Transformers have demonstrated remarkable capabilities in multi-step reasoning tasks. However, understandings of the underlying mechanisms by which they acquire these abilities through training remain limited, particularly from a theoretical standpoint. This work investigates how transformers learn to solve symbolic multi-step reasoning problems through chain-of-thought processes, focusing on path-finding in trees. We analyze two intertwined tasks: a backward reasoning task, where the model outputs a path from a goal node to the root, and a more complex forward reasoning task, where the model implements two-stage reasoning by first identifying the goal-to-root path and then reversing it to produce the root-to-goal path. Our theoretical analysis, grounded in the dynamics of gradient descent, shows that trained one-layer transformers can provably solve both tasks with generalization guarantees to unseen trees. In particular, our multi-phase training dynamics for forward reasoning elucidate how different attention heads learn to specialize and coordinate autonomously to solve the two subtasks in a single autoregressive path. These results provide a mechanistic explanation of how trained transformers can implement sequential algorithmic procedures. Moreover, they offer insights into the emergence of reasoning abilities, suggesting that when tasks are structured to take intermediate chain-of-thought steps, even shallow multi-head transformers can effectively solve problems that would otherwise require deeper architectures.

It has been demonstrated that carefully designed reasoning paradigms, like Chain-of-Thought (CoT) and Tree-of-Thought (ToT), can enhance the reasoning capabilities of small language models by detailed thinking and extensive thought searching, unbounded branching factors in the searching space create prohibitive reasoning consumption. However these methods fall into the trap of local optimum reasoning, which means the model lacks a global perspective while solving problems. We propose a novel reasoning paradigm called Reason from Future (RFF), which generates reasoning paths by bidirectional reasoning that combines top-down planning with bottom-up reasoning accumulation. The essence of RFF lies in its reverse reasoning mechanism, which prioritizes core logical relationships and imposes goal-oriented constraints on intermediate steps, thereby reducing the searching space and mitigating error accumulation inherent in sequential forward reasoning. Empirical evaluations across diverse experiments demonstrate that RFF outperforms conventional paradigms with higher accuracy and less searching space to solve complex tasks.

Large language model (LLM) based agents have shown great potential in following human instructions and automatically completing various tasks. To complete a task, the agent needs to decompose it into easily executed steps by planning. Existing studies mainly conduct the planning by inferring what steps should be executed next starting from the agent's initial state. However, this forward reasoning paradigm doesn't work well for complex tasks. We propose to study this issue in Minecraft, a virtual environment that simulates complex tasks based on real-world scenarios. We believe that the failure of forward reasoning is caused by the big perception gap between the agent's initial state and task goal. To this end, we leverage backward reasoning and make the planning starting from the terminal state, which can directly achieve the task goal in one step. Specifically, we design a BAckward Reasoning based agent (BAR). It is equipped with a recursive goal decomposition module, a state consistency maintaining module and a stage memory module to make robust, consistent, and efficient planning starting from the terminal state. Experimental results demonstrate the superiority of BAR over existing methods and the effectiveness of proposed modules.

Reverse thinking plays a crucial role in human reasoning. Humans can reason not only from a problem to a solution but also in reverse, i.e., start from the solution and reason towards the problem. This often enhances overall reasoning performance as it enables consistency checks between their forward and backward thinking. To enable Large Language Models (LLMs) to perform reverse thinking, we introduce Reverse-Enhanced Thinking (RevThink), a framework composed of data augmentation and learning objectives. In RevThink, we augment the dataset by collecting structured forward-backward reasoning from a teacher model, consisting of: (1) the original question, (2) forward reasoning, (3) backward question, and (4) backward reasoning. We then employ three objectives to train a smaller student model in a multi-task learning fashion: (a) generate forward reasoning from a question, (b) generate a backward question from a question, and (c) generate backward reasoning from the backward question. Experiments across 12 datasets covering commonsense, math, and logical reasoning show an average 13.53% improvement over the student model's zero-shot performance and a 6.84% improvement over the strongest knowledge distillation baselines. Moreover, our method demonstrates sample efficiency -- using only 10% of the correct forward reasoning from the training data, it outperforms a standard fine-tuning method trained on 10x more forward reasoning. RevThink also exhibits strong generalization to out-of-distribution held-out datasets.

While forward reasoning (i.e. find the answer given the question) has been explored extensively in the recent literature, backward reasoning is relatively unexplored. We examine the backward reasoning capabilities of LLMs on Math Word Problems (MWPs): given a mathematical question and its answer, with some details omitted from the question, can LLMs effectively retrieve the missing information? In this paper, we formally define the backward reasoning task on math word problems and modify three datasets to evaluate this task: GSM8k, SVAMP and MultiArith. Our findings show a significant drop in the accuracy of models on backward reasoning compared to forward reasoning across four SOTA LLMs (GPT4, GPT3.5, PaLM-2, and LLaMa-2). Utilizing the specific format of this task, we propose three novel techniques that improve performance: Rephrase reformulates the given problem into a forward reasoning problem, PAL-Tools combines the idea of Program-Aided LLMs to produce a set of equations that can be solved by an external solver, and Check your Work exploits the availability of natural verifier of high accuracy in the forward direction, interleaving solving and verification steps. Finally, realizing that each of our base methods correctly solves a different set of problems, we propose a novel Bayesian formulation for creating an ensemble over these base methods aided by a verifier to further boost the accuracy by a significant margin. Extensive experimentation demonstrates that our techniques successively improve the performance of LLMs on the backward reasoning task, with the final ensemble-based method resulting in a substantial performance gain compared to the raw LLMs with standard prompting techniques such as chain-of-thought.

Chain-of-Thought (CoT) prompting in large language models (LLMs) has shown promising performance on mathematical reasoning tasks. Recently, Self-Consistency samples a diverse set of reasoning chains with different answers and chooses the answer by majority voting. Though effective, its performance cannot be further improved by sampling more reasoning chains. To address this problem, we propose to integrate backward reasoning into answer verification. We first mask a number in the question by ${\bf x}$. The LLM is then asked to predict the masked number with a candidate answer $A$ embedded in the template: ``If we know the answer to the above question is $\{A\}$, what is the value of unknown variable ${\bf x}$?'' The LLM is expected to predict the masked number successfully if the provided candidate answer is correct. To further improve performance, we propose FOBAR (FOrward-BAckward Reasoning) to combine forward and backward reasoning for verifying candidate answers. Experiments are performed on six standard mathematical data sets and three LLMs (text-davinci-003, GPT-3.5-Turbo, GPT-4). Results show that FOBAR achieves state-of-the-art performance. In particular, FOBAR outperforms Self-Consistency which uses forward reasoning alone, demonstrating that combining forward and forward reasoning is better. It also outperforms existing verification methods, verifying the effectiveness of using the simple template in backward reasoning and the proposed combination.

Multi-hop Knowledge Base Question Answering (KBQA) aims to find the answer entities that are multiple hops away in the Knowledge Base (KB) from the entities in the question. A major challenge is the lack of supervision signals at intermediate steps. Therefore, multi-hop KBQA algorithms can only receive the feedback from the final answer, which makes the learning unstable or ineffective. To address this challenge, we propose a novel teacher-student approach for the multi-hop KBQA task. In our approach, the student network aims to find the correct answer to the query, while the teacher network tries to learn intermediate supervision signals for improving the reasoning capacity of the student network. The major novelty lies in the design of the teacher network, where we utilize both forward and backward reasoning to enhance the learning of intermediate entity distributions. By considering bidirectional reasoning, the teacher network can produce more reliable intermediate supervision signals, which can alleviate the issue of spurious reasoning. Extensive experiments on three benchmark datasets have demonstrated the effectiveness of our approach on the KBQA task.

Many techniques for representing knowledge have been proposed, but there have been few reports that compare their application This article presents an experimental comparison of four knowledge representation schemes: a simple production system, a structured production system, a frame system, and a logic system. The authors would like to express their appreciation to Dr Edward A. Feigenbaum and H Penny Nii of Stanford University for discussing the early results of this research during their visit to the authors' laboratory The authors also would like to acknowledge the support of Dr Jun Kawasaki, the general manager of Systems Development Laboratory, Hitachi, Ltd This information helps an expert system designer clarify the domain's characteristics and develop a conceptual system design. However, little information is provided for selecting adequate techniques after the system's function (input/output) is determined. The system was designed to interpret the X-ray powder diffraction spectra of rocks to determine their constituent minerals. This article focuses on expert system building tools; however, there may be many cases where no such tools are available.

Research in AI has built upon the tools and techniques of many different disciplines, including formal logic, probability theory, decision theory, management science, linguistics and philosophy. However, the application of these disciplines in AI has necessitated the development of many enhancements and extensions. Among the most powerful of these are the methods of computational logic. I will argue that computational logic, embedded in an agent cycle, combines and improves upon both traditional logic and classical decision theory. I will also argue that many of its methods can be used, not only in AI, but also in ordinary life, to help people improve their own human intelligence without the assistance of computers.

Many techniques for representing knowledge have been proposed, but there have been few reports that compare their application. This article presents an experimental comparison of four knowledge representation schemes: a simple production system, a structured production system. A frame system, and a logic system. We built four pilot expert systems to solve the same problem: risk management of a large construction project. Observations are made about hoe the structure of the domain knowledge affects the implementation of expert systems and their run time efficiency.