Implement Advanced Constraint Graph Solver with CSP/SAT Integration

[POSiTiiiV]

Overview

Instead of brute-force enumeration of Priority 2 tile configurations, implement a sophisticated constraint graph system with CSP/SAT solving. This approach will be more efficient, scalable, and mathematically sound.

Current Limitations of Brute-Force Approach

• Exponential complexity: 2^n configurations for n hidden tiles
• Redundant computation: Recalculating same constraints repeatedly
• Poor scalability: Becomes intractable with >15 tiles
• No incremental solving: Can't build on previous solutions

Proposed Constraint Graph Architecture

1. Constraint Graph Structure

class ConstraintGraph:
    def __init__(self):
        self.variables: dict[Tile, bool] = {}  # Hidden tiles as boolean variables
        self.constraints: list[Constraint] = []  # Mathematical constraints
        self.adjacency: dict[Tile, set[Tile]] = {}  # Variable dependencies
        self.domains: dict[Tile, set[bool]] = {}  # Possible values for each variable

2. Constraint Types

class SumConstraint:
    """Represents: sum(bomb_variables) = target_value"""
    def __init__(self, variables: list[Tile], target: int):
        self.variables = variables
        self.target = target
    
class GlobalBombConstraint:
    """Represents: total_bombs_remaining = known_value"""
    def __init__(self, remaining_bombs: int, all_variables: list[Tile]):
        self.remaining_bombs = remaining_bombs
        self.variables = all_variables

3. Advanced Solving Techniques

A. Constraint Propagation (Arc Consistency)

def propagate_constraints(self) -> bool:
    """
    Apply constraint propagation to reduce variable domains
    Returns True if changes were made, False if no progress
    """
    # AC-3 algorithm implementation
    # For each constraint, eliminate impossible values from variable domains
    # Continue until no more reductions possible

B. Unit Propagation (Boolean Constraint Propagation)

def unit_propagation(self) -> dict[Tile, bool]:
    """
    Find variables that can only have one value due to constraints
    Returns assignments that are logically forced
    """
    # If constraint sum = 0 → all variables must be False (safe)
    # If constraint sum = len(variables) → all variables must be True (bombs)
    # If domain of variable has only one value → assign it

C. Conflict-Driven Learning

def analyze_conflicts(self, failed_assignment: dict[Tile, bool]) -> Constraint:
    """
    When a partial assignment leads to contradiction, learn why
    Returns a new constraint (clause) to prevent similar failures
    """
    # Analyze which variable assignments led to the conflict
    # Generate learned clauses to prune search space

4. Integration with External Solvers

Option A: Z3 SMT Solver (Recommended)

from z3 import *

def solve_with_z3(self) -> dict[Tile, bool]:
    """Use Z3 theorem prover for optimal solving"""
    solver = Solver()
    
    # Create boolean variables for each hidden tile
    z3_vars = {tile: Bool(f"tile_{tile.row}_{tile.col}") 
               for tile in self.variables}
    
    # Add sum constraints
    for constraint in self.constraints:
        if isinstance(constraint, SumConstraint):
            solver.add(Sum([z3_vars[var] for var in constraint.variables]) == constraint.target)
    
    # Add global bomb count constraint
    solver.add(Sum(z3_vars.values()) == self.remaining_bombs)
    
    # Solve and extract assignments
    return self.extract_solution(solver, z3_vars)

Option B: Python-Constraint Library

from constraint import Problem, ExactSumConstraint

def solve_with_python_constraint(self) -> dict[Tile, bool]:
    """Use python-constraint CSP solver"""
    problem = Problem()
    
    # Add variables with binary domains
    for tile in self.variables:
        problem.addVariable(tile, [0, 1])  # 0=safe, 1=bomb
    
    # Add sum constraints
    for constraint in self.constraints:
        problem.addConstraint(ExactSumConstraint(constraint.target),
                            constraint.variables)
    
    return problem.getSolution()

5. Incremental Solving & Caching

Constraint Memoization

class ConstraintCache:
    def __init__(self):
        self.solved_regions: dict[frozenset[Tile], dict[Tile, float]] = {}
        self.constraint_signatures: dict[str, dict[Tile, bool]] = {}
    
    def get_cached_probabilities(self, region: set[Tile]) -> dict[Tile, float]:
        """Return cached probability results for a region if available"""
        
    def cache_solution(self, region: set[Tile], probabilities: dict[Tile, float]):
        """Store solved probabilities for future use"""

Incremental Updates

def update_constraint_graph(self, newly_revealed: set[Tile]):
    """
    Efficiently update constraint graph when new information is revealed
    Only recompute affected constraints, not entire graph
    """
    # Identify constraints that reference newly revealed tiles
    # Remove invalid constraints, add new ones
    # Trigger incremental solving only for affected regions

6. Probabilistic Reasoning

def compute_exact_probabilities(self) -> dict[Tile, float]:
    """
    Compute exact probabilities using model counting
    P(tile is bomb) = count(models where tile=bomb) / count(all models)
    """
    # Use #SAT (model counting) to get exact probabilities
    # Or sample from solution space if exact counting is too expensive

7. Decision Strategy Enhancements

Multi-Objective Optimization

def select_optimal_move(self, probabilities: dict[Tile, float]) -> tuple[str, int, int]:
    """
    Select move based on multiple criteria, not just probability
    """
    # Primary: Lowest bomb probability
    # Secondary: Maximum information gain (tiles likely to be revealed)
    # Tertiary: Strategic position (corner/edge preference)
    # Quaternary: Connectivity (opens up new regions)

Look-Ahead Analysis

def evaluate_move_consequences(self, candidate_tile: Tile) -> float:
    """
    Simulate revealing a tile and measure expected solving progress
    """
    # For each possible value the tile could have
    # Simulate the constraint graph update
    # Measure how many new deterministic moves become available
    # Weight by probability of each outcome

Implementation Benefits

Efficiency Gains

• Polynomial time complexity for most constraint types (vs exponential brute-force)
• Incremental solving: Only recompute affected regions
• Constraint caching: Reuse solutions for similar configurations
• Pruning: Eliminate impossible configurations early

Scalability Improvements

• Handles large regions: 50+ tiles with modern SAT solvers
• Parallel solving: Multiple constraint regions simultaneously
• Memory efficient: Sparse representation of constraint graph

Mathematical Soundness

• Exact probabilities: No approximation errors from sampling
• Completeness: Finds all valid solutions
• Optimality: Provably best moves when deterministic choice exists

Advanced Features

Endgame Optimization

def apply_global_constraints(self):
    """
    In endgame, use global bomb count for tighter constraints
    """
    # Add constraint: sum(all_remaining_variables) = remaining_bombs
    # This can eliminate many configurations in late game

Pattern Recognition

def detect_common_patterns(self) -> list[tuple[str, dict[Tile, bool]]]:
    """
    Recognize solved patterns (1-2-1, 1-2-2-1, etc.) and apply cached solutions
    """
    # Library of known Minesweeper patterns with their solutions
    # Pattern matching on constraint graph structure

Multi-Region Coordination

def coordinate_multiple_regions(self, regions: list[ConstraintGraph]) -> dict[Tile, bool]:
    """
    Solve multiple disconnected regions simultaneously considering global constraints
    """
    # Use global bomb count to link disconnected regions
    # Solve jointly for optimal information

Implementation Roadmap

Phase 1: Core Constraint Graph

• Implement basic ConstraintGraph class
• Add SumConstraint and GlobalBombConstraint
• Basic constraint propagation (AC-3)

Phase 2: External Solver Integration

• Add Z3 solver backend
• Implement solution extraction and probability computation
• Add constraint caching system

Phase 3: Advanced Reasoning

• Incremental solving and graph updates
• Multi-objective move selection
• Look-ahead analysis

Phase 4: Optimization & Patterns

• Pattern recognition library
• Multi-region coordination
• Performance profiling and optimization

Success Metrics

• Solving speed: 10x+ faster than brute-force on complex regions
• Win rate: Significant improvement on expert-level boards
• Scalability: Handle 50+ tile constraint regions efficiently
• Memory usage: Reasonable memory footprint even for large boards

This constraint graph approach transforms the AI from a tactical solver into a strategic reasoning system that can handle the most complex Minesweeper scenarios with mathematical rigor and computational efficiency.