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The current cycle of hype and anxiety concerning the benefits and risks to human society of Artificial Intelligence is fuelled, not only by the increasing use of generative AI and other AI tools by the general public, but also by claims made on behalf of such technology by popularizers and scientists. In particular, recent studies have claimed that Large Language Models (LLMs) can pass the Turing Test-a goal for AI since the 1950s-and therefore can "think". Large-scale impacts on society have been predicted as a result. Upon detailed examination, however, none of these studies has faithfully applied Turing's original instructions. Consequently, we conducted a rigorous Turing Test with GPT-4-Turbo that adhered closely to Turing's instructions for a three-player imitation game. We followed established scientific standards where Turing's instructions were ambiguous or missing. For example, we performed a Computer-Imitates-Human Game (CIHG) without constraining the time duration and conducted a Man-Imitates-Woman Game (MIWG) as a benchmark. All but one participant correctly identified the LLM, showing that one of today's most advanced LLMs is unable to pass a rigorous Turing Test. We c

This paper proposes to revisit the Turing test through the concept of normality. Its core argument is that the Turing test is a test of normal intelligence as assessed by a normal judge. First, in the sense that the Turing test targets normal/average rather than exceptional human intelligence, so that successfully passing the test requires machines to "make mistakes" and display imperfect behavior just like normal/average humans. Second, in the sense that the Turing test is a statistical test where judgments of intelligence are never carried out by a single "average" judge (understood as non-expert) but always by a full jury. As such, the notion of "average human interrogator" that Turing talks about in his original paper should be understood primarily as referring to a mathematical abstraction made of the normalized aggregate of individual judgments of multiple judges. Its conclusions are twofold. First, it argues that large language models such as ChatGPT are unlikely to pass the Turing test as those models precisely target exceptional rather than normal/average human intelligence. As such, they constitute models of what it proposes to call artificial smartness rather than artifi

Considering that Turing's original test was co-opted by Weizenbaum and that six of the most common criticisms of the Turing test are unfair to both Turing's argument and the historical development of AI.

Many Formal Languages and Automata Theory courses introduce students to Turing machine extensions. One of the most widely-used extensions endows Turing machines with multiple tapes. Although multitape Turing machines are an abstraction to simplify Turing machine design, students find them no less challenging. To aid students in understanding these machines, the FSM programming language provides support for their definition and execution. This, however, has proven insufficient for many students to understand the operational semantics of such machines and to understand why such machines accept or reject a word. To address this problem, three visualization tools have been developed. The first is a dynamic visualization tool that simulates machine execution. The second is a static visualization tool that automatically renders a graphic for a multitape Turing machine's transition diagram. The third is a static visualization tool that automatically renders computation graphs for multitape Turing machines. This article presents these tools and illustrates how they are used to help students design and implement multitape Turing machines. In addition, empirical data is presented that sugges

This paper investigates a predator-prey reaction-diffusion model incorporating predator-taxis and a prey refuge mechanism, subject to homogeneous Neumann boundary conditions. Our primary focus is the analysis of codimension-two Turing-Turing bifurcation and the calculation of its associated normal form for this model. Firstly, employing the maximum principle and Amann's theorem, we rigorously prove the local existence and uniqueness of classical solutions. Secondly, utilizing linear stability theory and bifurcation theory, we conduct a thorough analysis of the existence and stability properties of the positive constant steady state. Furthermore, we derive precise conditions under which the model undergoes a Turing-Turing bifurcation. Thirdly, by applying center manifold reduction and normal form theory, we derive the method for calculating the third-truncated normal form characterizing the dynamics near the Turing-Turing bifurcation point. Finally, we present numerical simulations to validate the theoretical findings, confirming the correctness of the analytical results concerning the bifurcation conditions and the derived normal form.

Hard attention Chain-of-Thought (CoT) transformers are known to be Turing-complete. However, it is an open problem whether softmax attention Chain-of-Thought (CoT) transformers are Turing-complete. In this paper, we prove a stronger result that length-generalizable softmax CoT transformers are Turing-complete. More precisely, our Turing-completeness proof goes via the CoT extension of the Counting RASP (C-RASP), which correspond to softmax CoT transformers that admit length generalization. We prove Turing-completeness for CoT C-RASP with causal masking over a unary alphabet (more generally, for letter-bounded languages). While we show this is not Turing-complete for arbitrary languages, we prove that its extension with relative positional encoding is Turing-complete for arbitrary languages. We empirically validate our theory by training transformers for languages requiring complex (non-linear) arithmetic reasoning.

The world has seen the emergence of machines based on pretrained models, transformers, also known as generative artificial intelligences for their ability to produce various types of content, including text, images, audio, and synthetic data. Without resorting to preprogramming or special tricks, their intelligence grows as they learn from experience, and to ordinary people, they can appear human-like in conversation. This means that they can pass the Turing test, and that we are now living in one of many possible Turing futures where machines can pass for what they are not. However, the learning machines that Turing imagined would pass his imitation tests were machines inspired by the natural development of the low-energy human cortex. They would be raised like human children and naturally learn the ability to deceive an observer. These ``child machines,'' Turing hoped, would be powerful enough to have an impact on society and nature.

Computer science theory provides many different measures of complexity of a system including Kolmogorov complexity, logical depth, computational depth, and Levin complexity. However, these measures are all defined only for deterministic Turing machines, i.e., deterministic dynamics of the underlying generative process whose output we are interested in. Therefore, by construction they cannot capture complexity of the output of stochastic processes - like those in the real world. Motivated by this observation, we combine probabilistic Turing machines with a prior over the inputs to the Turing machine to define a complete stochastic process of Turing machines. We call this a stochastic process Turing machine. We use stochastic process Turing machines to define a set of new generative complexity measures based on Turing machines, which we call stochastic depth. As we discuss, stochastic depth is related to other such measures including Kolmogorov complexity and Levin complexity. However, as we elaborate, it has many desirable properties that those others measures lack. In addition, stochastic depth is closely related to various thermodynamic properties of computational systems. Stochas

We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines that we call "branching Turing machines" (which includes most of the known examples of universal Turing machines) has positive topological entropy. Motivated by the recent construction of Turing complete Euler flows, we deduce that any Turing complete dynamics with a continuous encoding that simulates a universal branching machine is chaotic. On the other hand, we show that, unexpectedly, universal Turing machines with zero topological entropy (and even zero speed) can be constructed, unveiling the independence of chaos and universality at the symbolic level.

In this paper, I prove necessary and sufficient conditions for the existence of Turing instabilities in a general system with three interacting species. Turing instabilities describe situations when a stable steady state of a reaction system (ordinary differential equation) becomes an unstable homogeneous steady state of the corresponding reaction-diffusion system (partial differential equation). Similarly to a well-known inequality condition for Turing instabilities in a system with two species, I find a set of inequality conditions for a system with three species. Furthermore, I distinguish conditions for the Turing instability when spatial perturbations grow steadily and the Turing-Hopf instability when spatial perturbations grow and oscillate in time simultaneously.

Expanding upon the widely recognized notion of mathematical universality in Turing machines, a concept of thermodynamic universality in Turing machines is introduced. Under the physical Church-Turing thesis, the existence of a thermodynamically universal Turing machine (TUTM) is demonstrated. A TUTM not only has the capability to simulate the input-output behavior of any given Turing machine but also replicate the heat production of that machine up to an additive constant. The finding shows that the hypothesis that the physical world is simulated by Turing machines may not be completely absurd.

In the wake of the latest trends of artificial intelligence (AI), there has been a resurgence of claims and questions about the Turing test and its value, which are reminiscent of decades of practical "Turing" tests. If AI were quantum physics, by now several "Schrödinger's" cats would have been killed. It is time for a historical reconstruction of Turing's beautiful thought experiment. This paper presents a wealth of evidence, including new archival sources, and gives original answers to several open questions about Turing's 1950 paper, including its relation with early AI.

When two Turing modes interact, i.e., Turing-Turing bifurcation occurs, superposition patterns revealing complex dynamical phenomena appear. In this paper, superposition patterns resulting from Turing-Turing bifurcation are investigated in theory. Firstly, the third-order normal form locally topologically equivalent to original partial functional differential equations (PFDEs) is derived. When selecting 1D domain and Neumann boundary conditions, three normal forms describing different spatial patterns are deduced from original third-order normal form. Also, formulas for computing coefficients of these normal forms are given, which are expressed in explicit form of original system parameters. With the aid of three normal forms, spatial patterns of a diffusive predator-prey system with Crowley-Martin functional response near Turing-Turing singularity are investigated. For one set of parameters, diffusive system supports the coexistence of four stable steady states with different single characteristic wavelengths, which demonstrates our previous conjecture. For another set of parameters, superposition patterns, tri-stable patterns that a pair of stable superposition steady states coex

This research revisits the classic Turing test and compares recent large language models such as ChatGPT for their abilities to reproduce human-level comprehension and compelling text generation. Two task challenges -- summarization, and question answering -- prompt ChatGPT to produce original content (98-99%) from a single text entry and also sequential questions originally posed by Turing in 1950. We score the original and generated content against the OpenAI GPT-2 Output Detector from 2019, and establish multiple cases where the generated content proves original and undetectable (98%). The question of a machine fooling a human judge recedes in this work relative to the question of "how would one prove it?" The original contribution of the work presents a metric and simple grammatical set for understanding the writing mechanics of chatbots in evaluating their readability and statistical clarity, engagement, delivery, and overall quality. While Turing's original prose scores at least 14% below the machine-generated output, the question of whether an algorithm displays hints of Turing's truly original thoughts (the "Lovelace 2.0" test) remains unanswered and potentially unanswerabl

Grigore showed that Java generics are Turing complete by describing a reduction from Turing machines to Java subtyping. We apply Grigore's algorithm to Python type hints and deduce that they are Turing complete. In addition, we present an alternative reduction in which the Turing machines are simulated in real time, resulting in significantly lower compilation times. Our work is accompanied by a Python implementation of both reductions that compiles Turing machines into Python subtyping machines.

Can a Turing Machine simulate the human mind? If the Church-Turing thesis is assumed to be true, then a Turing Machine should be able to simulate the human mind. In this paper, I challenge that assumption by providing strong mathematical arguments against the Church-Turing thesis. First, I show that there are decision problems that are computable for humans, but uncomputable for Turing Machines. Next, using a thought experiment I show that a humanoid robot equipped with a Turing Machine as the control unit cannot perform all humanly doable physical tasks. Finally, I show that a quantum mechanical computing device involving sequential quantum wave function collapse can compute sequences that are uncomputable for Turing Machines. These results invalidate the Church-Turing thesis and lead to the conclusion that the human mind cannot be simulated by a Turing Machine. Connecting these results, I argue that quantum effects in the human brain are fundamental to the computing abilities of the human mind.

We propose an alternative to the Turing test that removes the inherent asymmetry between humans and machines in Turing's original imitation game. In this new test, both humans and machines judge each other. We argue that this makes the test more robust against simple deceptions. We also propose a small number of refinements to improve further the test. These refinements could be applied also to Turing's original imitation game.

Dark matter may be far more complicated than scientists once believed。 A new study suggests it could consist of at least two different kinds of particles that slowly separate over time, with heavier particles sinking toward the centers of galaxies and lighter ones drifting outward。 This simple idea could explain several puzzling cosmic observations

A new study suggests the brain begins making decisions much earlier than scientists previously thought。 Researchers found that even primary sensory regions are influenced by higher brain areas through rapid feedback loops, rather than simply passing information forward。 This more dynamic view of brain function could help engineers design future AI