SATNet is an award-winning MAXSAT solver that can be used to infer logical rules and integrated as a differentiable layer in a deep neural network. It had been shown to solve Sudoku puzzles visually from examples of puzzle digit images, and was heralded as an impressive achievement towards the longstanding AI goal of combining pattern recognition with logical reasoning. In this paper, we clarify SATNet's capabilities by showing that in the absence of intermediate labels that identify individual Sudoku digit images with their logical representations, SATNet completely fails at visual Sudoku (0% test accuracy). More generally, the failure can be pinpointed to its inability to learn to assign symbols to perceptual phenomena, also known as the symbol grounding problem, which has long been thought to be a prerequisite for intelligent agents to perform real-world logical reasoning. We propose an MNIST based test as an easy instance of the symbol grounding problem that can serve as a sanity check for differentiable symbolic solvers in general.

This paper provides a definitive, unifying framework for the Symbol Grounding Problem (SGP) by reformulating it within Algorithmic Information Theory (AIT). We demonstrate that the grounding of meaning is a process fundamentally constrained by information-theoretic limits, thereby unifying the Gödelian (self-reference) and No Free Lunch (statistical) perspectives. We model a symbolic system as a universal Turing machine and define grounding as an act of information compression. The argument proceeds in four stages. First, we prove that a purely symbolic system cannot ground almost all possible "worlds" (data strings), as they are algorithmically random and thus incompressible. Second, we show that any statically grounded system, specialized for compressing a specific world, is inherently incomplete because an adversarial, incompressible world relative to the system can always be constructed. Third, the "grounding act" of adapting to a new world is proven to be non-inferable, as it requires the input of new information (a shorter program) that cannot be deduced from the system's existing code. Finally, we use Chaitin's Incompleteness Theorem to prove that any algorithmic learning process is itself a finite system that cannot comprehend or model worlds whose complexity provably exceeds its own. This establishes that meaning is the open-ended process of a system perpetually attempting to overcome its own information-theoretic limitations.

Many experts argue that the future of artificial intelligence is limited by the field's

Recent advancements in large language models (LLMs) have revitalized philosophical debates surrounding artificial intelligence. Two of the most fundamental challenges - namely, the Frame Problem and the Symbol Grounding Problem - have historically been viewed as unsolvable within traditional symbolic AI systems. This study investigates whether modern LLMs possess the cognitive capacities required to address these problems. To do so, I designed two benchmark tasks reflecting the philosophical core of each problem, administered them under zero-shot conditions to 13 prominent LLMs (both closed and open-source), and assessed the quality of the models' outputs across five trials each. Responses were scored along multiple criteria, including contextual reasoning, semantic coherence, and information filtering. The results demonstrate that while open-source models showed variability in performance due to differences in model size, quantization, and instruction tuning, several closed models consistently achieved high scores. These findings suggest that select modern LLMs may be acquiring capacities sufficient to produce meaningful and stable responses to these long-standing theoretical challenges.

Summary and Contributions: EDIT after author response: I thank the authors for their thoughtful response, and am happy to see the authors will address the issues around the framing/tone of the paper. I also like the suggestion to include a discussion of the different definitions of end-to-end; this discussion is likely to be of great value to the community, given the overloaded definition of end-to-end. To address these claims/elaborate further -- * SATNet-specific: In cases where the logical rules aren't known, it's not possible to use a discrete solving module, since the rules need to be learned. That is, using a classifier and a separate Sudoku solver wouldn't work for the SATNet experiments, as the premise of the SATNet experiment was that the rules of Sudoku weren't known. This allows learning to, e.g., be more data-efficient.

SATNet is an award-winning MAXSAT solver that can be used to infer logical rules and integrated as a differentiable layer in a deep neural network. It had been shown to solve Sudoku puzzles visually from examples of puzzle digit images, and was heralded as an impressive achievement towards the longstanding AI goal of combining pattern recognition with logical reasoning. In this paper, we clarify SATNet's capabilities by showing that in the absence of intermediate labels that identify individual Sudoku digit images with their logical representations, SATNet completely fails at visual Sudoku (0% test accuracy). More generally, the failure can be pinpointed to its inability to learn to assign symbols to perceptual phenomena, also known as the symbol grounding problem, which has long been thought to be a prerequisite for intelligent agents to perform real-world logical reasoning. We propose an MNIST based test as an easy instance of the symbol grounding problem that can serve as a sanity check for differentiable symbolic solvers in general. We report on the causes of SATNet's failure and how to prevent them.

The remarkable performance of large language models (LLMs) on complex linguistic tasks has sparked a lively debate on the nature of their capabilities. Unlike humans, these models learn language exclusively from textual data, without direct interaction with the real world. Nevertheless, they can generate seemingly meaningful text about a wide range of topics. This impressive accomplishment has rekindled interest in the classical 'Symbol Grounding Problem,' which questioned whether the internal representations and outputs of classical symbolic AI systems could possess intrinsic meaning. Unlike these systems, modern LLMs are artificial neural networks that compute over vectors rather than symbols. However, an analogous problem arises for such systems, which we dub the Vector Grounding Problem. This paper has two primary objectives. First, we differentiate various ways in which internal representations can be grounded in biological or artificial systems, identifying five distinct notions discussed in the literature: referential, sensorimotor, relational, communicative, and epistemic grounding. Unfortunately, these notions of grounding are often conflated. We clarify the differences between them, and argue that referential grounding is the one that lies at the heart of the Vector Grounding Problem. Second, drawing on theories of representational content in philosophy and cognitive science, we propose that certain LLMs, particularly those fine-tuned with Reinforcement Learning from Human Feedback (RLHF), possess the necessary features to overcome the Vector Grounding Problem, as they stand in the requisite causal-historical relations to the world that underpin intrinsic meaning. We also argue that, perhaps unexpectedly, multimodality and embodiment are neither necessary nor sufficient conditions for referential grounding in artificial systems.

Many experts argue that the future of artificial intelligence is limited by the field's ability to integrate symbolic logical reasoning into deep learning architectures. The recently proposed differentiable MAXSAT solver, SATNet, was a breakthrough in its capacity to integrate with a traditional neural network and solve visual reasoning problems. For instance, it can learn the rules of Sudoku purely from image examples. Despite its success, SATNet was shown to succumb to a key challenge in neurosymbolic systems known as the Symbol Grounding Problem: the inability to map visual inputs to symbolic variables without explicit supervision ("label leakage"). In this work, we present a self-supervised pre-training pipeline that enables SATNet to overcome this limitation, thus broadening the class of problems that SATNet architectures can solve to include datasets where no intermediary labels are available at all. We demonstrate that our method allows SATNet to attain full accuracy even with a harder problem setup that prevents any label leakage. We additionally introduce a proofreading method that further improves the performance of SATNet architectures, beating the state-of-the-art on Visual Sudoku.