Codified Decision Trees (CDT)

• Codified Decision Trees are decision trees expressed in algebraic, matrix-vector formats that integrate temporal logic and behavioral rules for clear, interpretable decision-making.
• They enable efficient optimization and fast computation by codifying splits and predictions into parameterized templates, which are compatible with gradient-based tuning.
• CDTs facilitate adaptive, end-to-end differentiable learning and transparent rule validation, making them suitable for time-series analysis, narrative profiling, and dynamic model updates.

A Codified Decision Tree (CDT) is an algebraic, matrix-based or temporally/logically-structured realization of the classical decision tree, designed to admit efficient optimization, representation, and execution while retaining interpretability. CDT frameworks generalize the structure of conventional decision trees by codifying their tests, topology, and predictions into explicit matrix-vector forms, by unifying with temporal logic, or by organizing behavioral rules into verifiable, executable hierarchies. They serve several disparate but convergent goals: enabling differentiable end-to-end learning, efficiently encoding tree structure for fast computation, synthesizing symbolic rules for time-series or narrative data, and supporting transparent behavioral validation.

1. Matrix and Algebraic Representations of Decision Trees

CDT originated as a formalism to express all aspects of a binary decision tree in a parameterized, matrix–vector algebraic template. In Zhang's framework, a decision tree $T$ with $n_L$ internal nodes and $L$ leaves is represented by

• a selection matrix $S \in \mathbb{R}^{n_L \times n}$, where each row selects which input feature is tested at each node,
• a threshold vector $t \in \mathbb{R}^{n_L}$ for node splits,
• a bit-vector matrix $B \in \{0,1\}^{L \times n_L}$ describing which leaves survive each node decision,
• and a leaf value vector $v \in \mathcal{O}^L$.

The forward computation comprises:

• Test phase: Compute $h = \sigma(Sx - t)$, where $\sigma$ is a binarized ReLU or similar activation.
• Traversal phase: Select the surviving leaves via arithmetic code equivalent to logical AND of bitvectors; this is operationalized by $r = \text{diag}(p)(Bh + \mathbf{1})$ for leaf selection.
• Prediction: Assign $T(x) = v[\arg\max r]$.

This codification enables the exact emulation of a classical tree, compatibility with deep learning accelerators, and natural extension to oblique or differentiable trees by relaxing $S$, $B$, and $\sigma$ to continuous spaces (Zhang, 2021).

2. Differentiable Codified Decision Trees and Gradient-based Learning

By linking the tree encoding to continuous optimization, CDT admits end-to-end differentiable variants. In the Decision Machines framework, a tree is encoded as

• $S \in \{0,1\}^{M \times n}$, $t \in \mathbb{R}^M$, as above,
• $B \in \{-1,0,+1\}^{L \times M}$, where $B_{i,m}$ records left/right/absent test directions for leaf $i$ and node $m$,
• $v \in \mathbb{R}^L$ collects per-leaf outputs.

The evaluation proceeds via

$\tilde h = Sx - t \in \mathbb{R}^M,\quad h = \sgn(\tilde h) \in \{-1, +1\}^M,\quad D = \mathrm{diag}(\|B_1\|_1, \ldots, \|B_L\|_1),\quad p = D^{-1}Bh,$

finally predicting $T(x) = v_{\arg\max p}$.

By relaxing $\sgn$ to a smooth approximation, replacing $\arg\max$ with $\mathrm{softmax}$, and treating $S$, $t$, $B$, $v$ as learnable parameters, one obtains a soft attention-like interleaving of tree traversal and output averaging: $T(x) = v^\top \mathrm{softmax}(\tilde B h / \tau),$ which supports training by standard gradient methods with cross-entropy or regression objectives (Zhang, 2021). This establishes a formal congruence between classical tree walks and attention layers.

3. Concise Decision Trees and Temporal Logic Encoding

In the context of time-series classification, a distinct form of CDT arises: the Concise Decision Tree, integrating Signal Temporal Logic (STL) as the domain of primal node tests. Here, each internal node tests a PSTL primitive (e.g., $F_{[t_0,t_1]}(s_j > \pi)$), and a root-to-leaf path induces a conjunction (possibly negated) of such primitives, yielding a full STL formula for the classification boundary.

To avoid formulaic explosion and maintain interpretability, a CDT employs conciseness heuristics:

• collapsing temporally-adjacent or logically-similar primitives into a single node,
• joint optimization of primitive parameters to minimize empirical impurity.

Learning proceeds recursively, optimizing primitives at each split using a misclassification gain criterion weighted by robustness measures. Stopping criteria are either maximum tree depth or near-homogeneous node samples. Complexity per CDT is $C_E(N) = \Theta(N + 4\sum_{k=2}^N C_O(k))$ with $C_O(N) = \Omega(N)$ for primitive optimization (Aasi et al., 2021).

Boosted ensembles (BCDTs) utilize AdaBoost-style reweighting and aggregate CDTs into a weighted "wSTL" formula, with ensemble prediction by majority or by minimal-complexity rule in the case of a perfect classifier in the ensemble.

4. Data-driven Rule Induction for Executable Behavioral Logic

In agent behavior modeling, CDT has been adapted to induce executable, validated decision structures from large narrative corpora. Here, a CDT for an agent $x$ is a rooted tree $T_x = (N, E, Q, H, r)$:

• $N$ nodes, $E$ directed edges with edge labels $Q$ as Boolean scene conditions,
• each node $v$ carries a set $H(v)$ of codified behavior statements,
• traversal for a scene $s$ consists of recursively walking the tree via $Q$, collecting all $h \in H(\cdot)$ encountered.

The tree is learned by:

• clustering scene-action data,
• using LLMs to generate if–then candidate rules,
• validating candidate statements on held-out data via natural language inference (NLI) entailment tests,
• accepting, rejecting, or specializing rules based on accuracy and coverage thresholds.

Retrieval at execution time is via a deterministic tree-walk with $O(|E|)$ complexity, independent of model parameter count (Peng et al., 15 Jan 2026).

5. Empirical Performance and Applications

CDT demonstrates strong empirical performance in several regimes:

• For time-series tasks, BCDTs achieve test misclassification rates of $0$–$1\%$ with highly compact STL formulae (2–3 operators), outperforming human-mined logic and non-conciseness-regularized frameworks (Aasi et al., 2021).
• In narrative agent profiling, CDTs and their variant CDT-Lite surpass prior methods and human profiles on next-action entailment scores for both the Fine-grained Fandom and Bandori conversational benchmarks:

Method	Fandom	Bandori
Vanilla	55.57	65.50
Fine-tuning	45.68	62.86
RICL	56.01	68.86
ETA	56.91	72.25
Human Profile	58.33	71.28
Codified Human Profile	59.30	71.87
CDT	60.82	77.71
CDT-Lite	61.01	79.04

The interpretability, efficiency, and principled update guarantees of CDT facilitate their adoption for model inspection and safe deployment (Peng et al., 15 Jan 2026).

6. Interpretability, Differentiability, and Generalizations

CDT enables transparent rule inspection: every if–then decision, split condition, or logic primitive is explicitly encoded and modifiable. Matrix-based representations allow for the analysis of bit-vector sparsity, rank, or even direct learning of the split structure under differentiable constraints.

The framework is extensible:

• Replacing hard selection matrices with learned embeddings yields oblique or categorical splits,
• Real-valued relaxation of bitvectors or splitting functions enables probabilistic trees or full integration into neural architectures,
• Continuous approximations to tree logic admit end-to-end optimization, annealing to hard logic at convergence (Zhang, 2021, Zhang, 2021).

A plausible implication is that CDTs unify symbolic and connectionist approaches under a mathematically precise, extensible paradigm suitable for both specification and learning of complex decision boundaries.

7. Update, Validation, and Lifelong Adaptation

CDT algorithms support principled, localized updates as data arrives or as agent behavior changes. For instance, insertion of new observed behaviors or corrections involves re-growing only the affected subtree. Failed rules (by updated NLI entailment rates) can be pruned automatically without retraining the entire tree. In behavioral domains, this provides robust, transparent adaptation to shifting operational requirements or agent goals (Peng et al., 15 Jan 2026).

Overall, Codified Decision Trees constitute a versatile, theory-grounded family of representations for decision functions, time-series logic synthesis, and validated rule induction, supporting the requirements of interpretability, computability, and continuous optimization critical for modern machine learning and AI systems.