Frequent SubtreeMining on the Automata Processor ...es9bt/papers/ICS_2017_FTM_ppt.pdf · Boolean...

FrequentSubtree MiningontheAutomataProcessor:ChallengesandOpportunities

ElahehSadredini,RezaRahimi,Ke Wang,KevinSkadronDepartmentofComputerScience

UniversityofVirginia

Presentingin:InternationalConferenceonSupercomputing,June13-16,2017

Motivation

11/21/17 InternationalConferenceonSupercomputing2017 2

Motivation


Motivation


WhatIstheFrequentSubtree Mining(FTM)?Ø To efficiently enumerate all frequent subtrees in a forest (database of trees) according to a given

minimum supportØ The support of a subtree is the number of subtrees in D that contains one occurrence of SØ A subtree S is frequent if its support is more than or equal to a user specified minimum support

value


AC

A

B C

D B

A

BA

DB

C

B C

AD

Relative support threshold: 60%

WhatIstheFrequentSubtree Mining(FTM)?Ø To efficiently enumerate all frequent subtrees in a forest (database of trees) according to a given

minimum supportØ The support of a subtree is the number of subtrees in D that contains one occurrence of SØ A subtree S is frequent if its support is more than or equal to a user specified minimum support

value


AC

A

B C

D B

A

BA

CB

C

B C

AD

Relative support threshold: 60%

Support = 3

Is frequent:

Preliminaries

Inducedsubtree


Embeddedsubtree

B

A

C

B

A C

C

B

A

C

Tree

Subtree

0

1

2

3

6

5

4

B

A

C

B

A C

C

B

A

C

Tree

Subtree

0

1

2

3

6

5

4

IssueswiththeCurrentFTMSolutions(1)Pros Cons

BFS Massive pruningMemory efficient

Multi-pass of datasetslow

DFS Fast Little pruning opportunityMemory-hungry


BFSandDFSrefertocandidategenerationapproach,nottreetraversalJ

IssueswiththeCurrentFTMSolutions(2)


IssueswiththeCurrentFTMSolutions(2)


Wouldthatbeacceptabletoachievehundreds-X speedup attheexpenseofloosingacoupleofpercentaccuracy?

Contributionofthisresearch:

• Proposing a memory efficient and fast solution to the frequentsubtree mining problem on the Automata Processor

• Achieving 350X and more speed up, when allowing 7.5% falsepositive subtrees


TheRestofThisTalk• AutomataProcessor• FTMChallengesandOpportunitiesontheAP• ExperimentalEvaluation• Takeaways


TheAutomataProcessor(1)• TheMicronAutomataProcessor(AP)isareconfigurablenon-vonNeumann

architecture,whichimplementsnon-deterministicfiniteautomata(NFA)withBooleanlogicgatesandcountersinhardwarebasedonDRAMtechnology.

RowAdd

ress

(Inpu

tSym

bol)

RowAccessresultsin49,152match&routeoperations(thenBooleanANDwith“active”bit-vector)

RoutingMatrix

Automata Processor


TheAutomataProcessor(2)• Amassivelyparallel‘MISD’architecture• 1Gbps dataprocessing• Hardwareresourcesondevelopmentboard

– StateTransitionElements(STE):1.5M– ReportingSTEs:200K– CounterElements:25K– BooleanElements:74K


ApplicationsontheAP• Datamining

– Frequentitemset mining– Sequentialpatternmining

• Machinelearning– Randomforest– Entityresolution– String/treekernel

• Bioinformatics– Motifdiscovery– DNAalignment

• …


Challenges:ExactFTMontheAP

The AP supports regular languages

Tree can be represented using context-free-grammar [Ivn07]





The AP can not efficiently implement exact FTM



Exact solutions (e.g., stack implementation on the AP)InefficientDatabase dependent Impractical



The AP can not efficiently implement exact FTM


Opportunities:Pruning

• Four-stage pruning strategy– Subset pruning– Intersection pruning– Downward pruning – Connectivity pruning

• Kernel properties– Complementary pruning– Avoiding false negatives

Generatecandidatesofk-subtree

PrunecandidatesontheAP

K<maxK&&K-subtreenotempty

K=K+1

Yes

Writeresults

AP

CPU

FindfinalfrequentsubtreeontheCPU

CPU

Isexactsolutionneeded?

No

Yes

No


Background• Frequentitemset mining(FIM)


Trans. Items

1 Bread,Milk

2 Bread, Diaper, Beer, Eggs

3 Milk,Diaper, Beer, Coke

4 Bread,Milk, Diaper, Beer,Coke

5 Bread, Milk, Coke,Diaper

sup({Diaper,Milk})=3

Trans. Items

1 <{Bread,Milk},{Coke}>

2 <{Bread, Milk,Diaper}{Beer,Eggs}{Diaper}>

3 <{Milk} {Diaper} {Beer, Coke}>

4 <{Bread,Milk, Diaper}{Beer,Diaper}{Beer,Coke,Eggs}>

5 <{Bread, Milk}{Coke}{Diaper}{Eggs}>

• Sequentialpatternmining(SPM)

Bread Milk

Eggs

Kernel1:SubsetPruning• Maingoal:checksdownwardclosureproperty

*Wang,Ke,etal."AssociationruleminingwiththeMicronAutomataProcessor." ParallelandDistributedProcessingSymposium(IPDPS),IEEE,2015.11/21/17 InternationalConferenceonSupercomputing2017 21

Kernel2:IntersectionPruning• Maingoal:checksifallthesubsetsofacandidatehappensinthesametree


Kernel3:DownwardPruning• Maingoal:checksifancestordescendantrelationshipismet

* Wang, Ke, Elaheh Sadredini, and Kevin Skadron. "Sequential pattern mining with the Micron automata processor." Proceedings of the ACM International Conference on Computing Frontiers. ACM, 2016.


Kernel4:ConnectivityPruning• Maingoal:checksifsiblingrelationshipismet



Framework

1. MakingARMandSPMautomataforeachkernel2. Creatingappropriateinputstream3. ConfiguringtheautomataforeachkernelontheAPand

streamingthecorrespondinginputstream4. GettingthepotentialfrequentsubtreesfrotheAPoutputafter

applyingthekernels

PerformanceEvaluation• Platform

– CPU:Intel(R)Core™[email protected],Memory:32GB– GPU:TeslaK80,Memory24GB

• Dataset

• Apples-to-applescomparison


GPUImplementation• BFSApproach,because:

– NotbeboundbythefiniteGPUglobalmemory– Exposesmanyready-to-processcandidatesandprovideparallelism

• FTM-GPU– CandidategenerationontheCPU– SubsetpruningontheCPU– EnumerationontheGPU

• Treesinsharedmemory• Candidateinconstantmemory

• Sortingtheinputtrees– Decreasedivergence


Algorithmic&ArchitecturalContributions

APkerneloverCPUkernelspeedup:Subset=upto163X

Intersection:upto19XDownward:upto3144XConnectivity:upto2635X










Redbaroverblackbar:upto1.6XBlackbars:upto215XRedbars:upto353X

PruningEfficiency

Kerneleffectiveness:Subset=80%

Intersection:0.5%Downward:3.5%Connectivity:4.8%

11/21/1731

PruningEfficiency

Kerneleffectiveness:Subset=80%

Intersection:0.5%Downward:3.5%Connectivity:4.8%

11/21/1732

Removingintersectionkernel:APoverPatternMatcher:353Xà 2190X

Accuracy:86%à 83%

FTM-APvsOtherFTMAlgorithmsTrade-offbetweenspeed andaccuracy oftheAPsolutionvstheexistingFTMimplementation

33Dataset:CSLOGS


34Dataset:CSLOGS


35Dataset:CSLOGS


36Dataset:CSLOGS

Speedup

&'(_*++,--./0(,-12./

=upto353X

Memoryusage

'/..(<0./=&'(_*+

=upto5000X

ExactSolution:AP+TreeMinerD


ExactSolution:AP+TreeMinerD


PerformanceEvaluation:ExactSolution:AP+TreeMinerD

4.14X

5.8X

Intel Xeon CPU, 2.30GHz, 512 GB memory, 2.133GHz Dataset:TREEBANK

Summary• Proposeamulti-stagepruningframeworkontheAP

– ThefirstworktousetheAPasapruningmedia– Novelpruningkernels– Abetterscalabilityandstablebehavior– Proposeanexactsolution


Takeaways• Rethinkingthealgorithmwhenhavinganewhardwarearchitecture• AppliestospatialautomatacomputingarchitecturesuchasFPGAs• Thisapproachcanbeadoptedforothercomplexpatternminingproblems• Providessomeinsightforthearchitecturalchanges

References• Ke Wang, Elaheh Sadredini, and Kevin Skadron. "Hierarchical Pattern Mining with the Micron Automata Processor.“ International Journal of Parallel

Programming (IJPP), 2017.• Ke Wang, Elaheh Sadredini, and Kevin Skadron. "Sequential Pattern Mining with the Micron Automata Processor." ACM International Conference on

Computing Frontiers, May 2016• J. Wadden, V. Dang, N. Brunelle, T. Tracy II, D. Guo, E. Sadredini, K. Wang, C. Bo, G. Robins, M. Stan, K. Skadron. "ANMLZoo: A Benchmark Suite for

Exploring Bottlenecks in Automata Processing Engines and Architectures." IEEE International Symposium on Workload Characterization (IISWC), October2016.

• Elaheh Sadredini, Reza Rahimi, Ke Wang, and Kevin Skadron. "Frequent Subtree Mining on the Automata Processor: Opportunities and Challenges." ACMInternational Conference on Supercomputing (ICS), Chicago, June 2017

• Shirish Tatikonda, Srinivasan Parthasarathy, and Tahsin Kurc. ”TRIPS and TIDES: new algorithms for tree mining.” Proceedings of the 15th ACMInternational Conference on Information and Knowledge Management. ACM, 2006.

• Ke Wang, Elaheh Sadredini, and Kevin Skadron. ”Sequential pattern mining with the Micron automata processor.” Proceedings of the ACM InternationalConference on Computing Frontiers. ACM, 2016.

• Mohammed Javeed Zaki. ”Efficiently mining frequent trees in a forest: Algorithms and applications.” IEEE Transactions on Knowledge and DataEngineering 17.8 (2005): 1021-1035.

• Yun Chi, et al. ”Frequent subtree mining an overview.” Fundamenta Informaticae 21: 1001-1038, 2011. Paul Dlugosch, et al. ”An efficient and scalablesemiconductor architecture for parallel automata processing.” IEEE Transactions on Parallel and Distributed Systems, 25.12:3088-3098, 2014.

• Renta Ivncsy, and Istvn Vajk. ”Automata Theory Approach for Solving Frequent Pattern Discovery Problems.” World Academy of Science, Engineering andTechnology, International Journal of Computer, Electrical, Automation, Control and Information Engineering 1(8): 2556-2561, 2007.

• Fedja Hadzic, Henry Tan, and Tharam S. Dillon. Mining of data with complex structures. Vol. 333. Springer-Verlag, 2011.• John L. Hennessy, and David A. Patterson. Computer architecture: a quantitative approach. Elsevier, 2011.• Ke Wang, et al. ”Association rule mining with the Micron Automata Processor.” Parallel and Distributed Processing Symposium (IPDPS), IEEE

International, 2015.• Wang, Ke, Kevin Angstadt, Chunkun Bo, Nathan Brunelle, Elaheh Sadredini, Tommy Tracy II, Jack Wadden, Mircea Stan, and Kevin Skadron. "An overview

of micron's automata processor." In Proceedings of the Eleventh IEEE/ACM/IFIP International Conference on Hardware/Software Codesign and SystemSynthesis, p. 14. ACM, 2016.


Thank you J

Questions?


BackupSlides


B

A

D

A

C B

StringEncoding:ABC– BA– – – D

C

A

B

BD

B

A

D

ADCA

D

StringEncoding:AA – CD– BDA– C– – BD– A– – – – BD

PDA-basedSubtreeMining:AnExampleTree

Embeddedsubtree


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– BDA– C– – BD– A– – – – BD

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀

DeterministicPDA


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:A A – CD– BDA– C– – BD– A– – – – BD

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


A

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – C D– BDA– C– – BD– A– – – – BD

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD – BDA– C– – BD– A– – – – BD

D

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– B DA– C– – BD– A– – – – BD

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– BD A– C– – BD– A– – – – BD

D

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– BDA – C– – BD– A– – – – BD

A

D

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


D

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– BDA– C – – BD– A– – – – BD

<C,2>

D

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


D

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– BDA– C– – B D– A– – – – BD

<B,4>

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– BDA– C– – BD – A– – – – BD

D

<B,4>

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


<B,4>

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– BDA– C– – BD– A – – – – BD

<A,5>

<B,4>

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


<B,4>

<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


<B,1>

C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


C

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

Inputtree:AA – CD– BDA– C– – BD– A– – – – B D

B

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


𝑞?,? 𝒒𝟏,𝟏 𝒒𝟐,𝟐 𝒒𝟑,𝟑

𝒒𝟗,𝟏 𝒒𝟖,𝟐 𝒒𝟕,𝟑 𝒒𝟔,𝟒

𝒒𝟓,𝟑

𝒒𝟒,𝟐A,*/<A,0>* B,*/<B,1>* C,*/<C,2>*

A,*/<A,5>*

-,C/𝜀-,<C,*>/𝜀

-,B/𝜀-,<B,*>/𝜀


<D,9>

B

<A,0>

*

𝜆\A,*/𝜆\A*-,*/𝜀

𝒒𝟏𝟎,𝟏 D,*/<D,9>*

B,*/<B,4>*

-,A/𝜀-,<A,*>/𝜀

-,B/𝜀-,<B,*>/𝜀

𝜆\B,*/𝜆\B*-,*\<A,0>/𝜀

𝜆\C,*/𝜆\C*-,*\<B,1>/𝜀

𝜆,*\𝜆*-,*\{C,<C,*>}/𝜀

𝜆\B,*/𝜆\B*-,*\<B,1>/𝜀

𝜆\A,*/𝜆\A*-,*\<B,4>/𝜀

𝜆,*\𝜆*-,*\{A,<A,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆,*\𝜆*-,*\{B,<B,*>}/𝜀

𝜆\D,*/𝜆\D*-,*\<A,0>/𝜀

𝜆,*/*-,*/𝜀

-,<A,0>/ 𝜀-,<B,1>/ 𝜀

-,<B,1>/ 𝜀

-,<B,4>/ 𝜀

-,<A,0>/ 𝜀


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