8,243 research outputs found
Zeno machines and hypercomputation
This paper reviews the Church-Turing Thesis (or rather, theses) with
reference to their origin and application and considers some models of
"hypercomputation", concentrating on perhaps the most straight-forward option:
Zeno machines (Turing machines with accelerating clock). The halting problem is
briefly discussed in a general context and the suggestion that it is an
inevitable companion of any reasonable computational model is emphasised. It is
hinted that claims to have "broken the Turing barrier" could be toned down and
that the important and well-founded role of Turing computability in the
mathematical sciences stands unchallenged.Comment: 11 pages. First submitted in December 2004, substantially revised in
July and in November 2005. To appear in Theoretical Computer Scienc
Digital Genesis: Computers, Evolution and Artificial Life
The application of evolution in the digital realm, with the goal of creating
artificial intelligence and artificial life, has a history as long as that of
the digital computer itself. We illustrate the intertwined history of these
ideas, starting with the early theoretical work of John von Neumann and the
pioneering experimental work of Nils Aall Barricelli. We argue that
evolutionary thinking and artificial life will continue to play an integral
role in the future development of the digital world.Comment: Extended abstract of talk presented at the 7th Munich-Sydney-Tilburg
Philosophy of Science Conference: Evolutionary Thinking, University of
Sydney, 20-22 March 2014. Presentation slides from talk available at
http://www.tim-taylor.com/papers/digital-genesis-presentation.pd
Proof of Church's Thesis
We prove that if our calculating capability is that of a universal Turing
machine with a finite tape, then Church's thesis is true. This way we
accomplish Post (1936) program.Comment: 6 page
Quantum Random Self-Modifiable Computation
Among the fundamental questions in computer science, at least two have a deep
impact on mathematics. What can computation compute? How many steps does a
computation require to solve an instance of the 3-SAT problem? Our work
addresses the first question, by introducing a new model called the ex-machine.
The ex-machine executes Turing machine instructions and two special types of
instructions. Quantum random instructions are physically realizable with a
quantum random number generator. Meta instructions can add new states and add
new instructions to the ex-machine. A countable set of ex-machines is
constructed, each with a finite number of states and instructions; each
ex-machine can compute a Turing incomputable language, whenever the quantum
randomness measurements behave like unbiased Bernoulli trials. In 1936, Alan
Turing posed the halting problem for Turing machines and proved that this
problem is unsolvable for Turing machines. Consider an enumeration E_a(i) =
(M_i, T_i) of all Turing machines M_i and initial tapes T_i. Does there exist
an ex-machine X that has at least one evolutionary path X --> X_1 --> X_2 --> .
. . --> X_m, so at the mth stage ex-machine X_m can correctly determine for 0
<= i <= m whether M_i's execution on tape T_i eventually halts? We demonstrate
an ex-machine Q(x) that has one such evolutionary path. The existence of this
evolutionary path suggests that David Hilbert was not misguided to propose in
1900 that mathematicians search for finite processes to help construct
mathematical proofs. Our refinement is that we cannot use a fixed computer
program that behaves according to a fixed set of mechanical rules. We must
pursue methods that exploit randomness and self-modification so that the
complexity of the program can increase as it computes.Comment: 50 pages, 3 figure
Monoidal computer III: A coalgebraic view of computability and complexity
Monoidal computer is a categorical model of intensional computation, where
many different programs correspond to the same input-output behavior. The
upshot of yet another model of computation is that a categorical formalism
should provide a much needed high level language for theory of computation,
flexible enough to allow abstracting away the low level implementation details
when they are irrelevant, or taking them into account when they are genuinely
needed. A salient feature of the approach through monoidal categories is the
formal graphical language of string diagrams, which supports visual reasoning
about programs and computations.
In the present paper, we provide a coalgebraic characterization of monoidal
computer. It turns out that the availability of interpreters and specializers,
that make a monoidal category into a monoidal computer, is equivalent with the
existence of a *universal state space*, that carries a weakly final state
machine for any pair of input and output types. Being able to program state
machines in monoidal computers allows us to represent Turing machines, to
capture their execution, count their steps, as well as, e.g., the memory cells
that they use. The coalgebraic view of monoidal computer thus provides a
convenient diagrammatic language for studying computability and complexity.Comment: 34 pages, 24 figures; in this version: added the Appendi
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