11.1 SAT solvers

SAT solvers are crucial tools in formal hardware verification, tackling the Boolean satisfiability problem. They determine if a given Boolean formula can be satisfied, enabling automated checking of hardware designs against specified properties.

These solvers use algorithms like DPLL and CDCL to efficiently search for satisfying assignments. They employ techniques such as conflict-driven learning, variable selection heuristics, and clause learning to handle complex formulas and improve performance in hardware verification tasks.

Boolean satisfiability problem

• Fundamental problem in formal verification of hardware involves determining if a given Boolean formula can be satisfied
• Plays crucial role in various aspects of hardware design and verification processes
• Forms basis for many automated reasoning techniques used in circuit analysis and validation

SAT in formal verification

Top images from around the web for SAT in formal verification

• 

  Logic Circuits View original

  Is this image relevant?

• 

  example1 - Electronics-Lab.com View original

  Is this image relevant?

• 

  absorption - Electronics-Lab.com View original

  Is this image relevant?

• 

  Logic Circuits View original

  Is this image relevant?

• 

  example1 - Electronics-Lab.com View original

  Is this image relevant?

1 of 3

Top images from around the web for SAT in formal verification

• 

  Logic Circuits View original

  Is this image relevant?

• 

  example1 - Electronics-Lab.com View original

  Is this image relevant?

• 

  absorption - Electronics-Lab.com View original

  Is this image relevant?

• 

  Logic Circuits View original

  Is this image relevant?

• 

  example1 - Electronics-Lab.com View original

  Is this image relevant?

1 of 3

• Enables automated checking of hardware designs against specified properties
• Translates verification problems into Boolean formulas for efficient analysis
• Supports verification of combinational and sequential circuits by encoding behavior as SAT instances

NP-completeness of SAT

• Belongs to class of problems without known polynomial-time solutions
• Exhibits exponential worst-case time complexity for general instances
• Serves as reduction target for many other NP-complete problems in hardware verification

Structure of SAT solvers

DPLL algorithm

• Forms foundation of modern SAT solvers with backtracking search
• Employs unit propagation to simplify formulas during search process
• Utilizes pure literal elimination to further reduce problem complexity
• Implements decision heuristics to guide variable assignments

CDCL algorithm

• Enhances DPLL with conflict-driven clause learning
• Analyzes conflicts to derive new clauses and prune search space
• Employs non-chronological backtracking to efficiently explore solution space
• Utilizes activity-based heuristics for variable and clause management

Conflict-driven learning

• Generates learned clauses from conflict analysis to prevent repeated mistakes
• Implements resolution-based derivation of conflict clauses
• Utilizes unique implication points (UIPs) for effective clause learning
• Manages learned clause database to balance memory usage and solver performance

SAT solver components

Variable selection heuristics

• VSIDS (Variable State Independent Decaying Sum) prioritizes recently active variables
• Phase saving retains previous assignments to guide future decisions
• Implements random walks to escape local minima in search space
• Utilizes dynamic restarts to adapt search strategy based on solver progress

Clause learning techniques

• 1-UIP (Unique Implication Point) scheme generates concise learned clauses
• Implements clause minimization to reduce size of learned clauses
• Utilizes on-the-fly subsumption to eliminate redundant clauses
• Employs resolution-based strengthening to enhance learned clause quality

Backtracking strategies

• Non-chronological backtracking jumps to relevant decision levels
• Implements backjumping to skip irrelevant portions of search tree
• Utilizes conflict analysis to determine optimal backtrack points
• Employs trail saving to preserve partial assignments during backtracking

Optimizations in SAT solvers

Two-watched literal scheme

• Efficiently detects unit clauses and conflicts during propagation
• Reduces number of memory accesses required for clause watching
• Implements lazy data structure updates to minimize computational overhead
• Supports incremental solving by maintaining watch lists across multiple SAT calls

Restart policies

• Luby restart sequence balances short and long runs
• Implements dynamic restarts based on learned clause quality
• Utilizes rapid restarts in early solving stages to quickly explore search space
• Employs adaptive restart strategies that adjust based on problem characteristics

Clause deletion strategies

• LBD (Literal Block Distance) metric identifies high-quality learned clauses
• Implements activity-based clause deletion to remove less useful clauses
• Utilizes clause subsumption to eliminate redundant learned clauses
• Employs size-bounded clause deletion to control growth of clause database

Advanced SAT techniques

Incremental SAT solving

• Reuses learned information across multiple related SAT instances
• Implements assumption-based solving for efficient problem modifications
• Utilizes incremental preprocessing techniques to maintain solver state
• Supports push/pop operations for managing incremental problem formulations

Parallel SAT solving

• Implements portfolio-based approaches with diverse solver configurations
• Utilizes clause sharing to exchange learned information between threads
• Employs work stealing strategies to balance computational load
• Implements parallel preprocessing techniques to enhance overall solver performance

Portfolio-based approaches

• Combines multiple SAT solving strategies to tackle diverse problem instances
• Implements algorithm selection techniques to choose optimal solver configuration
• Utilizes machine learning for predicting best-performing solver strategies
• Employs online portfolio adaptation based on runtime performance metrics

SAT vs SMT solvers

Differences in capabilities

• SMT solvers handle richer theories beyond Boolean logic (arithmetic, arrays)
• SAT solvers focus exclusively on Boolean satisfiability problems
• SMT solvers integrate theory-specific decision procedures with SAT solving
• SAT solvers generally exhibit better performance for pure Boolean problems

Use cases in hardware verification

• SAT solvers excel in combinational equivalence checking and bounded model checking
• SMT solvers handle verification tasks involving arithmetic operations and data structures
• SAT-based approaches often used for initial abstraction refinement in verification flows
• SMT solvers support verification of higher-level hardware descriptions and properties

Applications in hardware verification

Bounded model checking

• Unrolls transition relation for fixed number of steps to check safety properties
• Encodes bounded reachability problems as SAT instances
• Implements incremental BMC to efficiently explore increasing bound depths
• Utilizes SAT-based interpolation for unbounded model checking

Equivalence checking

• Verifies functional equivalence between RTL and gate-level implementations
• Employs miter circuits to encode equivalence checking problems as SAT instances
• Implements structural hashing to identify equivalent sub-circuits
• Utilizes SAT sweeping for simplifying equivalence checking problems

Property verification

• Encodes safety and liveness properties as SAT formulas
• Implements property-directed reachability (IC3/PDR) using SAT queries
• Utilizes SAT-based abstraction refinement for handling complex properties
• Employs Craig interpolation for generating inductive invariants

Limitations of SAT solvers

Scalability challenges

• Exponential worst-case complexity limits effectiveness for large problem instances
• Memory consumption of learned clauses can become prohibitive for complex formulas
• Performance degradation observed for problems with high structural complexity
• Difficulty in handling global constraints that span across many variables

Handling of non-Boolean constraints

• Limited expressiveness for encoding arithmetic operations efficiently
• Challenges in representing and reasoning about floating-point arithmetic
• Inefficient encoding of cardinality constraints and pseudo-Boolean formulas
• Difficulty in handling quantified formulas and higher-order logic

Integration with other techniques

SAT-based model checking

• Implements k-induction using SAT solvers for proving safety properties
• Utilizes SAT-based interpolation for computing over-approximations of reachable states
• Employs SAT solving in IC3/PDR algorithm for generating inductive clauses
• Implements SAT-based abstraction refinement for handling complex state spaces

SAT in theorem proving

• Utilizes SAT solvers as decision procedures in SMT-based theorem provers
• Implements DPLL(T) framework for integrating SAT solving with theory reasoning
• Employs SAT-based techniques for finite model finding in first-order logic
• Utilizes SAT solving for checking consistency of axiom sets in automated reasoning

Industrial SAT solvers

Popular SAT solver implementations

• MiniSat serves as foundation for many modern SAT solvers
• Glucose implements aggressive clause deletion and rapid restarts
• CryptoMiniSat specializes in cryptographic problems and XOR constraints
• Lingeling employs advanced preprocessing and inprocessing techniques

Benchmarking and competitions

• SAT Competition evaluates solver performance on diverse benchmark sets
• Implements application, hard combinatorial, and random benchmark tracks
• Utilizes cloud computing platforms for large-scale solver evaluation
• Employs careful crafting of benchmark suites to challenge state-of-the-art solvers

Future directions

Machine learning in SAT solving

• Implements neural network-based branching heuristics for variable selection
• Utilizes reinforcement learning for adaptive restart and clause deletion policies
• Employs machine learning techniques for algorithm selection and configuration
• Implements learned clause quality prediction for more effective clause management

Quantum SAT solvers

• Explores quantum annealing approaches for solving SAT problems
• Implements hybrid classical-quantum algorithms for large-scale SAT instances
• Utilizes quantum-inspired algorithms to enhance classical SAT solving techniques
• Investigates potential speedups for specific classes of SAT problems using quantum computing