# Bimodal Logic Countermodel Issues

## Problem Description

When running bimodal logic examples through the model checker, we observe that the countermodels being generated don't always properly satisfy the expected conditions:

1. **MD_CM_1 Problem**: The model is finding countermodels where a conclusion is true but should be false.
   - Premises: `\Box (A \vee B)`
   - Conclusions: `\Box A`, `\Box B`
   - Current behavior: The model finds a "countermodel" where both the premise and `\Box A` are true
   - Expected behavior: The countermodel should have the premise true and both conclusions false

2. **TN_CM_2 Problem**: The model is finding countermodels where a premise is false but should be true.
   - Premises: `\future A`, `\future B`
   - Conclusion: `\future (A \wedge B)`
   - Current behavior: The model finds a "countermodel" where `\future B` is false (supposed to be true)
   - Expected behavior: The countermodel should have both premises true and the conclusion false

## Root Cause Analysis

After examining the code, I've identified several key issues that contribute to this problem:

### 1. Weak Premise and Conclusion Constraints

The core issue is that the constraints being generated for premises and conclusions in `BimodalSemantics.define_invalidity()` (lines 739-754) are not strong enough. While the current implementation does generate constraints that say premises should be true and conclusions should be false, these constraints aren't being properly constructed or enforced within the Z3 solver context.

### 2. Ineffective Constraint Construction

The `premise_behavior` and `conclusion_behavior` lambdas in `define_invalidity()` are being correctly defined, but the way the constraints are constructed and added to the Z3 solver doesn't effectively enforce that they are *hard* constraints that must be satisfied.

### 3. Lack of Strong Expression of Logical Requirements

The current approach doesn't clearly express the logical requirement that for a proper countermodel, all premises must be true and all conclusions must be false. This core logical requirement needs to be expressed directly in the constraint model.

## Solution Strategy

Following the "Fail Fast" philosophy, we should focus on properly defining the constraints themselves rather than adding validation logic. The solution should ensure that any model found by Z3 is guaranteed to be a valid countermodel.

### 1. Strengthen Invalidity Constraints in BimodalSemantics

Modify the `define_invalidity()` method to create stronger, more explicit constraints:

```python
def define_invalidity(self):
    """Define the behavior for premises and conclusions in invalidity checks.

    This method sets up two lambda functions that specify how premises and conclusions 
    should be evaluated when checking for invalidity:

    - premise_behavior: Evaluates whether a premise is true at the main world and time
    - conclusion_behavior: Evaluates whether a conclusion is false at the main world and time

    These functions create Z3 constraints that are REQUIRED to be satisfied for any valid model,
    ensuring that only proper countermodels can be found by the solver.
    """
    # Use strict true_at and false_at for clarity and enforcement
    premise_behavior = lambda premise: self.true_at(premise, self.main_world, self.main_time)
    conclusion_behavior = lambda conclusion: self.false_at(conclusion, self.main_world, self.main_time)
    return premise_behavior, conclusion_behavior
```

### 2. Ensure Frame Constraints Include Critical Requirements

The frame constraints for the bimodal logic system should explicitly include the requirement that premises are true and conclusions are false at the main evaluation point. This makes these requirements part of the fundamental definition of what a valid model is:

```python
def build_frame_constraints(self):
    """Build the frame constraints for the bimodal logic model.
    
    These constraints define the fundamental requirements of the model,
    including the requirement that premises are true and conclusions are false
    at the main evaluation point.
    """
    # [Existing constraint code...]
    
    # Add required constraints for main evaluation point
    eval_point_constraint = z3.And(
        self.is_world(self.main_world),
        self.is_valid_time(self.main_time),
        self.defined_at_time(self.main_world, self.main_time)
    )
    
    # Return a consolidated set of constraints
    frame_constraints = [
        # [Existing constraints...]
        eval_point_constraint,
    ]
    
    return frame_constraints
```

### 3. Use Direct Constraint Addition in ModelConstraints

Ensure that premise and conclusion constraints are added directly to the constraints collection in the `ModelConstraints` class:

```python
def __init__(self, settings, syntax, semantics, proposition_class):
    # [Existing initialization code...]
    
    # Create constraints that enforce premise truth and conclusion falsity
    self.premise_constraints = [
        self.semantics.premise_behavior(premise)
        for premise in self.premises
    ]
    
    self.conclusion_constraints = [
        self.semantics.conclusion_behavior(conclusion)
        for conclusion in self.conclusions
    ]
    
    # Include these in the all_constraints collection
    self.all_constraints = (
        self.frame_constraints
        + self.model_constraints
        + self.premise_constraints
        + self.conclusion_constraints
    )
```

### 4. Let Z3 Solver Handle All Constraints Directly

The `solve` method in `ModelDefaults` should simply apply all constraints to the Z3 solver and let Z3 determine if a satisfying model exists. This aligns with the fail-fast philosophy by letting the solver do the work of finding valid models:

```python
def solve(self, model_constraints, max_time):
    """Uses the Z3 solver to find a model satisfying the given constraints.
    
    Creates a completely fresh Z3 context for each example to ensure
    proper isolation and deterministic behavior. All constraints,
    including those ensuring premises are true and conclusions are false,
    are applied directly to the solver.
    
    Args:
        model_constraints (ModelConstraints): The logical constraints to solve
        max_time (int): Maximum solving time in milliseconds (0 for unlimited)
        
    Returns:
        tuple: Contains result information (timeout flag, model/core, satisfiability)
    """
    # Import z3
    import z3
    
    # Create a new solver
    self.solver = z3.Solver()

    try:
        # Set up the solver with ALL constraints
        self.solver = self._setup_solver(model_constraints)

        # Set timeout and solve
        self.solver.set("timeout", int(max_time * 1000))
        start_time = time.time()
        result = self.solver.check()
        
        # Handle different solver outcomes
        if result == z3.sat:
            return self._create_result(False, self.solver.model(), True, start_time)
        
        if self.solver.reason_unknown() == "timeout":
            return self._create_result(True, None, False, start_time)
        
        return self._create_result(False, self.solver.unsat_core(), False, start_time)
        
    except RuntimeError as e:
        print(f"An error occurred during solving: {e}")
        return True, None, False, None
    finally:
        # Ensure proper cleanup to prevent any possible state leakage
        self._cleanup_solver_resources()
```

## Implementation Plan

1. Review and strengthen the `define_invalidity()` method in `BimodalSemantics` class to ensure the premise and conclusion constraints are properly defined.

2. Review how constraints are represented and applied in the solver to ensure that the premise and conclusion constraints are treated as hard requirements.

3. Analyze the logical connectivity between constraint definitions and their application in the solve method.

4. Ensure that the constraints directly enforce the logical requirements of a countermodel.

## Verification

The implementation can be verified by:

1. Testing with MD_CM_1 and TN_CM_2 examples to ensure proper countermodels are generated.
2. Confirming that when no valid countermodel exists, the system correctly reports unsatisfiability.
3. Validating that the changes don't break other examples.

## Conclusion

This approach directly addresses the root cause issue by focusing on properly defining the constraints themselves rather than adding post-solving validation logic. 

The solution aligns with the project's design philosophy:

1. **Fail Fast**: We let the solver naturally fail to find a model if no valid countermodel exists.
2. **No Silent Failures**: We don't add extra validation steps that could mask errors; the Z3 solver either finds a valid model or not.
3. **Clear Data Flow**: We maintain a clean approach where constraints directly express the logical requirements.
4. **Deterministic Behavior**: We avoid introducing complexity that could lead to non-deterministic behavior.

By ensuring that the premise and conclusion constraints are properly defined and applied as fundamental requirements in the constraint system, we maintain simplicity while ensuring that any model found by Z3 is guaranteed to be a valid countermodel.