Claude E. Shannon. A universal Turing machine with two internal states. Automata studies, edited by C. E. Shannon and J. McCarthy, Annals of Mathematics studies no. 34, lithoprinted, Princeton University Press, Princeton1956, pp. 157–165.

1971 ◽  
Vol 36 (3) ◽  
pp. 532-532
Author(s):  
Patrick C. Fischer
Keyword(s):  
Turing Machine ◽  
Princeton University ◽  
Universal Turing Machine ◽  
Internal States

Computing with Neurons: A Summary

1998 ◽  
Author(s):  
Christof Koch
Keyword(s):  
Turing Machine ◽  
Neural Coding ◽  
Ionic Currents ◽  
Finite Alphabet ◽  
Universal Turing Machine ◽  
Internal States ◽  
Voltage Dependent ◽  
Finite State ◽  
Limited Precision ◽  
The Brain

We now have arrived at the end of the book. The first 16 chapters dealt with linear and nonlinear cable theory, voltage-dependent ionic currents, the biophysical origin of spike initiation and propagation, the statistical properties of spike trains and neural coding, bursting, dendritic spines, synaptic transmission and plasticity, the types of interactions that can occur among synaptic inputs in a passive or active dendritic arbor, and the diffusion and buffering of calcium and other ions. We attempted to weave these disparate threads into a single tapestry in Chaps. 17-19, demonstrating how these elements interact within a single neuron. The penultimate chapter dealt with various unconventional biophysical and biochemical mechanisms that could instantiate computations at the molecular and the network levels. It is time to summarize. What have we learned about the way brains do or do not compute? The brain has frequently been compared to a universal Turing machine (for a very lucid account of this, see Hofstadter, 1979). A Turing machine is a mathematical abstraction meant to clarify what is meant by algorithm, computation, and computable. Think of it as a machine with a finite number of internal states and an infinite tape that can read messages composed with a finite alphabet, write an output, and store intermediate results as memory. A universal Turing machine is one that can mimic any arbitrary Turing machine. We are here not interested in the renewed debate as to whether or not the brain can, in principle, be treated as such a machine (Lucas, 1964; Penrose, 1989), but whether this is a useful way to conceptualize nervous systems in this manner. Because brains have limited precision, only finite amounts of memory and do not live forever, they cannot possibly be like “real” Turing machines. It is therefore more appropriate to ask: to what extent can brains be treated as finite state machines or automata! Such a machine only has finite computational and memory resources (Hopcroft and Ullman, 1979). The answer has to be an ambiguous “it depends.”

scholarly journals Undecidability in quantum thermalization

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Naoto Shiraishi ◽  
Keiji Matsumoto
Keyword(s):  
Turing Machine ◽  
Thermal Equilibrium ◽  
General Theorem ◽  
Nearest Neighbour ◽  
Many Body ◽  
Initial State ◽  
One Dimensional ◽  
Universal Turing Machine ◽  
Many Body Systems ◽  
Neighbour Interaction

AbstractThe investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

Interactive Computation and Artificial Epistemologies

2021 ◽  
pp. 026327642110485
Author(s):  
Luciana Parisi
Keyword(s):  
Turing Machine ◽  
Mathematical Formalism ◽  
Symbolic Logic ◽  
Formal Models ◽  
Universal Turing Machine ◽  
Interactive Computation

What is algorithmic thought? It is not possible to address this question without first reflecting on how the Universal Turing Machine transformed symbolic logic and brought to a halt the universality of mathematical formalism and the biocentric speciation of thought. The article draws on Sylvia Wynter’s discussion of the sociogenic principle to argue that both neurocognitive and formal models of automated cognition constitute the epistemological explanations of the origin of the human and of human sapience. Wynter’s argument will be related to Gilbert Simondon’s reflections on ‘technical mentality’ to consider how socio-techno-genic assemblages can challenge the biocentricism and the formalism of modern epistemology. This article turns to ludic logic as one possible example of techno-semiotic languages as a speculative overturning of sociogenic programming. Algorithmic rules become technique-signs coinciding not with classic formalism but with interactive localities without re-originating the universality of colonial and patriarchal cosmogony.

Solvable and Unsolvable Problems (1954)

2004 ◽  
Author(s):  
Alan Turing
Keyword(s):  
Turing Machine ◽  
Popular Science ◽  
Systematic Method ◽  
Starting Position ◽  
Universal Turing Machine ◽  
Science News ◽  
Mathematical Problems ◽  
Substitution Type ◽  
Sharpened Form ◽  
Unsolvable Problems

In Chapter 1 Turing proves the existence of mathematical problems that cannot be solved by the universal Turing machine. There he also advances the thesis, now called the Church–Turing thesis, that any systematic method for solving mathematical problems can be carried out by the universal Turing machine. Combining these two propositions yields the result that there are mathematical problems which cannot be solved by any systematic method—cannot, in other words, be solved by any algorithm. In ‘Solvable and Unsolvable Problems’ Turing sets out to explain this result to a lay audience. The article first appeared in Science News, a popular science journal of the time. Starting from concrete examples of problems that do admit of algorithmic solution, Turing works his way towards an example of a problem that is not solvable by any systematic method. Loosely put, this is the problem of sorting puzzles into those that will ‘come out’ and those that will not. Turing gives an elegant argument showing that a sharpened form of this problem is not solvable by means of a systematic method (pp. 591–2). The sharpened form of the problem involves what Turing calls ‘the substitution type of puzzle’. An typical example of a substitution puzzle is this. Starting with the word BOB, is it possible to produce BOOOB by replacing selected occurrences of the pair OB by BOOB and selected occurences of the triple BOB by O? The answer is yes: . . . BOB → BBOOB → BBOBOOB → BOOOB . . .Turing suggests that any puzzle can be re-expressed as a substitution puzzle. Some row of letters can always be used to represent the ‘starting position’ envisaged in a particular puzzle, e.g. in the case of a chess problem, the pieces on the board and their positions. Desired outcomes, for example board positions that count as wins, can be described by further rows of letters, and the rules of the puzzle, whatever they are, are to be represented in terms of permissible substitutions of groups of letters for other groups of letters.

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