Logic-Embedded Training Objectives

• Logic-embedded training objectives integrate symbolic logic constraints directly into neural loss functions to guide model reasoning and generalization.
• They employ unified representations—such as converting circuits into AIG formats or mapping temporal logic to continuous embeddings—to ensure structural consistency and semantic optimization.
• Architectural designs, including specialized GNNs and Transformer decoders, incorporate inductive biases that preserve logical flow and enable accurate semantic mapping.

Logic-Embedded Training Objectives

Logic-embedded training objectives refer to supervision or loss functions in machine learning models that explicitly encode logical semantics or constraints, thereby aligning the model’s learning behavior with underlying symbolic, formal, or structural logical properties. Rather than treating logic as merely a source of external constraint, these approaches incorporate logical information—often in the form of formula embeddings, gate-level structural effects, or probabilistic logic supervision—directly into the core training objective. The result is a fusion of symbolic logic and gradient-based learning, enabling models to reason about logical structures, optimize within semantic spaces, and generalize to problems governed by formal rules.

1. Unified Representation Formats for Logic Integration

Logic-embedded training objectives require input data that reflect both the structural and functional aspects of logical systems. One practical strategy is the transformation of logical artifacts—such as circuits or formulae—into formats amenable to neural processing:

• In DeepGate (Li et al., 2021), logic circuits are converted into the And-Inverter Graph (AIG) format. All gates are standardized to two types (AND, inverter), providing consistent node semantics and a strong inductive bias. This homogenization ensures that learned representations encode both the combinatorial logic of gates and the graph-theoretical structure of real-world electronic design automation (EDA) circuits.
• For temporal logic formulae, continuous semantic embeddings are constructed via kernel methods that reflect robustness over distributions of signals (Candussio et al., 10 Jul 2025). This approach maps symbolic formulae (e.g., in Signal Temporal Logic, STL) into vector spaces where nearby points correspond to semantically similar requirements or behaviors.

Unified representations are foundational to logic-embedded training objectives, as they guarantee that learning signals are grounded in both function and structure and are generalizable across instances.

2. Construction of Logic-Embedded Training Objectives

Logic-embedded training objectives are designed to reflect the semantic content of logical systems, either through direct regression on logical properties or through discriminative tasks in embedding spaces:

• In DeepGate, the training objective targets the signal probability at each circuit node: the probability that the gate outputs logic '1' under random input. The dense supervision task is formalized through an average L1 loss,

$$\mathcal{L} = \frac{1}{N} \sum_{v \in \mathcal{V}} |y_v - \hat{y}_v|$$

where $y_v$ is the simulated signal probability and $\hat{y}_v$ is the model’s prediction (Li et al., 2021). Signal probabilities couple logical function (the composition of gate outputs) with propagation across the DAG circuit structure.

• In logic-based requirement mining for temporal logic, the objective is a supervised two-class separation based on robustness properties:

$$G(\varphi) = \frac{\mathbb{E}_{\bm{X}_p}[R_{\varphi}(\bm{X}_p)] - \mathbb{E}_{\bm{X}_n}[R_{\varphi}(\bm{X}_n)]}{\sigma_{\bm{X}_p}(R_{\varphi}(\bm{X}_p)) + \sigma_{\bm{X}_n}(R_{\varphi}(\bm{X}_n))}$$

where $R_\varphi$ is the robustness, and expectations and standard deviations are taken over positive and negative trajectory classes (Candussio et al., 10 Jul 2025). The training process optimizes formula embeddings to maximize $G$, capturing both qualitative and quantitative separation.

By embedding logical semantics directly into the supervised loss, these objectives enable models to learn functions aligned with symbolic reasoning and properties, not just statistical correlations.

3. Architectural Inductive Biases Guided by Logic

To fully exploit logic-embedded objectives, neural architectures often include bespoke inductive biases reflecting logic computation:

• DeepGate's GNN applies directed, topological message passing, attention focused on controlling values (e.g., ‘0’ dominating AND gates), explicit skip connections for reconvergent motifs using positional encoding, and bidirectional propagation layers that emulate implication and backtracking found in formal reasoning (Li et al., 2021). Node updates are implemented via GRUs, supporting recursive logic computation.
• In temporal logic decoding, the architecture employs a decoder-only Transformer with cross-attention from the embedding vector, token-level autoregressive generation matching STL syntax, and validity constraints at inference (Candussio et al., 10 Jul 2025).

These inductive biases ensure that learned representations and propagation mechanisms are logic-aware, enhancing both expressivity and generalization.

4. Semantic Embeddings and Invertibility

Central to logic-embedded training is the construction and utilization of semantic embeddings:

• For STL, kernel-based embeddings are used, defined via

$$k(\varphi, \psi) = \langle \rho(\varphi, \cdot), \rho(\psi, \cdot) \rangle = \int_{\xi \in \mathcal{T}} \rho(\varphi, \xi) \rho(\psi, \xi) d\mu_0(\xi)$$

with $D$-dimensional vectors $k(\varphi)$ built from anchor formulae (Candussio et al., 10 Jul 2025). These embeddings are semantically consistent: formulae with similar behaviors are embedded close together.

• The invertibility challenge is addressed by training a Transformer decoder to map continuous embeddings back to symbolic formulae. Strict invertibility is not possible—distinct syntaxes may share semantics—but feasible inversion (constructing an equivalent or semantically close formula) is sufficient for most learning tasks.

Embedding logic into the vector space enables continuous optimization and bridging between symbolic and sub-symbolic learning.

5. Evaluation Protocols and Downstream Applications

Logic-embedded learning schemes are evaluated not only for statistical fidelity but also for semantic and syntactic integrity:

• DeepGate demonstrates strong generalization through accurate signal probability prediction in large, unseen circuits, substantially outperforming standard GNN baselines (Li et al., 2021).
• In temporal logic embedding and decoding, syntactic validity rates reach 85–100%, and semantic metrics (cosine similarity over robustness vectors) exceed 0.95 after sufficient training. Decoded formulae are often syntactically simpler yet remain semantically close to target references (Candussio et al., 10 Jul 2025). Out-of-distribution scenarios are managed via natural penalties for invalid decodings.
• Downstream tasks include interpretable requirement mining (optimization directly in embedding space using Bayesian optimization), where learned formulae separate time-series trajectory classes, and neural circuit representation for multi-task EDA.

These protocols confirm that logic-embedded objectives yield outputs that are not only statistically sound, but also actionable and interpretable in terms of formal logic.

6. Tabular Summary of Logic-Embedded Learning Pipelines

Step	DeepGate (Li et al., 2021)	Temporal Logic Decoding (Candussio et al., 10 Jul 2025)
Representation	AIG format, one-hot node types	Kernel-based semantic embeddings
Objective	Signal probability regression (L1 loss)	Cross-entropy over formula tokens; maximize $G$
Architecture	Logic-aware GNN (directional, attention, GRU, skip)	Decoder-only Transformer (cross-attention)
Evaluation	Per-node probability accuracy, generalization	Semantic/syntactic validity, robustness similarity
Application	Multi-circuit EDA tasks	Requirement mining, semantic optimization

7. Significance and Methodological Implications

Logic-embedded training objectives mark a convergence between symbolic systems (logic circuits, formal specifications) and neural methods, expanding the scope of machine learning into domains where semantics, structure, and interpretability are essential. By supervising models on dense, function-centric logic properties, and integrating architecture-level logic computation, these methods improve generalization, enable powerful semantic optimization, and facilitate interpretable AI applications. A plausible implication is that such approaches may become foundational for a broad class of tasks in electronic design, automated reasoning, and specification-driven machine learning.