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Time to Reason: Scalable Neurosymbolic Learning for LTLf via Fuzzy Semantics

This paper introduces DiffLTLf, a scalable neurosymbolic framework that integrates formally defined fuzzy semantics for LTLf to enable automata-free temporal learning, demonstrating competitive performance and superior scalability compared to existing state-of-the-art approaches.

Original authors: Riccardo Andreoni, Andrei Buliga, Alessandro Daniele, Paolo Felli, Chiara Ghidini, Marco Montali, Massimiliano Ronzani

Published 2026-08-18

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CC BY 4.0

Original authors: Riccardo Andreoni, Andrei Buliga, Alessandro Daniele, Paolo Felli, Chiara Ghidini, Marco Montali, Massimiliano Ronzani

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of artificial intelligence, there is a long-standing divide between two powerful ways of thinking. On one side are deep learning systems, which are excellent at recognizing patterns in raw data like images or sounds, but often struggle to explain their reasoning or follow strict rules. On the other side are symbolic systems, which use logic and rules to reason clearly and reliably, but often fail when faced with the messy, uncertain reality of the real world. Researchers have been trying to bridge this gap for years, creating "neurosymbolic" systems that combine the pattern-recognition power of neural networks with the logical rigor of symbolic rules. This fusion is crucial for building AI that is not only smart but also trustworthy, fair, and safe. A particularly difficult challenge in this field involves time. Many real-world processes, from manufacturing lines to medical monitoring, unfold over a sequence of events. To understand these processes, an AI must not only recognize what it sees in a single moment but also understand how those moments relate to one another over time, such as knowing that a specific step must always happen before another.

A team of researchers has developed a new approach to solve this problem of teaching AI to reason about time without getting bogged down by computational complexity. Their work focuses on a specific type of logical rule used to describe sequences of events, known as linear temporal logic on finite traces. Traditionally, to teach a computer to follow these rules, scientists would first translate the rules into a complex machine called a finite-state automaton. This machine acts like a map, guiding the AI through every possible state of a sequence. However, this method has a major flaw: as the rules become more complex or the sequences get longer, the map grows so large and intricate that it becomes impossible to navigate efficiently. The researchers found a way to bypass this bottleneck entirely. Instead of building a massive map, they taught the AI to evaluate the rules directly, using a mathematical concept called fuzzy semantics. This approach allows the system to handle uncertainty and partial truths, assigning a degree of "truth" to a rule rather than just a simple yes or no.

The researchers tested three different ways of applying this fuzzy logic to see which one worked best. They discovered that the choice of mathematical method mattered immensely. One method, which treats truth values somewhat like probabilities, proved to be the most robust. It allowed the AI to learn effectively even when the rules were complicated and the sequences of images were long. In contrast, the other two methods, which relied on different mathematical definitions of truth, struggled as the tasks became harder, often failing to find the correct solution. When compared to the traditional map-based methods, this new direct approach achieved similar or even better accuracy in recognizing patterns and following rules. More importantly, it did so much faster. While the traditional methods slowed down significantly as the rules grew more complex, the new system maintained a steady speed, making it far more scalable for real-world applications.

To verify these findings, the team created a challenging test environment involving streams of images representing different activities, such as cutting, welding, or packaging. The AI had to learn to identify these activities and determine if a sequence of actions followed a specific rule, such as "do not package until inspection is complete." The system was only told whether the entire sequence was correct or incorrect, without being told which specific image was which activity. This is a difficult learning task known as weakly supervised symbol grounding. The results showed that the new system could successfully learn to identify the activities and judge the sequences correctly. When the researchers increased the complexity of the rules or the length of the image sequences, the traditional methods began to falter, taking much longer to compute and sometimes losing accuracy. The new system, however, remained efficient and accurate, proving that it is possible to teach AI to reason about time without relying on the heavy, slow machinery of traditional automata.

This work suggests that the future of temporal reasoning in AI may not lie in building bigger and more complex maps, but in refining how we interpret the rules themselves. By choosing the right mathematical framework, researchers can create systems that are not only smarter but also more practical for the dynamic, time-sensitive tasks that define so much of our modern world. The study confirms that direct fuzzy evaluation is a viable and superior alternative to existing methods, offering a path forward for building AI that can understand the flow of time with both precision and efficiency.

Technical Summary: Time to Reason: Scalable Neurosymbolic Learning for LTLf via Fuzzy Semantics

Problem Statement

Neurosymbolic (NeSy) Artificial Intelligence seeks to integrate Deep Learning (DL) with symbolic reasoning to address challenges in data efficiency, fairness, and safety. While significant progress has been made in propositional and first-order logics, extending NeSy frameworks to Temporal Logics, specifically Linear Time Temporal Logic on finite traces (LTLf), remains an open challenge.

Existing temporal NeSy approaches (e.g., FuzzyA, NeSyA) typically rely on compiling LTLf specifications into Deterministic Finite Automata (DFA). This approach introduces two critical limitations:

Scalability: The size of the constructed automaton can be double-exponential relative to the formula size.
Sequential Bottleneck: Evaluating the automaton state requires recurrent computation over the sequence, preventing parallelization over time.

Furthermore, while differentiable semantics (fuzzy or probabilistic) have been proposed to interpret temporal formulas, there is no unified framework systematically analyzing how the choice of semantics (e.g., Gödel, Product, Łukasiewicz) impacts theoretical properties (equivalences, dualities) and predictive performance. Current benchmarks often rely on intermediate supervision signals or automata-specific constraints, limiting the assessment of alternative integration methods.

Methodology: The $\partial$ LTLf Framework

The authors propose $\partial$ LTLf, a novel neurosymbolic framework that evaluates LTLf specifications directly on fuzzy traces without intermediate automaton construction.

1. Theoretical Foundation: Fuzzy LTLf Semantics

The paper formally defines three distinct fuzzy semantics for LTLf, denoted as $fltl^x_f$ , based on different t-norms:

Gödel Semantics ( $fltl^G_f$ ): Uses min/max for conjunction/disjunction. The paper proves that classical LTLf equivalences (e.g., between Weak Until $W$ and Until $U$ ) hold fully under this semantics due to distributivity.
Product Semantics ( $fltl^P_f$ ): Treats truth values as probabilistic information, using multiplication for conjunction and probabilistic sum for disjunction.
Łukasiewicz Semantics ( $fltl^L_f$ ): Uses bounded sum and max-min operations.

A key theoretical finding is that while operators like $F$ (Eventually) and $G$ (Globally) can be derived from $X$ and $U$ in all three semantics, the equivalence for Weak Until ( $W$ ) holds only under Gödel semantics. Under Product and Łukasiewicz semantics, $W$ and Strong Release ( $M$ ) must be defined natively as they lose interdefinability with standard operators.

2. The $\partial$ LTLf Architecture

The framework addresses the Weakly Supervised Symbol Grounding task. Given a sequence of raw observations (e.g., images) and a binary label indicating whether the sequence satisfies a temporal specification $\Phi$ , the model learns to map observations to symbolic concepts.

The pipeline consists of three differentiable components:

Perception Module ( $f_\theta$ ): A neural network mapping raw inputs to fuzzy truth values for atomic propositions, generating a fuzzy trace $\lambda_\theta$ .
Temporal Logic Module: Evaluates the formula $\Phi$ directly on $\lambda_\theta$ using the chosen fuzzy semantics. This module is implemented as a recursive tensor operation (Algorithm 1) that processes the syntactic tree of $\Phi$ from leaves to root. It handles padding via boundary operators ( $\nabla_0, \nabla_1$ ) to ensure valid evaluation for sequences of varying lengths.
Loss Function: Computes binary cross-entropy between the fuzzy satisfaction value $v(\Phi, \lambda_\theta)$ and the ground-truth sequence label.

Crucially, $\partial$ LTLf is model-based: the temporal specification acts as a differentiable layer within the model during inference, rather than a loss constraint applied post-hoc.

Key Contributions

Systematic Theoretical Analysis: The paper provides the first unified definition of fuzzy semantics for LTLf, analyzing the preservation of classical equivalences and dualities. It demonstrates that Gödel semantics preserves full interdefinability, while Product and Łukasiewicz semantics require native definitions for certain operators.
Scalable Framework ( $\partial$ LTLf): A novel framework that bypasses automaton construction, enabling parallel evaluation of temporal formulas on fuzzy traces. This eliminates the double-exponential blowup and sequential evaluation bottlenecks of DFA-based methods.
Enhanced Evaluation Protocol: The authors introduce a benchmark that isolates temporal reasoning by removing relational constraints and intermediate supervision signals (e.g., automaton states). The protocol varies both formula complexity (number of conjoined sub-formulas) and sequence length to rigorously test scalability.

Experimental Results

The framework was evaluated on multi-stream image sequences (MNIST and Fashion-MNIST) against state-of-the-art DFA-based methods (FuzzyA, NeSyA) and purely neural baselines.

Impact of Semantics (RQ1): The choice of fuzzy semantics significantly impacts performance. Product semantics proved robust across increasing formula complexities, maintaining high symbol grounding accuracy (>85% even with 8 classes). In contrast, Gödel and Łukasiewicz semantics suffered from performance degradation and high variance as complexity increased, attributed to vanishing gradients (single-passing nature of min/max in Gödel and saturation issues in Łukasiewicz).
Comparison with State-of-the-Art (RQ2):
- Accuracy: $\partial$ LTLf (with Product semantics) achieved performance on par with, and in high-complexity/long-sequence scenarios, superior to DFA-based methods. Purely neural baselines failed to learn the task, confirming the necessity of symbolic constraints.
- Scalability: $\partial$ LTLf demonstrated superior computational efficiency. While DFA-based methods showed super-linear growth in training time as formula complexity increased (due to automaton size) and linear growth with sequence length (due to sequential state updates), $\partial$ LTLf remained nearly invariant to formula complexity and scaled linearly with sequence length.
- Efficiency: At high sequence lengths (200 steps) and high complexity, $\partial$ LTLf was up to three orders of magnitude faster per epoch than NeSyA and significantly faster than FuzzyA.

Significance and Claims

The paper claims that direct fuzzy interpretation of LTLf formulas is a competitive and scalable alternative to existing automata-based temporal NeSy frameworks.

Scalability: By avoiding automaton construction, $\partial$ LTLf overcomes the computational barriers that limit the application of temporal NeSy to complex specifications and long sequences.
Design Insight: The work establishes that the choice of fuzzy semantics is a critical design decision, with Product semantics offering a favorable trade-off between gradient informativeness and computational efficiency in temporal settings.
Foundation for Future Work: The systematic theoretical analysis of fuzzy LTLf and the proposed evaluation protocol provide a solid foundation for developing future temporal neurosymbolic systems, moving beyond preliminary proposals toward robust, scalable integration of temporal logic and deep learning.

The authors conclude that their results validate direct fuzzy evaluation as a practical solution for learning under temporal constraints, particularly in scenarios where scalability and computational efficiency are paramount.