Universidade Técnica de Lisboa Parallel MiniSAT - ALGOS Group

129 Pages·2007·0.7 MB·English

Universidade Te´cnica de Lisboa
Instituto Superior Te´cnico
Parallel MiniSAT
Lu´ıs Filipe Cola¸co Messias Gil
Diploma Thesis
Degree in Computer Science
Supervisors from INESC–ID:
Prof. Lu´ıs Miguel Silveira
Prof. Paulo Flores
Supervisor from DM–IST:
Prof. Jaime Ramos
July 2007
i
Agradecimentos
Se um homem esvazia a sua carteira para
a sua cabe¸ca, ningu´em lhe pode ficar com
nada. Um investimento em conhecimento
paga sempre o melhor proveito.
Benjamin Franklin
Em primeiro lugar quero agradecer aos meus orientadores Lu´ıs Miguel Sil-
veira,PauloFloreseJaimeRamostodooapoio,orienta¸c˜ao,sugest˜oesepaciˆencia
concedidos para que este trabalho pudesse ser conclu´ıdo com ˆexito.
Agrade¸co ao Professor Am´ılcar Sernadas pela ajuda na escolha do TFC e
por me ter indicado o professor Lu´ıs Miguel Silveira como especialista em com-
puta¸c˜ao paralela.
AoDr. NiklasE´enoesclarecimentodedu´vidasrelacionadascomofunciona-
mento do MiniSAT.
Aos meus amigos Ana Sofia Gra¸ca, Bruno Pereira, Carlos Tamulonis, H´elio
Pais,PauloAbranteseSandraMarquespeloscoment´ariosesugest˜oessobreeste
relat´orio e o programa.
Agrade¸co ao Professor David Matos por tirar du´vidas e resolver os proble-
mas com a grid.
Tamb´em quero agradecer ao INESC–ID o facto de ter colocado `a minha
disposi¸c˜ao os meios computacionais necess´arios para desenvolver e testar o pro-
grama.
Finalmente quero agradecer aos meus pais que me ofereceram esta licen-
ciatura. Sem eles nada teria sido poss´ıvel. Obrigado pelo apoio e incentivo
para alcan¸car sempre os objectivos propostos, o carinho e aten¸c˜ao dados nos
momentos dif´ıceis, e o mais importante de tudo, o esfor¸co e sacrif´ıcio para que
eu tivesse sempre uma boa educa¸c˜ao.
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Acknowledgements
If a man empties his purse into his head,
no man can take it away from him. An in-
vestment in knowledge always pays the best
interest.
Benjamin Franklin
First of all, I want to thank my supervisors Lu´ıs Miguel Silveira, Paulo Flo-
resandJaimeRamosforallthehelp, guidance, suggestionsandpatiencegiven,
allowing me to finish this work with success.
I also want to thank Professor Am´ılcar Sernadas for helping me choose my
final year project and by suggesting professor Lu´ıs Miguel Silveira as an expert
in parallel computing.
Thanks to Dr. Niklas E´en for having answered some of our doubts about
MiniSAT.
TomyfriendsAnaSofiaGra¸ca,BrunoPereira,CarlosTamulonis,H´elioPais,
PauloAbranteseSandraMarquesforthecommentsandsuggestionsaboutthis
report and the program.
Thanks to Professor David Matos for answering my questions and solving
the problems with the grid.
I also want to thank INESC–ID for providing the computational means to
develop and test the program.
Finally,lastbutnottheleast,Iwanttothanktomyparentsthatofferedme
this degree. Without them nothing would be possible. Thanks for the support
for me to always reach all the proposed goals, the care and attention on the
rough times, and most of all, the effort and sacrifice so that I would always get
a good education.
iii
Resumo
OproblemaSAT,verificarseumaf´ormulabooleanapodeserverdadeira,ocorre
em diversas ´areas da ciˆencia e engenharia como por exemplo automa¸c˜ao de
desenho electr´onico, pesquisa, planeamento e verifica¸c˜ao formal.
Existem v´arios programas para resolver o problema que utilizam os algo-
ritmos mais avan¸cados que a ciˆencia conhece. Um deles ´e o MiniSAT que ´e
amplamente utilizado.
Este trabalho apresenta a pesquisa, planeamento e implementa¸c˜ao de uma
vers˜ao paralela do MiniSAT que utiliza a tecnologia MPI (Message Passing
Interface) para ser executada numa grid ou cluster de computadores. S˜ao de-
scritas as principais caracter´ısticas do programa: modos de busca, remo¸c˜ao de
hip´oteses e partilha de cl´ausulas aprendidas. Tamb´em´e avaliada a eficiˆencia da
paraleliza¸c˜ao e do esquema de distribui¸c˜ao de carga entre os processadores.
Palavras chave: Computa¸c˜ao Paralela, SAT-Solver, Satisfiabilidade.
iv
Abstract
The SAT problem, determine if a given boolean formula can be true, is largely
usedinseveralareasofscienceandengineeringlikeelectronicdesignautomation,
search, planning and formal verification.
There are several programs to search for satisfiability, called SAT-solvers,
thatusesomeofthemostadvancedtechniquesknown. OneofthemisMiniSAT
that is widely used.
This work presents a research, planning and implementation of a parallel
version of MiniSAT with MPI (Message Passing Interface) technology to be
executed in clusters or grids of computers. Are described the main features of
the program: search modes, assumptions pruning and share of learnt clauses.
Is also made an analysis of its performance and load balance.
Keywords: Parallel Computing, SAT-Solver, Satisfiability.
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Contents
1 Introduction 1
2 Logic and complexity 2
2.1 Boolean Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Computational complexity . . . . . . . . . . . . . . . . . . . . . . 4
3 Parallel computing 6
3.1 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.2 Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.1 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.2 Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 The SAT-Solver 15
4.1 General algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 MiniSAT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Parallel implementation 19
5.1 PCAM design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1.1 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1.2 Communication . . . . . . . . . . . . . . . . . . . . . . . . 20
5.1.3 Agglomeration . . . . . . . . . . . . . . . . . . . . . . . . 20
5.1.4 Mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.2 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . 20
5.2.1 Variables selection . . . . . . . . . . . . . . . . . . . . . . 21
5.2.2 Assumptions generation . . . . . . . . . . . . . . . . . . . 21
5.2.3 Assumptions pruning. . . . . . . . . . . . . . . . . . . . . 26
5.2.4 Sharing learnt clauses . . . . . . . . . . . . . . . . . . . . 26
5.2.5 Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.6 Automatic settings . . . . . . . . . . . . . . . . . . . . . . 28
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5.3 Application’s work flow . . . . . . . . . . . . . . . . . . . . . . . 29
5.4 Modules of the program . . . . . . . . . . . . . . . . . . . . . . . 31
5.5 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Experimental results and performance analysis 33
6.1 Grid resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.3 Goals and difficulties . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.4 Time measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.5 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.5.1 Communication delay . . . . . . . . . . . . . . . . . . . . 38
6.5.2 Modes and options . . . . . . . . . . . . . . . . . . . . . . 38
6.5.3 Granularity . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.5.4 Load distribution . . . . . . . . . . . . . . . . . . . . . . . 41
7 Conclusion 42
A User manual 45
A.1 How to use this manual . . . . . . . . . . . . . . . . . . . . . . . 45
A.2 System requirements and installation . . . . . . . . . . . . . . . . 45
A.3 Quick start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.4 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.5 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.7 Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.8 Hints for better performance . . . . . . . . . . . . . . . . . . . . 52
A.9 Error messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.10FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B Performance tables 56
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List of Figures
2.1 Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1 Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Task Farm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.1 Davis-Putnam algorithm . . . . . . . . . . . . . . . . . . . . . . . 15
5.1 Task Farm organization . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Function test4SAT() . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Program’s main() . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.1 Timed operations (gray fill) . . . . . . . . . . . . . . . . . . . . . 37
A.1 Program’s options . . . . . . . . . . . . . . . . . . . . . . . . . . 48
A.2 Configuration file . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
viii
List of Tables
4.1 Mapping of modules in the file system . . . . . . . . . . . . . . . 18
5.1 Message for Equal mode . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Message for Progressive mode, assuming n<k literals . . . . . . 27
5.3 Result message . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.4 Set of learnt clauses . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.5 Model of the formula . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.6 Mapping of modules in the file system . . . . . . . . . . . . . . . 32
6.1 Benchmark files . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
B.1 Sequential vs Parallel with master and local worker . . . . . . . . 58
B.2 Sequential vs Parallel with master and remote worker . . . . . . 59
B.3 Average time of each file taken by the sequential MiniSAT . . . 59
B.4 Average of each file in 3, 6 and 9 workers . . . . . . . . . . . . . 60
B.5 Average of each SAT file in 3, 6 and 9 workers . . . . . . . . . . 60
B.6 Average of each UNSAT file in 3, 6 and 9 workers . . . . . . . . 61
B.7 Best performances without options . . . . . . . . . . . . . . . . . 61
B.8 Worst performances without options . . . . . . . . . . . . . . . . 62
B.9 Best performances with conflicts . . . . . . . . . . . . . . . . . . 63
B.10 Worst performances with conflicts . . . . . . . . . . . . . . . . . 64
B.11 Best performances sharing learnt clauses . . . . . . . . . . . . . . 65
B.12 Worst performances sharing learnt clauses . . . . . . . . . . . . . 66
B.13 Best performances with different granularity . . . . . . . . . . . . 67
B.14 Worst performances with different granularity . . . . . . . . . . . 68
B.15 Tests with 6 variables to fpga10 11 uns rcr.cnf . . . . . . . . . 69
B.16 Tests with 6 variables to fpga10 12 uns rcr.cnf . . . . . . . . . 70
B.17 Tests with 6 variables to fpga10 13 uns rcr.cnf . . . . . . . . . 71
B.18 Tests with 6 variables to frb40-19-1.cnf . . . . . . . . . . . . . 72
B.19 Tests with 6 variables to frb40-19-2.cnf . . . . . . . . . . . . . 73
B.20 Tests with 6 variables to frb40-19-3.cnf . . . . . . . . . . . . . 74
B.21 Tests with 6 variables to frb40-19-4.cnf . . . . . . . . . . . . . 75
B.22 Tests with 6 variables to frb40-19-5.cnf . . . . . . . . . . . . . 76
B.23 Tests with 6 variables to hole11.cnf . . . . . . . . . . . . . . . . 77
B.24 Tests with 6 variables to mod2-3cage-unsat-9-11.cnf . . . . . 78
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