Boolean Satisfiability Present and Future

1
Boolean Satisfiability Present and Future

• Lintao Zhang
• Microsoft Research SVC

2
SAT Introduction

• Deciding the satisfiability of Boolean formulas
• SAT Propositional ( usually in Conjunctive
  Normal Form (CNF) )
• QBF With quantifiers
• Many driving forces
• Verification
• Electronic Design Automation (EDA)
• Artificial Intelligence
• Operational Research
• Physics ???
• Where is SAT used in our context?
• A basic building block for the reasoning engine
• Reasoning on Boolean domain
• Driving branching and backtracking for search
• Model Checking
• Decision Procedures
• Etc.

3
SAT Solvers Current State-of-the-Art

• Tremendous performance improvements on
  Propositional SAT solvers in the last decade
• Speed increased by at least 3 orders of magnitude
• We regularly solve SAT instances with tens of
  thousands variables and clauses
• Example
• Microprocessor verification, Bounded Model
  Checking, 14 Cycles
• 1 million variables, 10 million literals
  initially
• 200 million literals in added clauses
• 30 million literals finally
• 4 million clauses initially
• 200k clauses added
• 1.5 million decisions
• 3 hours run time

4
SAT Solvers Current State-of-the-Art

• Non performance related improvements of SAT
  solvers
• Explanatory SAT solver
• UNSAT Unsatisfiable core extraction
• SAT Minimal model extraction
• Interpolation using SAT solver
• Checkable proof of unsatisfiability
• On line progress report
• In the last several years, SAT solver changed
  from a toy with only academic interest to become
  a powerful and feature rich deduction method
  capable of industrial strength applications.

5
Can we increase the SAT solver speed even more?

• Yes. Better heuristics and better implementations
  are proposed everyday
• But the improvements are incremental
• DPLL is the dominating algorithm for SAT solving
  for more than 40 years
• Basic DPLL algorithm is proposed in 1962
• Learning and non-chronological backtracking
  (1996)
• Careful implementation and tuning (2001)
• 2-Literal Watching BCP, VSIDS decision heuristic,
  1-UIP learning, etc.
• Whats going on?
• Better decision/branching heuristics
• Better learning and garbage collection schemes
• Leverage the structure of the Boolean formula
• Whats next?
• We cannot bet on any breakthrough without
  fundamentally change the way the algorithm works
• i.e. Dont bet on another 3 orders of magnitude
  speed up in the next 10 years.

6
SAT Solver as a White Box

• Traditional usage of SAT as a Black Box
• Translate the problem into a monolithic CNF
  instance
• Throw it to the SAT solver and pray
• Used to work well
• To squeeze the last bit of performance out of a
  SAT solver, current applications need to treat
  SAT as a White Box
• We need to use the knowledge about the
  application to help the SAT solver
• Input to the SAT solver may be a Boolean formula
  combined with
• Branching suggestions
• Domain specific implications
• Continuous interaction because of abstract and
  refinement
• SAT engines need to be expandable, with cleanly
  defined interfaces.
• What can the applications offer?

7
QBF Quantified Boolean Formulas

• Many applications stress SAT solver because of
  quantifications
• Eliminate quantification by expansion the
  formula becomes exponentially large
• Eliminate quantification by solution enumeration
  SAT solver is bad at this
• Unlike SAT, QBF just started to attract interests
  from researchers in the last couple of years
• No predominant algorithm
• DPLL search
• Resolution
• BDD
• And many other approaches
• Quantification elimination is hard (exponential
  in space), while deciding the satisfiability of a
  QBF may not be.
• Will there be a breakthrough in the near future?
• Much more probable than a breakthrough in SAT