In recent years, the research of deep neural network models has witnessed significant progress as various large language models are proposed and evolved at a fast pace. However, currently these models still cannot achieve reasonable performance in logical and mathematical reasoning without much human intervention, although more than 10 billion trainable parameters are placed in their latest neural networks. The failure of these sophisticated models in some seemly simple math problems inspired us to propose a non-Turing computer structure to address this challenge. The number of previous works on non-Turing machines is relatively small ([1,2,3,4,5,6,7]), which mostly discuss it conceptually and psychologically without giving any specific workable architecture like the Turing one. Among these, [2,7] argued that analog or natural computations should be used to conduct calculations involving continuous real numbers. The potential adaptivity of this type of models, however, is only restricted to the range between the discrete approximation of a continuous number and its unrepresented portion. Moreover, it lacks the interactions among the discrete domain and the continuous domain to solve the problem in their model as well. Recently, [8] and its variants investigated so-called Neural Turing Machines, which uses neural networks like LSTM [9] as the controller to make the entire system differentiable for end-to-end training. Compared with this structure, the abstract controller in the proposed non-Turing machine (which will be called “Ren’s machine" for simplicity below) can still be discrete/binary-based for logic reasoning. This novel structure allows the flexibility of switching back and forth between the abstract domain and the specific domain to better approach a solution to a given problem with rules summarized or learned from thought/tool experiments during the process. What’s more, it also enables the interactive collaborations of specific contents and abstract contents on both types of tapes to command and instruct the controller for completing a task.

The components and their relations of Ren’s machine are elaborated in Figure 1. This structure consists of a controller and two types of read/write heads and tapes, one in the specific domain and the other in the abstract domain. The interactions inter and intra the specific domain and the abstract domain are realized by different types of mappings annotated in this figure. These mappings can be implemented by applying prior knowledge or learned rules (e.g., neural networks or memorized tables) or simply rendering using a built-in function on the screen. Note that the writing and reading processes utilizing the heads in the specific domain are affected by the environment in the real world. As a result, some information from the outside world is also involved during these processes.

In this experiment, we aim to autonomously find a method of computing the product of any positive integers by using very basic algebraic rules that are already known. The following diagram shows the workflow of Ren’s machine to calculate an expression where it learned the distributive rule of multiplication by experiments generated in the specific domain and used the rule to get the correct result of the expression.

In this process, when it is recognized that a rule is technically applicable to any objects in either the abstract domain or the specific domain, the rule will be called and applied to them as a trial to move towards a solution even it seems that applying this rule to the objects is `meaningless’ (certainly a search strategy like reinforcement learning could be added for more `efficient’ look up). New rules are summarized or learned from thought or tool experiments generated from prior knowledge and existing rules that have already been learned so far. For example, in the computing process demonstrated in Figure 2, the distributive rule is learned by the cooperation of specific observations and abstract prior knowledge. Specifically, Ren’s machine can compute the multiplication of any small integers like $12\times 30$, $10\times 30$, $2\times 30$ using multiple additions by the original definition of multiplication. Then it finds that actually the equation $\boxed{12\times 30=10\times 30+2\times 30}$ holds true. It then generates an example in the specific domain accordingly by simply displaying the equation on the screen. Using similar procedure, it can generate a large number of examples (images) in the specific domain showing that the distributive property exists. Then neural networks can be trained to learn the transformation between the images on the left side of the equation and those on the right side of the equation. It is shown in the experiment that such neural networks (e.g. LLaMA [10], ChatGPT [11]) are able to generalize to other pairs of integers that are unseen in the training, even when the integers are very large. After applying the transformation in the visual domain, we can use computer vision models like YOLO [12] to convert the equation back into the abstract domain. This procedure eventually enables Ren’s machine to autonomously learn a general method of calculating the multiplications of arbitrary positive integers.

We conduct the test of computing the multiplications in 20 pairs of positive integers listed as follows:

12*34,	123*345,	1234*3456,	12345*34567,	123456*345678
23*45,	234*456,	2345*4567,	23456*45678,	234567*456789
34*56,	345*567,	3456*5678,	34567*56789,	345678*567891
45*67,	456*678,	4567*6789,	45678*67891,	456789*678912

The results in this experiment show that the accuracy ($100\%$) achieved by the proposed method on Ren’s machine is significantly higher than that ($40\%$) achieved by the state-of-the-art large language model used in Microsoft Copilot [13].

In this paper, we propose a non-Turing architecture, Ren’s machine, that solves a problem by autonomously learning rules and building knowledge bases from generated experiments in the process of approaching an answer. It can input and output by using two types of read/write heads in both the specific domain and the abstract domain. The mappings among these domains can be conducted based on learned rules such as neural networks, memorized tables, or simply built-in rendering functions. In this way the controller can be commanded/instructed cooperatively by both the contents on the tape in the abstract domain and those in the specific domain. In the experiments, it is demonstrated that this flexible mechanism enables Ren’s machine to solve the multiplication problem of positive integers in a self-directed manner with higher accuracy rate, compared with other state-of-the-art LLM methods. More advantages of Ren’s machine could be further explored by coping with the problems that cannot be solved by Turing machine intrinsically. As we know, the halting problem is the problem of determining whether an arbitrary program will terminate, or continue to run forever. This problem is proven to be undecidable on Turing machine, meaning that there exists no general algorithm on Turing machine that can solve the halting problem for all possible pairs of program and input. On the other hand, on Ren’s machine, since the controller is connected with the specific domain using another type of read/write heads, it can also be commanded/instructed by the contents shown on the tape in the specific domain. Thus the halting problem can be addressed with significantly high accuracy, if one includes a clock in the specific domain to count the time that a program has been running for, or uses neural networks to recognize the text patterns of never-ending programs, or counts the number of steps from their printouts on the screen, etc. In this regard, instead of pretending to be human in conversation, if a machine can actively generate rules from thought/tool experiments, for generalizing them to approach an answer of the problem it tries to solve, then it can be closer to the real artificial general intelligence.