Bridging Logic and Learning: Decoding Temporal Logic Embeddings via Transformers

Continuous representations of logic formulae allow us to integrate symbolic knowledge into data-driven learning algorithms. If such embeddings are semantically consistent, i.e. if similar specifications are mapped into nearby vectors, they enable continuous learning and optimization directly in the semantic space of formulae. However, to translate the optimal continuous representation into a concrete requirement, such embeddings must be invertible. We tackle this issue by training a Transformer-based decoder-only model to invert semantic embeddings of Signal Temporal Logic (STL) formulae. STL is a powerful formalism that allows us to describe properties of signals varying over time in an expressive yet concise way. By constructing a small vocabulary from STL syntax, we demonstrate that our proposed model is able to generate valid formulae after only 1 epoch and to generalize to the semantics of the logic in about 10 epochs. Additionally, the model is able to decode a given embedding into formulae that are often simpler in terms of length and nesting while remaining semantically close (or equivalent) to gold references. We show the effectiveness of our methodology across various levels of training formulae complexity to assess the impact of training data on the model's ability to effectively capture the semantic information contained in the embeddings and generalize out-of-distribution. Finally, we deploy our model for solving a requirement mining task, i.e. inferring STL specifications that solve a classification task on trajectories, performing the optimization directly in the semantic space.

stl2vec: Semantic and Interpretable Vector Representation of Temporal Logic

Integrating symbolic knowledge and data-driven learning algorithms is a longstanding challenge in Artificial Intelligence. Despite the recognized importance of this task, a notable gap exists due to the discreteness of symbolic representations and the continuous nature of machine-learning computations. One of the desired bridges between these two worlds would be to define semantically grounded vector representation (feature embedding) of logic formulae, thus enabling to perform continuous learning and optimization in the semantic space of formulae. We tackle this goal for knowledge expressed in Signal Temporal Logic (STL) and devise a method to compute continuous embeddings of formulae with several desirable properties: the embedding (i) is finite-dimensional, (ii) faithfully reflects the semantics of the formulae, (iii) does not require any learning but instead is defined from basic principles, (iv) is interpretable. Another significant contribution lies in demonstrating the efficacy of the approach in two tasks: learning model checking, where we predict the probability of requirements being satisfied in stochastic processes; and integrating the embeddings into a neuro-symbolic framework, to constrain the output of a deep-learning generative model to comply to a given logical specification.

Retrieval-Augmented Mining of Temporal Logic Specifications from Data

The integration of cyber-physical systems (CPS) into everyday life raises the critical necessity of ensuring their safety and reliability. An important step in this direction is requirement mining, i.e. inferring formally specified system properties from observed behaviors, in order to discover knowledge about the system. Signal Temporal Logic (STL) offers a concise yet expressive language for specifying requirements, particularly suited for CPS, where behaviors are typically represented as time series data. This work addresses the task of learning STL requirements from observed behaviors in a data-driven manner, focusing on binary classification, i.e. on inferring properties of the system which are able to discriminate between regular and anomalous behaviour, and that can be used both as classifiers and as monitors of the compliance of the CPS to desirable specifications. We present a novel framework that combines Bayesian Optimization (BO) and Information Retrieval (IR) techniques to simultaneously learn both the structure and the parameters of STL formulae, without restrictions on the STL grammar. Specifically, we propose a framework that leverages a dense vector database containing semantic-preserving continuous representations of millions of formulae, queried for facilitating the mining of requirements inside a BO loop. We demonstrate the effectiveness of our approach in several signal classification applications, showing its ability to extract interpretable insights from system executions and advance the state-of-the-art in requirement mining for CPS.

ECATS: Explainable-by-design concept-based anomaly detection for time series

Deep learning methods for time series have already reached excellent performances in both prediction and classification tasks, including anomaly detection. However, the complexity inherent in Cyber Physical Systems (CPS) creates a challenge when it comes to explainability methods. To overcome this inherent lack of interpretability, we propose ECATS, a concept-based neuro-symbolic architecture where concepts are represented as Signal Temporal Logic (STL) formulae. Leveraging kernel-based methods for STL, concept embeddings are learnt in an unsupervised manner through a cross-attention mechanism. The network makes class predictions through these concept embeddings, allowing for a meaningful explanation to be naturally extracted for each input. Our preliminary experiments with a simple CPS-based dataset show that our model is able to achieve great classification performance while ensuring local interpretability.

Towards Invertible Semantic-Preserving Embeddings of Logical Formulae

Logic is the main formal language to perform automated reasoning, and it is further a human-interpretable language, at least for small formulae. Learning and optimising logic requirements and rules has always been an important problem in Artificial Intelligence. State of the art Machine Learning (ML) approaches are mostly based on gradient descent optimisation in continuous spaces, while learning logic is framed in the discrete syntactic space of formulae. Using continuous optimisation to learn logic properties is a challenging problem, requiring to embed formulae in a continuous space in a meaningful way, i.e. preserving the semantics. Current methods are able to construct effective semantic-preserving embeddings via kernel methods (for linear temporal logic), but the map they define is not invertible. In this work we address this problem, learning how to invert such an embedding leveraging deep architectures based on the Graph Variational Autoencoder framework. We propose a novel model specifically designed for this setting, justifying our design choices through an extensive experimental evaluation. Reported results in the context of propositional logic are promising, and several challenges regarding learning invertible embeddings of formulae are highlighted and addressed.

Graph Neural Networks for Propositional Model Counting

Graph Neural Networks (GNNs) have been recently leveraged to solve several logical reasoning tasks. Nevertheless, counting problems such as propositional model counting (#SAT) are still mostly approached with traditional solvers. Here we tackle this gap by presenting an architecture based on the GNN framework for belief propagation (BP) of Kuch et al., extended with self-attentive GNN and trained to approximately solve the #SAT problem. We ran a thorough experimental investigation, showing that our model, trained on a small set of random Boolean formulae, is able to scale effectively to much larger problem sizes, with comparable or better performances of state of the art approximate solvers. Moreover, we show that it can be efficiently fine-tuned to provide good generalization results on different formulae distributions, such as those coming from SAT-encoded combinatorial problems.