First-Order Temporal Logic Tensor Networks

Logic Tensor Networks (LTN) provide a neurosymbolic framework in which first-order logic is interpreted through tensor operations, enabling logical constraints to be integrated with differentiable learning. However, the original formulation of LTN is primarily suited to data represented as flat collections of individuals, and does not explicitly capture structural organization such as temporal order, sequential position, or graph connectivity. We introduce sLTN, an extension of LTN that makes structural dimensions first-class elements of the language. Structural dimensions represent named tensor axes associated with domain-specific organization, such as time steps, sequence positions, or graph nodes. They can be quantified explicitly, related through structural relations, and used to express temporal, sequential, and relational constraints directly at the logical level. We formalize the syntax and fuzzy tensor semantics of sLTN and show that, in the absence of structural dimensions, the framework recovers the original LTN semantics as a special case. We further describe a PyTorch implementation based on a declarative signature, formula parsing, and tensorial interpretation. The framework is illustrated on representative temporal and sequential reasoning examples. This paper serves as a companion to the sltn library, available at https://github.com/logictensornetworks/sltn.

Despite significant advances, large language models (LLMs) continue to exhibit brittle performance on complex temporal reasoning tasks. This failure mode is widely attributed to inherent deficits in autoregressive logical deduction. In this paper, we challenge this prevailing narrative, demonstrating that temporal reasoning is not the fundamental bottleneck; rather, the locus of failure lies in unstructured text-to-event representation. We introduce a novel neuro-symbolic question-answering framework governed by a Probabilistic Inconsistency Signal (PIS) that explicitly isolates perceptual errors from reasoning failures. By lifting unstructured text into explicit event graphs and interval constraints, our architecture strictly decouples semantic extraction from a symbolic reasoning engine. To robustly detect structural breaks, the PIS elegantly unifies symbolic credal intervals with epistemic neural uncertainty extracted via Evidential Deep Learning on LLM hidden states. Empirical evaluations reveal a striking paradigm shift: when provided with correct structural representations, our system's explicit proof traces achieve perfect 1.0 accuracy (4000/4000) and strictly zero false positives/negatives on temporal arithmetic benchmarks. On broader, noise-injected QA settings, the framework maintains a competitive 75.1% accuracy while enabling deterministic, step-level failure localization. Ultimately, by isolating the representation bottleneck from the reasoning substrate, this work reframes temporal QA from an algorithmic reasoning challenge to a structural alignment problem, charting a verifiable path forward for reliable neuro-symbolic AI.

In this work we study offline reinforcement learning (RL) under temporally extended task constraints expressed in Linear Temporal Logic over finite traces (LTLf). Recently, transformer-based approaches such as Trajectory Transformers and Decision Transformers have been adopted to address RL as a sequence modeling problem. However, these methods optimize purely for reward and do not account for high-level temporal requirements. Here, we introduce a neurosymbolic framework that injects LTLf background knowledge into such transformer-based RL policies. Our approach compiles LTLf formulas into deterministic finite automata (DFAs) and integrates them into the learning process through a differentiable representation and a logic-based loss function. In particular, we derive differentiable satisfaction signals from DFA progression and use them as a regularization term during training. The resulting method is architecture-agnostic across different models. We evaluate the proposed framework on navigation environments with specification suites covering combinations of safety and reachability temporal properties. Experimental results show that incorporating background knowledge not only improves constraint satisfaction, but also maintains competitive return compared to vanilla baselines.