The pursuit of Artificial General Intelligence (AGI) demands AI systems that not only perceive but also reason in a human-like manner. While symbolic systems pioneered early breakthroughs in logic-based reasoning, such as MYCIN and DENDRAL, they suffered from brittleness and poor scalability. Conversely, modern deep learning architectures have achieved remarkable success in perception tasks, yet continue to fall short in interpretable and structured reasoning. This dichotomy has motivated growing interest in Neural–Symbolic AI, a paradigm that integrates symbolic logic with neural computation to unify reasoning and learning. This survey provides a comprehensive and technically grounded overview of AI reasoning in the deep learning era, with a particular focus on Neural–Symbolic AI. Beyond a historical narrative, we introduce a formal definition of AI reasoning and propose a novel three-dimensional taxonomy that organizes reasoning paradigms by representation form, task structure, and application context. We then systematically review recent advances—including Differentiable Logic Programming, abductive learning, program induction, logic-aware Transformers, and LLM-based symbolic planning—highlighting their technical mechanisms, capabilities, and limitations. In contrast to prior surveys, this work bridges symbolic logic, neural computation, and emergent generative reasoning, offering a unified framework to understand and compare diverse approaches. We conclude by identifying key open challenges such as symbolic–continuous alignment, dynamic rule learning, and unified architectures, and we aim to provide a conceptual foundation for future developments in general-purpose reasoning systems.

The key aim of compressed sensing is to stably recover a $K$ -sparse signals $\x$ from a linear model $\y=\A\x+\v$ , where $\v$ is a noise vector. Minimization of $\|\x\|_1-\|\x\|_2$ is a recently proposed effective recovery method. In this paper, we show that if the mutual coherence $\mu$ of $\A$ satisfies $\mu<\frac{1}{3K}$ , then this method can stably recover any $K$ -sparse signal $\x$ based on $\y$ and $\A$ . As far as we know, this is the first sufficient condition based on mutual coherence for this method.