E : enable variables.

I CE (new): enabledcompletion valuevariables.

I HUE (new): flag variablesthat signal whether theuser has set a feature’senabled state.

I UE : user set enabledvalue variables.

I V: set of data valuevariables.

I CV (new): data valuecompletion variables.

I HUV (new) : flagvariables that signalwhether the user has seta feature’s data value.

I UV : user set data valuevariables.

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1) Build M1

I Satisfiability constraint ΦM1:

Φ1 ∧ ΦSE1∧ ΦSV1

∧ ΦHUE1∧ ΦHUV1

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1) Build M1




I ΦSE : source of a feature’s enabled value.

n∧i=1

ei = (hei ?uei : cei )

where ei ∈ E , hei ∈ HUE , uei ∈ UE and cei ∈ CE .

I ΦSV : analogous, but manipulates variables related to V.

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1) Build M1




I ΦHUE: controls which features cannot have the enabled value

set by the user. ∧i∈{k | fk∈F , ¬overridableE(fk )}

hei = false

where overridableE(fi ) is as defined before.

I ΦHV : analogous, but manipulates variables related to V.

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2) Copy User-Values to M2

I Copy constraint (ΦCOPY):(n∧

i=1

(oei2 = hei1)

)∧

(m∧i=1

(ovi2 = hvi1)

)∧(

n∧i=1

(uei2= uei1

)

)∧

(m∧i=1

(uvi2= uvi1

)

)

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3) Counter Example for Default Completion in M2

I Φ2 constraint was defined as

ΦFM2 ∧ ΦCTC2

I Counter example constraint (ΦCOUNTER-EXAMPLE-M2):

(ΦFM2 ∧ ¬ΦCTC2) ∧ (ΦSE2∧ ΦSV2

) ∧ (ΦDE2∧ ΦDV2

)

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4) Putting Everything Together

ΦM1 ∧ ΦCOPY ∧ ΦCOUNTER-EXAMPLE-M2

I If SMT Solver returns SAT, a counter example is readilyavailable.

I Otherwise, the SMT returns UNSAT: no counter exampleexists and the set of defaults is valid for all correct partialconfigurations.

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Case Study: CDL

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CDL Specifics

I CDL requires a new constraint: ΦDo (stating a feature’s legalvalues).

I Completion of M1 should be based on default-defaults.

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Analysis of CDL Models

I Analysis of existing CDL models, given this logical framework,is straightforward.

I In CDL, each feature has an enabled and data value part(n = m).

I Simple models were manually translated to Z3 SMT Solver asa proof of concept.

I Will it scale to the whole set of models?. . .

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Questions?

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analysis of defaults in real world variability models

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