Evaluating SAT and SMT Solvers on Large-Scale Sudoku Puzzles

Modern SMT solvers have revolutionized the approach to constraint satisfaction problems by integrating advanced theory reasoning and encoding techniques. In this work, we evaluate the performance of modern SMT solvers in Z3, CVC5 and DPLL(T) against a standard SAT solver in DPLL. By benchmarking these solvers on novel, diverse 25x25 Sudoku puzzles of various difficulty levels created by our improved Sudoku generator, we examine the impact of advanced theory reasoning and encoding techniques. Our findings demonstrate that modern SMT solvers significantly outperform classical SAT solvers. This work highlights the evolution of logical solvers and exemplifies the utility of SMT solvers in addressing large-scale constraint satisfaction problems.

Causal Language Modeling Can Elicit Search and Reasoning Capabilities on Logic Puzzles

Causal language modeling using the Transformer architecture has yielded remarkable capabilities in Large Language Models (LLMs) over the last few years. However, the extent to which fundamental search and reasoning capabilities emerged within LLMs remains a topic of ongoing debate. In this work, we study if causal language modeling can learn a complex task such as solving Sudoku puzzles. To solve a Sudoku, the model is first required to search over all empty cells of the puzzle to decide on a cell to fill and then apply an appropriate strategy to fill the decided cell. Sometimes, the application of a strategy only results in thinning down the possible values in a cell rather than concluding the exact value of the cell. In such cases, multiple strategies are applied one after the other to fill a single cell. We observe that Transformer models trained on this synthetic task can indeed learn to solve Sudokus (our model solves $94.21\%$ of the puzzles fully correctly) when trained on a logical sequence of steps taken by a solver. We find that training Transformers with the logical sequence of steps is necessary and without such training, they fail to learn Sudoku. We also extend our analysis to Zebra puzzles (known as Einstein puzzles) and show that the model solves $92.04 \%$ of the puzzles fully correctly. In addition, we study the internal representations of the trained Transformer and find that through linear probing, we can decode information about the set of possible values in any given cell from them, pointing to the presence of a strong reasoning engine implicit in the Transformer weights.

Bridging Logic and Learning: A Neural-Symbolic Approach for Enhanced Reasoning in Neural Models

Neural-symbolic learning, an intersection of neural networks and symbolic reasoning, aims to blend neural networks' learning capabilities with symbolic AI's interpretability and reasoning. This paper introduces an approach designed to improve the performance of neural models in learning reasoning tasks. It achieves this by integrating Answer Set Programming (ASP) solvers and domain-specific expertise, which is an approach that diverges from traditional complex neural-symbolic models. In this paper, a shallow artificial neural network (ANN) is specifically trained to solve Sudoku puzzles with minimal training data. The model has a unique loss function that integrates losses calculated using the ASP solver outputs, effectively enhancing its training efficiency. Most notably, the model shows a significant improvement in solving Sudoku puzzles using only 12 puzzles for training and testing without hyperparameter tuning. This advancement indicates that the model's enhanced reasoning capabilities have practical applications, extending well beyond Sudoku puzzles to potentially include a variety of other domains. The code can be found on GitHub: https://github.com/Fadi2200/ASPEN.

PuzzleBench: Can LLMs Solve Challenging First-Order Combinatorial Reasoning Problems?

Recent works have explored the use of LLMs for reasoning tasks focussing on relatively simple problems, such as logical question answering. In our work, we wish to tackle more complicated problems, significantly expanding the capabilities of these models. Particularly, we explore whether LLMs can solve challenging first-order combinatorial reasoning problems, an example being the popular puzzle Sudoku. These problems have an underlying first-order structure described by a general description in natural language and can be instantiated to instances of varying sizes. Moreover these problems are computationally intensive requiring several reasoning steps to reach the solution. We present PuzzleBench a dataset of 31 such challenging puzzles. We observe that LLMs even when aided by symbolic solvers perform rather poorly on our benchmark. In response we propose a new approach, Puzzle-LM which combines LLMs with both symbolic solvers and program interpreters enabling them to reason about such challenging problems. We also show how feedback from smaller solved instances can help improve this reasoning ability.

Assessing SATNet's Ability to Solve the Symbol Grounding Problem

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. Naive applications of SATNet on this test lead to performance worse than that of models without logical reasoning capabilities. We report on the causes of SATNet's failure and how to prevent them.

Neuro-Symbolic Sudoku Solver

Deep Neural Networks have achieved great success in some of the complex tasks that humans can do with ease. These include image recognition/classification, natural language processing, game playing etc. However, modern Neural Networks fail or perform poorly when trained on tasks that can be solved easily using backtracking and traditional algorithms. Therefore, we use the architecture of the Neuro Logic Machine (NLM) and extend its functionality to solve a 9X9 game of Sudoku. To expand the application of NLMs, we generate a random grid of cells from a dataset of solved games and assign up to 10 new empty cells. The goal of the game is then to find a target value ranging from 1 to 9 and fill in the remaining empty cells while maintaining a valid configuration. In our study, we showcase an NLM which is capable of obtaining 100% accuracy for solving a Sudoku with empty cells ranging from 3 to 10. The purpose of this study is to demonstrate that NLMs can also be used for solving complex problems and games like Sudoku. We also analyze the behaviour of NLMs with a backtracking algorithm by comparing the convergence time using a graph plot on the same problem. With this study we show that Neural Logic Machines can be trained on the tasks that traditional Deep Learning architectures fail using Reinforcement Learning. We also aim to propose the importance of symbolic learning in explaining the systematicity in the hybrid model of NLMs.

Learning Reliable Logical Rules with SATNet

Bridging logical reasoning and deep learning is crucial for advanced AI systems. In this work, we present a new framework that addresses this goal by generating interpretable and verifiable logical rules through differentiable learning, without relying on pre-specified logical structures. Our approach builds upon SATNet, a differentiable MaxSAT solver that learns the underlying rules from input-output examples. Despite its efficacy, the learned weights in SATNet are not straightforwardly interpretable, failing to produce human-readable rules. To address this, we propose a novel specification method called "maximum equality", which enables the interchangeability between the learned weights of SATNet and a set of propositional logical rules in weighted MaxSAT form. With the decoded weighted MaxSAT formula, we further introduce several effective verification techniques to validate it against the ground truth rules. Experiments on stream transformations and Sudoku problems show that our decoded rules are highly reliable: using exact solvers on them could achieve 100% accuracy, whereas the original SATNet fails to give correct solutions in many cases. Furthermore, we formally verify that our decoded logical rules are functionally equivalent to the ground truth ones.

Large Language Model Guided Tree-of-Thought

In this paper, we introduce the Tree-of-Thought (ToT) framework, a novel approach aimed at improving the problem-solving capabilities of auto-regressive large language models (LLMs). The ToT technique is inspired by the human mind's approach for solving complex reasoning tasks through trial and error. In this process, the human mind explores the solution space through a tree-like thought process, allowing for backtracking when necessary. To implement ToT as a software system, we augment an LLM with additional modules including a prompter agent, a checker module, a memory module, and a ToT controller. In order to solve a given problem, these modules engage in a multi-round conversation with the LLM. The memory module records the conversation and state history of the problem solving process, which allows the system to backtrack to the previous steps of the thought-process and explore other directions from there. To verify the effectiveness of the proposed technique, we implemented a ToT-based solver for the Sudoku Puzzle. Experimental results show that the ToT framework can significantly increase the success rate of Sudoku puzzle solving. Our implementation of the ToT-based Sudoku solver is available on GitHub: \url{https://github.com/jieyilong/tree-of-thought-puzzle-solver}.

A Solver-Free Framework for Scalable Learning in Neural ILP Architectures

There is a recent focus on designing architectures that have an Integer Linear Programming (ILP) layer within a neural model (referred to as Neural ILP in this paper). Neural ILP architectures are suitable for pure reasoning tasks that require data-driven constraint learning or for tasks requiring both perception (neural) and reasoning (ILP). A recent SOTA approach for end-to-end training of Neural ILP explicitly defines gradients through the ILP black box (Paulus et al. 2021) - this trains extremely slowly, owing to a call to the underlying ILP solver for every training data point in a minibatch. In response, we present an alternative training strategy that is solver-free, i.e., does not call the ILP solver at all at training time. Neural ILP has a set of trainable hyperplanes (for cost and constraints in ILP), together representing a polyhedron. Our key idea is that the training loss should impose that the final polyhedron separates the positives (all constraints satisfied) from the negatives (at least one violated constraint or a suboptimal cost value), via a soft-margin formulation. While positive example(s) are provided as part of the training data, we devise novel techniques for generating negative samples. Our solution is flexible enough to handle equality as well as inequality constraints. Experiments on several problems, both perceptual as well as symbolic, which require learning the constraints of an ILP, show that our approach has superior performance and scales much better compared to purely neural baselines and other state-of-the-art models that require solver-based training. In particular, we are able to obtain excellent performance in 9 x 9 symbolic and visual sudoku, to which the other Neural ILP solver is not able to scale.

Puzzle Solving without Search or Human Knowledge: An Unnatural Language Approach

The application of Generative Pre-trained Transformer (GPT-2) to learn text-archived game notation provides a model environment for exploring sparse reward gameplay. The transformer architecture proves amenable to training on solved text archives describing mazes, Rubik's Cube, and Sudoku solvers. The method benefits from fine-tuning the transformer architecture to visualize plausible strategies derived outside any guidance from human heuristics or domain expertise. The large search space ($>10^{19}$) for the games provides a puzzle environment in which the solution has few intermediate rewards and a final move that solves the challenge.