# Analysis of Unsatisfiable Temporal Countermodels: TN_CM_1 and TN_CM_2

## Introduction

This document analyzes the causes of unsatisfiability in the bimodal logic temporal countermodels TN_CM_1 and TN_CM_2. Both examples are intended to demonstrate countermodels for temporal operators, but fail to produce valid models.

## Problem Overview

The two unsatisfiable countermodels are:

1. **TN_CM_1**: Testing `A` → `\Future A` 
   - Premise: `A`
   - Conclusion: `\Future A`

2. **TN_CM_2**: Testing `\future A, \future B` → `\future (A ∧ B)`
   - Premises: `\future A`, `\future B`
   - Conclusion: `\future (A ∧ B)`

Both examples result in "UNSATISFIABLE" from the Z3 solver, meaning no countermodel can be constructed.

## Root Cause Analysis

After examining the unsatisfiable core constraints, the root cause appears to be an interaction between:

1. The frame constraints (particularly the skolem abundance constraint)
2. The temporal operator semantics
3. The time domain construction

### Analysis of TN_CM_1

The core conflict in TN_CM_1 can be traced to:

```
FRAME CONSTRAINTS:
1. ForAll([abundance_source_id, time_shift],
       Implies(And(is_world(abundance_source_id),
                  0 != time_shift,
                  defined_at_time(abundance_source_id, time_shift)),
              And(is_world(world_shifted_from(abundance_source_id)),
                  ...)))

PREMISES CONSTRAINTS:
1. And(defined_at_time(0, 0),
    truth_condition(world_function(0)[0], A))

CONCLUSIONS CONSTRAINTS:
1. Not(ForAll(future_true_time,
           Implies(And(defined_at_time(0, future_true_time),
                      0 < future_true_time),
                  And(defined_at_time(0, future_true_time),
                      truth_condition(world_function(0)[future_true_time], A)))))
```

The problem involves the interaction between these constraints:

1. The premise makes A true at time 0 in world 0
2. The conclusion requires that there's at least one future time where A is false in world 0
3. The skolem abundance constraint requires that for every defined time in world 0, there must exist another world that is a time-shifted version of world 0

What's happening is not that A must be true at all future times in world 0, but rather that the combination of constraints creates a complex dependency between worlds. For any future time where A would be false, the skolem abundance requirement would create a time-shift relationship that would eventually conflict with the premise that A is true at time 0.

The `is_shifted_by` function (lines 437-447) requires that corresponding time points in shifted worlds must have identical states:
```python
z3.Implies(
    self.defined_at_time(source_world, time),
    z3.Select(source_array, time) == z3.Select(target_array, shifted_time)
)
```

This creates a chain of constraints that propagates truth values across the model in a way that makes it impossible to satisfy the conjunction of all the requirements.

### Analysis of TN_CM_2

For TN_CM_2, the conflict is similar but more complex:

```
PREMISES CONSTRAINTS:
1. Not(ForAll(future_true_time,
           Implies(And(defined_at_time(0, future_true_time),
                      0 < future_true_time),
                  And(defined_at_time(0, future_true_time),
                      Not(truth_condition(world_function(0)[future_true_time], A))))))

2. Not(ForAll(future_true_time,
           Implies(And(defined_at_time(0, future_true_time),
                      0 < future_true_time),
                  And(defined_at_time(0, future_true_time),
                      Not(truth_condition(world_function(0)[future_true_time], B))))))

CONCLUSIONS CONSTRAINTS:
1. Not(Not(ForAll(future_true_time,
               Implies(And(defined_at_time(0, future_true_time),
                          0 < future_true_time),
                      Not(And(And(defined_at_time(0, future_true_time),
                                  truth_condition(world_function(0)[future_true_time], A)),
                              And(defined_at_time(0, future_true_time),
                                  truth_condition(world_function(0)[future_true_time], B))))))))
```

The premises require that:
- There exists at least one future time where A is true
- There exists at least one future time where B is true

The conclusion (when simplified) requires that:
- There exists a future time where both A and B are true

This seems satisfiable at first glance - if A is true at time 1 and B is true at time 2, we should be able to make both true at time 3. However, the skolem abundance constraints and world shifting requirements create a situation where it's impossible to satisfy all constraints simultaneously.

For instance, if A is true at time 1, the skolem abundance constraint requires another world that has the corresponding times shifted, and all corresponding states must be identical. The same applies for B at time 2. When these constraints are combined with the need to have both A and B true at some time, yet also satisfy the complex web of relationships between shifted worlds, Z3 cannot find a satisfiable model.

## Detailed Technical Explanation

The key issue is how the bimodal semantics implements world shifting and time domains:

1. The `skolem_abundance` constraint (lines 354-372 in semantic.py) enforces that for any world with a defined time point, there must exist another world that is a time-shifted version of it.

2. The `is_shifted_by` function (lines 437-465) creates strong equivalence requirements between corresponding states in shifted worlds.

3. The `shifted_times` function (lines 467-476) requires that time domains between shifted worlds correspond exactly.

These requirements create a highly constrained system where truth values must propagate in specific ways across the model. For the temporal countermodels, this results in constraints that cannot be simultaneously satisfied.

## Recommended Solution

To make these countermodels satisfiable, consider the following changes:

1. **Modify the skolem abundance constraint**: 
   - Consider making it less restrictive, perhaps by only requiring it for certain kinds of worlds or time points
   - Alternatively, make it a "soft constraint" rather than a hard requirement

2. **Revise the world shifting mechanism**:
   - Modify `is_shifted_by` to be less strict about state correspondence
   - Consider allowing shifted worlds to have different truth values for atomic propositions
   - Maintain task transition integrity but allow more flexibility in state correspondence

3. **Update temporal operator implementations**:
   - Ensure that the `FutureOperator` and related temporal operators properly handle the intended semantics
   - Review how these operators interact with the frame constraints

4. **Alternative solution - frame constraint optimization**:
   - Consider only enforcing the skolem abundance constraint for a limited number of time shifts
   - Allow for "time cutoffs" beyond which strict correspondence is not required

## Conclusion

The unsatisfiability of TN_CM_1 and TN_CM_2 stems from the interaction between several sophisticated constraints in the bimodal logic system. While each constraint makes sense in isolation, their combination creates a system that is too rigidly structured to admit the intended countermodels.

This reflects a common challenge in formal semantics: balancing the need for well-formed models against the flexibility required to represent interesting logical scenarios. By carefully revising the frame constraints and temporal operator semantics, it should be possible to enable these countermodels while maintaining the essential features of the bimodal logic system.