1. Thursday Seminar 2017-10-12
Masahiro Sakai (酒井 政裕)
Writing a SAT solver
as a hobby project

2. Outline
• Decision procedures and Me
• My hobby project
• About SAT and SMT
• Some results
• Conclusion

3. About me
• Masahiro Sakai (酒井 政裕)
• joined PFN this May
as an Engineer
• My technical interest includes:
– Functional Programming (esp. Haskell)
– Program Verification and Decision Procedures for it
– and Machine Learning (of course!)

4. Decision procedures
• Decision problem: yes/no answer problems
– e.g. Is intersection of given two regular language empty?
• Model Checking
– Does a finite state system M satisfy
a temporal logic specification φ ?
– it is just a language emptiness of M ∩ ¬φ,
if we represent both M and φ as automaton
– and can be decided using automata algorithm
• Likewise, many program verification task can be reduced to
simple decision problems,
• Fast decision procedures have accelerated program verification.
All possible
behavior of M
Behaviors that
violate φ
Counter
example

5. Decision procedures and Me
• When I was a grad student, ~10 years ago, I listened to
talks about model checking tools.
• Later, I was working on program verification,
using decision procedures as off-the-shelf tools
• I was impressed by their strength and usefulness, and
got interested in their algorithms and implementations
• Therefore I started writing toy level implementations …

6. My hobby project

7. My hobby project : toysolver
• Several years ago
– I started writing some code snippet just to
understand those algorithms
• Over the years,
– I have added more and more algorithms,
and the codebase grew
• Now
– it contains not a few algorithms and it’s ~30 kloc

8. Some of problems and algorithms
• Presburger Arithmetic
– Omega Test
– Cooper’s Algorithm
• Real Arithmetic
– Fourier-Motzkin variable
elimination
– Simplex method
– Gröbner basis
(Buchberger)
– Quantifier Elimination
• (Mixed) Integer Programming
– Branch-and-bound
– Cutting plane (Gomory’s Cut)
– Conti-Traverso
• SAT / MaxSAT / Pseudo Boolean
– DPLL / CDCL
• SMT
– Uninterpreted function
(Congruence Closure)
– Bitvectors
Most of them are toy-level, but the SAT solver is modestly fast.

9. https://github.com/msakai/toysolver http://hackage.haskell.org/package/toysolver
It’s available online

10. Wait !?
http://hackage.haskell.org/package/toysolver
……

11. Why Haskell?
1. Because it’s the language I used most, and it’s handy for me if
usable from Haskell easily
2. It’s suitable for symbolic computation like formula manipulation.
3. I have two questions:
– Haskell has a good compiler and runtime suitable for usual
application domain, But…
– “How about computation intensive application like SAT?”
– “Is good functional interface possible?”

12. Why I’m writing it?
• Developing a state-of-the-art implementation
• Learning algorithms
• Familiarizing other libraries and tools
• Handy implementation for my self

13. Reason: Learning algorithms
• Learning cleaver algorithm is itself fun
• Implementation is the best way to understand it
• In addition to that,
• the knowledge of algorithms was useful when I use
other (state-of-the-art) implementation as a off-the-
shelf tool

14. Reason: Familiarizing other libraries and tools
• Writing and maintaining such project is a good
opportunity to try new libraries, frameworks and services
• For example,
– I have learned several Haskell libraries,
– and also integration of GitHub, Traivs CI, AppVeyor,
Coveralls.io, etc.

15. Reason: Handy implementation for myself
• Having handy implementation for myself is useful for
– experimenting new algorithms,
– solving puzzles (e.g. Sudoku, Number Link, etc.)
– using in a programming competition
• In particular, there are less such library available for
Haskell

16. About SAT and SMT

17. What is SAT?
• SAT = Boolean SATisfiability problem
– “Is there an assignment that makes given formula true?”
• Examples:
– (P∨Q)∧(P∨￢Q)∧(￢P∨￢Q) is satisfiable with
{P ↦ True, Q ↦ False}
– (P∨Q)∧(P∨￢Q)∧(￢P∨￢Q)∧(￢P∨Q) is unsatisfiable
• SAT is NP complete, but state-of-the-art SAT-solver can often solve
problems with millions of variables / constraints.
• Has many applications

18. Some Applications of SAT (and SMT)
• Software/Hardware verification
– Model checking, Test-case generation, …
• Theorem proving
• Puzzles: Sudoku, Number link, Nonogram, etc.
• Program Synthesis
• and more

19. What is SMT?
• Weakness of SAT: Really low-level representation
– Encoding problems into SAT sometimes blows-up
– SAT solver cannot leverage high-level knowledge
• SMT = Satisfiability Modulo Theories
– An approach to overcome the weakness of SAT
– Problem Example:
Is there array a, function f, integers i, j such that
“0 ≤ i ∧ i < 10 ∧ (2i+1=j ∨ read(a,i)=0) ∧
f(read(write(a,i,3), j-2)) ≠ f(j-i+1)”?

20. SMT = SAT solver + Theory solvers
• SAT solver is responsible for Boolean reasoning
• Theory solvers are responsible for handling specific
functions/relations etc.
SAT
Solver
Arithmetic
Solver:
+, ×, ≤
BitVector
Solver
Uninterpreted
Function Solver:
f, g, =
Array Solver
read, write
…

21. What is interesting about SAT and SMT? (for me)
• Theory of mathematical logic is connected to practical
application
• For example,
– In SAT, its CDCL algorithm can be seen as doing
concurrently
• search satisfying assignment (model theoretic)
• constraint propagation and conflict learning (proof
theoretic deduction)
– In SMT, theory combination closely related to Craig
interpolation theorem

22. My SAT solver: toysat
p cnf 250 1065
-159 -234 197 0
-71 13 194 0
45 -218 38 0
191 -129 -88 0
117 -164 -29 0
107 53 115 0
167 111 -57 0
-115 94 98 0
25 -51 -165 0
247 31 -64 0
156 228 11 0
64 199 -162 0
1 173 -54 0
$ toysat UF250.1065.100/uf250-01.cnf
……
c #vars 250
c #constraints 1065
c Solving starts ...
c ============================[ Search Statistics ]================
c Time | Restart | Decision | Conflict | LEARNT | Fixed | Removed
c | | | | Limit GC | Var | Constra
c ==========================================================
c 0.0s | 0 | 0 | 0 | 438 0 | 0 | 0
……
c 4.0s | 11 | 23520 | 19665 | 1259 37 | 0 | 0
c #cpu_time = 3.872s
c #wall_clock_time = 3.957s
c #decision = 23520
c #random_decision = 105
c #conflict = 19665
c #restart = 11
s SATISFIABLE
v -1 -2 3 -4 5 -6 7 -8 9 -10
v -11 -12 -13 14 15 16 17 -18 19 20
v 21 22 -23 -24 25 26 -27 28 29 -30
v 31 32 -33 34 -35 36 37 -38 39 -40
…
¬x159∨¬x234∨x197

23. My SMT solver: toysmt
(set-option :produce-models true)
(set-logic QF_UFLRA)
(declare-sort U 0)
(declare-fun x () Real)
(declare-fun f (U) Real)
(declare-fun P (U) Bool)
(declare-fun g (U) U)
(declare-fun c () U)
(declare-fun d () U)
(assert (= (P c) (= (g c) c)))
(assert (ite (P c) (> x (f d)) (< x (f d))))
(check-sat)
(get-model)
(exit)
$ toysmt QF_UFLRA.smt2
success
…
sat
((define-fun P ((x!1 U)) Bool
(ite (= x!1 (as @3 U)) true false))
(define-fun c () U (as @3 U))
(define-fun d () U (as @4 U))
(define-fun f ((x!1 U)) Real
(ite (= x!1 (as @4 U)) 0 (/ 555555 1)))
(define-fun g ((x!1 U)) U
(ite (= x!1 (as @3 U)) (as @3 U) (as @-1 U)))
(define-fun x () Real (/ 1 10)))

24. Some results?

25. Some results?
• Several talks
• Solver competitions
– Pseudo Boolean Competition 2012
– Max-SAT Evaluation 2013
– SMT-COMP 2016
– …
• How was writing SAT solver in Haskell?

26. I have had talks based on what I learned

27. Pseudo Boolean Competition 2012 (PB'12) competition result:
• Ranked high in some minor divisions:
– placed 2nd in PARTIAL-BIGINT-LIN and SOFT-
BIGINT-LIN categories
– placed 4th in PARTIAL-SMALLINT-LIN and SOFT-
SMALLINT-LIN categories
– placed 8th in OPT-BIGINT-LIN category

28. SMT-COMP 2016
QF_LRA (Main Track) http://smtcomp.sourceforge.net/2016/results-QF_LRA.shtml?v=1467876482
‘toysmt’ ended up dead last.
But without wrong results! (Thanks to QuickCheck!)

29. How was writing SAT solver in Haskell?
• How about writing computation intensive application like SAT?
– Not so bad, but with some frustration:
– e.g. Haskell allocate memory like breathing, but I want to
avoid memory allocation in tight loop. It’s possible if we
write carefully, but …
• Is good functional interface possible?
– Not tried yet. My SAT solver is based on imperative API
and imperative implementation.

30. Conclusion
• As a hobby project, I have been writing toy-level
implementation of various algorithms
• In particular, algorithms for SAT and related problems
• It’s mainly for fun, but I have gained good experience
from the activity

31. Thank you!
And any questions or comments?