318 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
The origins of the halting problem
[EN] The halting problem is a prominent example of undecidable problem and its formulation and undecidability proof is usually attributed to Turing's 1936 landmark paper. Copeland noticed in 2004, though, that it was so named and, apparently, first stated in a 1958 book by Martin Davis. We provide additional arguments partially supporting this claim as follows: (i) with a focus on computable (real) numbers with infinitely many digits (e.g., pi), in his paper Turing was not concerned with halting machines; (ii) the two decision problems considered by Turing concern the ability of his machines to produce specific kinds of outputs, rather than reaching a halting state, something which was missing from Turing's notion of computation; and (iii) from 1936 to 1958, when considering the literature of the field no paper refers to any "halting problem" of Turing Machines until Davis' book. However, there were important preliminary contributions by (iv) Church, for whom termination was part of his notion of computation (for the lambda-calculus), and (v) Kleene, who essentially formulated, in his 1952 book, what we know as the halting problem now.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32, PROMETEO/2019/098.Lucas Alba, S. (2021). The origins of the halting problem. Journal of Logical and Algebraic Methods in Programming. 121:1-9. https://doi.org/10.1016/j.jlamp.2021.1006871912
A Stochastic Complexity Perspective of Induction in Economics and Inference in Dynamics
Rissanen's fertile and pioneering minimum description length principle (MDL) has been viewed from the point of view of statistical estimation theory, information theory, as stochastic complexity theory -.i.e., a computable approximation to Kolomogorov Complexity - or Solomonoff's recursion theoretic induction principle or as analogous to Kolmogorov's sufficient statistics. All these - and many more - interpretations are valid, interesting and fertile. In this paper I view it from two points of view: those of an algorithmic economist and a dynamical system theorist. >From these points of view I suggest, first, a recasting of Jevons's sceptical vision of induction in the light of MDL; and a complexity interpretation of an undecidable question in dynamics.
Instruction sequence processing operators
Instruction sequence is a key concept in practice, but it has as yet not come
prominently into the picture in theoretical circles. This paper concerns
instruction sequences, the behaviours produced by them under execution, the
interaction between these behaviours and components of the execution
environment, and two issues relating to computability theory. Positioning
Turing's result regarding the undecidability of the halting problem as a result
about programs rather than machines, and taking instruction sequences as
programs, we analyse the autosolvability requirement that a program of a
certain kind must solve the halting problem for all programs of that kind. We
present novel results concerning this autosolvability requirement. The analysis
is streamlined by using the notion of a functional unit, which is an abstract
state-based model of a machine. In the case where the behaviours exhibited by a
component of an execution environment can be viewed as the behaviours of a
machine in its different states, the behaviours concerned are completely
determined by a functional unit. The above-mentioned analysis involves
functional units whose possible states represent the possible contents of the
tapes of Turing machines with a particular tape alphabet. We also investigate
functional units whose possible states are the natural numbers. This
investigation yields a novel computability result, viz. the existence of a
universal computable functional unit for natural numbers.Comment: 37 pages; missing equations in table 3 added; combined with
arXiv:0911.1851 [cs.PL] and arXiv:0911.5018 [cs.LO]; introduction and
concluding remarks rewritten; remarks and examples added; minor error in
proof of theorem 4 correcte
The P Versus NP Problem Through Cellular Computing with Membranes
We study the P versus NP problem through membrane systems.
Language accepting P systems are introduced as a framework allowing
us to obtain a characterization of the P = NP relation by the
polynomial time unsolvability of an NP–complete problem by means of a
P system.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0
A Primer on the Tools and Concepts of Computable Economics
Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society
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
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