State-merging DFA induction algorithms withmandatory merge constraints

From MSM to ASM

Bernard Lambeau, Christophe Damas and Pierre Dupont

Computing Science and Engineering DepartmentUniversité catholique de Louvain – Belgium

September 23, 2008

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 1.

Motivations

Requirements Engineering (RE)It has been claimed that the hardest part in building a softwaresystem is deciding precisely what the system should doOne can automate parts of this RE process by learning behaviormodels from scenariosScenarios are strings of possible events which can be generalizedto form a language of acceptable behaviorsSuch languages are conveniently represented by finite-statemachines

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 2.

ScenariosA train system example

Scenarios describe interactions between the software-to-be andits environmentScenarios are typical examples of system usage provided by anend-user involved in the requirements elicitation process

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 3.

Synthesis of behavior models and DFA inductionA win-win situation

Regular languages are considered to be powerful enoughDFAs offer a convenient representation for model checking andcode generationThe typical size of such DFAs is about 20 . . . 100 states

I ⇒ hard to design exactly by a software analystI ⇒ not problematic for state-of-the-art DFA induction algorithm

(RPNI, BlueFringe)

Typical alphabet size ≈ 10 . . . 20The end-user can really be used as an oracle in practice

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 4.

State-merging induction with membership queries

Our previous work: the QSM algorithm [Damas et al. 05],[Dupont et al. 08]

An extension to RPNI or BlueFringe (also known as redBlue) withmembership queries

The limited amount of positive and negative scenarios provided initiallyby an end-user can be enriched by asking membership queries

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 5.

A high-level Message Sequence ChartFlow-charting of various scenarios

This information defines Mandatory Merge Constraints between somestates of the prefix tree acceptor (PTA)

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(UCL Machine Learning Group) From MSM to ASM September 23, 2008 6.

A high-level Message Sequence ChartFlow-charting of various scenarios

This information defines Mandatory Merge Constraints between somestates of the prefix tree acceptor (PTA)

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emergency stop emergency open

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 6.

State-merging DFA induction

Prefix-Tree Acceptor (PTA)

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⇓Quotient automaton

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(UCL Machine Learning Group) From MSM to ASM September 23, 2008 7.

State-merging DFA induction algorithm

Algorithm STATE-MERGING DFA INDUCTION ALGORITHMInput: A positive and negative sample (S+, S−)

Output: A DFA A consistent with (S+, S−)

// Compute a PTA, let N denote the number of its statesPTA← Initialize(S+, S−); π ← {{0}, {1}, ..., {N − 1}}

// Main state-merging loopwhile (Bi , Bj )← ChoosePair(π) do

πnew ← Merge(π, Bi , Bj )if Compatible(PTA/πnew , S−) then

π ← πnew

return PTA/π

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 8.

The Merge function also reduces non-determinism

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Merging 10 and 6

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Merging 5 and 2 (for determinization)

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 9.

Tree invariant

Tree invariant propertyAt least one of the 2 states implied in a merging operation is theroot of a (sub)-treeTrue for RPNI, BlueFringe (= redBlue), etcSimplification of the actual implementation

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(UCL Machine Learning Group) From MSM to ASM September 23, 2008 10.

Incompatibility constraints

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Augmented PTA with positively accepting states (= grey) andnegatively accepting states (= black)The Merge function reduces non-determinism and checks suchcoloring constraints

I States having different colors may not be mergedI States having the same color can be merged

Coloring constraints define incompatibility between states frompositive and negative information or additional domain knowledge[Coste et al. 04], [Dupont et al. 08]

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 11.

Mandatory merge constraints

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Another kind of domain knowledge defines mandatory mergeconstraints between states sharing the same labelsLabeling constraints are the logical counterpart to the coloringconstraints

I States with the same label must be mergedI States with different labels can be merged

A fully labeled PTA does not define a trivial induction problemI Without coloring constraints (such as those provided by the

negative sample) all states will be merged

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 12.

MSM algorithm

Algorithm MSMInput: A non-empty initial positive and negative sample (S+, S−)

Input: Labeling and coloring constraintsOutput: A DFA A consistent with (S+, S−) and all constraints

// Compute a PTA, let N denote the number of its statesPTA← Initialize(S+, S−); π ← {{0}, {1}, ..., {N − 1}}

// Merge all states according to labeling constraintswhile (Bi , Bj )← FindSameBlocks(π) do

π ← Merge(π, Bk , Bl )

// Main state-merging loopwhile (Bi , Bj )← ChoosePair(π) do

tryπ ← Merge(π, Bi , Bj )

catch avoid// inconsistency between coloring and labeling constraints

return PTA/π

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 13.

MSM does not satisfy the tree invariant property

MSM is a straightforward extension to standard state-mergingalgorithmsHowever...

Labeling constraints can force to merge states such that theresulting automaton has a general graph structureThe tree invariant property is no longer satisfiedRecursive merging to reduce non-determinism naturally stopseven for general graphs

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 14.

MSM does not satisfy the tree invariant property

MSM is a straightforward extension to standard state-mergingalgorithmsHowever...

Labeling constraints can force to merge states such that theresulting automaton has a general graph structureThe tree invariant property is no longer satisfiedRecursive merging to reduce non-determinism naturally stopseven for general graphs

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 14.

Experiments on synthetic data

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 15.

Requirements engineering case study

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Algorithm |lab| AccuracyRPNI - 0.55BlueFringe - 0.83MSM 0 0.55

3 0.716 0.7310 0.8815 0.90

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 16.

DFA induction from a positive DFA and a negativesample

Algorithm ASMInput: A positive DFA A+ and a negative sample S−Output: A DFA A consistent with (A+, S−)

// Augment the automaton A+ with states

// marked/added from S−

M ← Augment(A+, S−)

// Compute the natural order on M

π ← NatOrder(M)

// Main state-merging loop

π ← Generalize(π)

return M/π

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 17.

DFA induction from positive and negative DFAs

Algorithm ASM∗

Input: A positive DFA A+ and a negative DFA A− such that L(A+) ∩ L(A−) = ∅Output: A DFA A consistent with (A+, A−)

// Augment the automaton A+ with states

// marked/added from S−

M ← Product(A+, A−)

// Compute the natural order on M

π ← NatOrder(M)

// Main state-merging loop

π ← Generalize(π)

return M/π

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 18.

Product DFA

A+ A− M

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 19.

Take home message

Mandatory merge constraints are introduced to model domainknowledge, for instance, from a Requirements EngineeringperspectiveMandatory merge constraints form the logical counterpart toincompatibility constraintsThe MSM algorithm deals with both types of constraintsMSM is a straightforward extension to RPNI or BlueFringe but

I without satisfying the tree-invariant propertyI using recursive merging extended to general graphs

MSM gives rise to ASM∗ to induce DFAs from prior positive andnegative DFAs

I interesting from a practical viewpointI may require a new theoretical framework

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 20.

Future work

MSM implementation with the BlueFringe search order (easy)

MSM as an extension to QSM for active learning with queries(somewhat more challenging)

Other applicative contexts where mandatory merge constraintsare natural

Further analyze ASM∗I theoretically: characteristic samples?I practically: experimental protocol?

(UCL Machine Learning Group) From MSM to ASM September 23, 2008 21.