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.
What’s going on?
Better decision/branching heuristics
Better learning and garbage collection schemes
Leverage the structure of the Boolean formula
What’s next?
We cannot bet on any breakthrough without fundamentally change the way the algorithm works
i.e. Don’t 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