ECE 647 TERM PROJECT

1. ECE 647 TERM PROJECT MiniSAT parallelization ShahadatHossain Saud Wasly

2. Agenda • Introduction • MiniSAT Background • MiniSAT Algorithm • Objectives • Difficulties • Implemented changes • Results • Conclusion • Questions

3. The SAT Problem + MiniSAT • Boolean Satisfiablity is a mathematical problem to find if a given set of variables can be assigned in a such a way that all given constraints are met • SAT problems commonly arise in EDA applications • MiniSAT is an open source SAT solver created in 2003 • Designed to be extensible; Original code was ~700 lines • Current code ~1k lines • SAT competition • Main track: Optimize any SAT solver • Parallel track: parallelize any SAT solver • MiniSAT Hack track: minor changes to miniSAT

4. Format and Terminology • SAT problems are presented in Conjunctive Normal Form (CNF) • F=(x1 v ¬x2 v ¬x3) ∧ (¬ x1 x5 ¬x7 x20) ∧ (...) ∧ ... • Variables: x1, x2, x3, ... • Literals: A variable and its negation • x1 AND ¬ x1, x2 AND ¬ x2, ... • Clause: A collection of variables that have an OR relationship • (x1 v ¬x2 v ¬x3) • A clause can be true, false, free, or asserting

5. Internal Structures c simple_v3_c2.cnf c p cnf 3 2 1 -3 0 2 3 -1 0 • Clause Database: Contains all clauses (input constraints) and all assigned literals • Learnts: Vector of literals that have been learnt from assertions • Trail: Literal path taken so far • Watches: Vector of literals to watch since they are closely related to those that have been changed • Order Heap: Priority heap • Decision Level: The level an assertion has been propagated to; used for backtracking

6. MiniSAT Algorithm • MiniSAT uses 2 methods to solve a SAT problem: • Conflict-driven backtracking • Program makes assertions on variables • When clause becomes asserting, MiniSAT ‘learns’ a literal • When a conflict arises, backtrack and ‘unlearn’ literals • Dynamic Variable Ordering* • When literal is asserted, related literal is moved to top of stack • Exit Condition • All literals assigned (SAT), no conflicts • Backtracked to root of tree (UNSAT)

7. Parallelization • Why Parallelize? • Large problems can take hours to solve • On modern multi-core computers, MiniSAT isn’t taking advantage of total processing power • MiniSAT is essentially a trial-and-error algorithm, should benefit from multiple copies of same program • MiniSAT occasionally learns constraints, data that be shared between threads

8. Proposed Improvements • Parallelize base program • Task parallel vs. Data parallel • Parallelize tasks (with synchronization) or copy entire program • Assign different heuristics to different threads • Current MiniSAT uses activity based (dynamic) prioritization • Excellent for single thread, but cannot be extended • Divide Literal branches and assign a sequential heuristic* • Assign a random heuristic within a given solution space* • Hybrid heuristic combining all 3 • Share common database between threads** • Difficult to implement because of backtracking • Only clauseDB is global, and frequent R/W will become a bottleneck

9. Implementation • Currently have multiple threads working in parallel • Threads are synchronized for exit • Thread creation is dynamic, based on the number of cores in the system • Preventing OS from changing core assignment • Currently have 3 working heuristics • Single processor execution results in original miniSAT (activity based stack heuristic) • 2nd thread will be a FIFO search algorithm • Following threads will be random search heuristics* • All new heuristics are activity based, but have different prioritization • Random decisions are randomized based on thread • When there is no chain activity to follow, miniSAT makes a random decision • Random decision is seeded with thread ID and is thus unrelated from thread to thread

10. Challenges • Testing • CNF files can be extremely large and time consuming to solve • SAT competition benchmark files can have over 200,000 variables and take 30+ mins to solve • Poorly Written Code • Algorithms are clearly defined in documentation, but actual code has very few comments • Variable names are not always descriptive • Thread and data synchronization • Data structures are large and smaller structures are prone to ‘backtracking’ • Multiple solutions • A SAT problem may have more than 1 possible solution • Verifying a solution?

11. Tradeoffs - Algorithm • Initial activity based heuristic has variable performance • Because of the way MiniSAT is programmed, changing the input variable order has a significant impact on processing time • Ordered Heap is a stack implementation, so removing from the bottom adds considerable processing time • Virtually requires a rebuilding of the stack every time • Ideally, a few threads will execute sequentially, while other threads “learn” randomly for the main threads • Requires a moderate level of communication between threads

12. Tradeoffs - Parallelization • Frequent access to shared/protected data reduces processing speed • For long literals and deep propagations, parallelization should be useful • More threads result in more memory usage • Leads to frequent garbage collection • Reduces work space of every thread

13. Work Remaining • Introduce data sharing/synchronization • Small data structures (e.g: learnts list)cannot be shared because of backtracking • Sharing learnt would require tracking of which thread has learned what, so the appropriate part can be ‘unlearnt’ • After unlearning, the entire list would have to be recreated • Requires substantial changes to existing code and algorithm • Large data structures (e.g: clauseDB) are ideal for sharing • Would reduce the overall memory usage • Would slow down processing time because of frequent access • Actual database structure is not well defined

14. Ideal Implementation • Functions currently built to search branches independently • With data synchronization, separate branches can help each other by learning new information and introducing new constraints

15. Preliminary Results • The sequential heuristic can beat the miniSAT heuristic in some cases • Because of random variables introduced, the execution time of multiple threads is variable • Tested small problem on 2 cores • Execution time depends on particular CNF file used • Sequential heuristic is occasionally better (by as much as 20%) • Tested small problem on 6 cores (ecelinux), but system has memory/quota issues • Time tracking is potentially incorrect on multicore • Re-implemented to reflect actual time

16. Preliminary Results 2 Original miniSAT output Parallelized miniSAT output

17. Conclusion • Synchronization of parallel miniSAT is an extremely complex problem • Provides excellent exposure to SAT solving algorithm (DPLL algorithm) and practical problems faced while implementing • 4 competitors in 2009 SAT competition • No solver could solve hard difficulty SAT problems within time limit

18. SAT competition Results • MiniSAT performance is hard to beat!! • Only 1 contestant from the parallel group placed in the top 20 (ManySAT) • Modifications can result in previously solvable problems becoming unsolvable

19. Reference • 2009 Sat Competition Results www.satcompetition.org/2009/sat09comp-slides.pdf • POSIX thread (pthread) libraries http://www.yolinux.com/TUTORIALS/LinuxTutorialPosixThreads.html

20. Questions