Formal Framework

Disclaimer: The definitions below are proposed and speculative. They represent the current best formulation of PKT’s core ideas, not proven results. Where something is a conjecture rather than a theorem, it is labeled as such.

1. The Knowledge Tensor

Definition. The Knowledge Tensor at time $t$ is a multidimensional array:

\[\mathcal{T}^{(t)} \in \mathbb{R}^{d_1 \times d_2 \times \cdots \times d_n}\]

where each mode (dimension) represents a distinct aspect of knowledge:

Mode Interpretation Example
$d_1$ Concepts / entities “electron,” “gravity,” “Paris”
$d_2$ Relations / predicates “is-a,” “causes,” “located-in”
$d_3$ Confidence / belief strength $[0, 1]$ — how strongly the relation is held
$d_4$ Temporal context When the belief was formed or applies
$d_5, \ldots, d_n$ Additional structure Modality, source, abstraction level

Each entry $\mathcal{T}^{(t)}_{i_1, i_2, \ldots, i_n}$ represents the belief strength for a specific knowledge claim — e.g., “the concept electron stands in the relation has-charge with confidence 0.97 in the context of quantum mechanics.”

This is deliberately more expressive than a knowledge graph (which is a rank-2 adjacency matrix) or a knowledge base (which stores discrete facts). The tensor captures graded, structured, multi-relational knowledge.

2. Inductive Learning

The inductive process builds $\mathcal{T}$ from data. Given a dataset $\mathcal{D}$, the model learns parameters $\theta$ that populate the tensor:

\[\mathcal{T}^{(t)} = f_\theta(\mathcal{D})\]

where $f_\theta$ is a neural encoder (e.g., a transformer or graph neural network). The inductive loss encourages the tensor to faithfully represent the data:

\[\mathcal{L}_{\text{inductive}}(\theta) = -\mathbb{E}_{x \sim \mathcal{D}} \left[ \log p_\theta(x) \right]\]

This is standard — any self-supervised or supervised objective fits here. The key point is that inductive learning alone produces a tensor that reflects statistical patterns in $\mathcal{D}$, including any biases, correlations, or outright falsehoods present in the data.

3. The Falsification Operator

Definition. The Falsification Operator is a function:

\[\mathcal{F}: \mathbb{R}^{d_1 \times \cdots \times d_n} \times \mathcal{R} \rightarrow \mathbb{R}^{d_1 \times \cdots \times d_n}\]

where $\mathcal{R}$ is a set of deductive rules. The operator takes a Knowledge Tensor and a rule set, and returns a pruned tensor:

\[\mathcal{T}' = \mathcal{F}(\mathcal{T}, \mathcal{R})\]

Proposed semantics. For each rule $r \in \mathcal{R}$ and each tensor entry, $\mathcal{F}$ checks whether the entry is consistent with $r$. If not, the entry is projected to zero:

\[\mathcal{F}(\mathcal{T}, \mathcal{R})\_{i\_1, \ldots, i\_n} = \begin{cases} \mathcal{T}\_{i\_1, \ldots, i\_n} & \text{if } \forall r \in \mathcal{R}: \text{consistent}(\mathcal{T}\_{i\_1, \ldots, i\_n}, r) \\\\ 0 & \text{otherwise} \end{cases}\]

This is a hard operation — not a soft penalty. Entries that violate deductive rules are eliminated, not merely discouraged.

What counts as a “rule”?

Rules in $\mathcal{R}$ could include:

The rule set is externally provided, not learned. This is a deliberate design choice: the rules represent the deductive component of knowledge, which in the Popperian framework comes from theory, not from data.

Open problem: differentiability

The falsification operator as defined above is not differentiable — it involves a hard threshold (zero out or keep). This creates a tension with gradient-based training. Three possible resolutions:

  1. Straight-through estimator: Use the hard operator in the forward pass but approximate gradients in the backward pass (Bengio et al., 2013).
  2. Alternating optimization: Alternate between gradient-based inductive steps and non-differentiable falsification steps (similar to EM algorithms).
  3. Smooth approximation: Replace the hard threshold with a steep sigmoid, making $\mathcal{F}$ approximately differentiable. But this compromises the “hard falsification” claim — it becomes a soft constraint with a very high penalty, which is what existing systems already do.

This is the central technical challenge of PKT. Resolution 2 (alternating optimization) is the most promising, as it preserves the hard falsification property while remaining compatible with neural training.

4. Tensor Decomposition as Pruning

After falsification, the tensor $\mathcal{T}’$ may be sparse and noisy. Tensor decomposition provides a principled way to extract the essential structure.

The CP decomposition (Hitchcock, 1927; Kolda & Bader, 2009) approximates a tensor as a sum of rank-one components:

\[\mathcal{T}' \approx \sum_{k=1}^{R} \lambda_k \, \mathbf{a}_k^{(1)} \otimes \mathbf{a}_k^{(2)} \otimes \cdots \otimes \mathbf{a}_k^{(n)}\]

where $R$ is the rank, $\lambda_k$ are weights, and $\mathbf{a}_k^{(j)}$ are factor vectors. Choosing a low rank $R$ forces the decomposition to retain only the dominant patterns, discarding noise and hallucination.

Conjecture. Low-rank decomposition of the falsified tensor $\mathcal{T}’$ produces a representation that is both logically consistent (by construction, since inconsistent entries were zeroed) and statistically parsimonious (by the rank constraint). This combination may be a useful formal definition of “knowledge” — structured belief that has survived both empirical and logical tests.

5. The Combined Loss Function

The full PKT training objective combines inductive and falsification losses:

\[\mathcal{L} = \mathcal{L}_{\text{inductive}} + \lambda \, \mathcal{L}_{\text{falsification}}\]

where:

\[\mathcal{L}_{\text{falsification}} = \sum_{i_1, \ldots, i_n} \mathcal{T}_{i_1, \ldots, i_n}^2 \cdot \mathbb{1}\left[\neg \text{consistent}(\mathcal{T}_{i_1, \ldots, i_n}, \mathcal{R})\right]\]

Note: This loss function is a soft approximation of the hard falsification operator. In practice, training would likely use this differentiable loss, with periodic applications of the hard operator $\mathcal{F}$ to enforce strict consistency. The interplay between soft guidance (during gradient steps) and hard enforcement (between epochs) is a key area for future work.

6. The Learning Cycle

Putting it all together, PKT proposes an iterative learning process:

Step 1. Induction. Learn $\mathcal{T}^{(t)}$ from data by minimizing $\mathcal{L}$.

Step 2. Falsification. Apply $\mathcal{F}$ to get $\mathcal{T}’^{(t)} = \mathcal{F}(\mathcal{T}^{(t)}, \mathcal{R})$.

Step 3. Decomposition. Compute low-rank approximation of $\mathcal{T}’^{(t)}$ to extract core structure.

Step 4. Refinement. Use the decomposed tensor as initialization for the next inductive step.

Repeat until convergence.

Convergence Conjecture. Under suitable conditions on $\mathcal{R}$ and $\mathcal{D}$, the iteration above converges to a fixed point $\mathcal{T}^$ that is both statistically faithful to $\mathcal{D}$ and logically consistent with $\mathcal{R}$.*

This conjecture is unproven. Conditions for convergence likely require:

Proving (or disproving) this conjecture is the most important theoretical milestone for PKT.

References

Next: Open Questions — the deeper philosophical puzzles this framework raises.