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According to the interactive view of computation, communication happens during the computation, not before or after it. This approach, distinct from concurrency theory and the theory of computation, represents a paradigm shift that changes our understanding of what is computation and how it is modeled. Interaction machines extend Turing machines with interaction to capture the behavior of concurrent...
In this paper, we show that closed-form analytic maps and flows can simulate Turing machines in an error-robust manner. The maps and ODEs defining the flows are explicitly obtained and the simulation is performed in real time.
Recent developments in the theory of infinite time Turing machines include the solution of the infinitary P versus NP problem and the rise of infinitary computable model theory.
We consider formal recombination operations presented in terms of string, graphs and permutations. These operations are faithful to the molecular operations of gene assembly introduced by Prescott, Ehrenfeucht, and Rozenberg [17]. The results mentioned here on the formal operations are mostly stated in the recent book [8], where one also finds the original references.
Symmetric Enumeration reducibility (≤se) is a subrelation of Enumeration reducibility (≤e) in which both the positive and negative information content of sets is compared. In contrast with Turing reducibility (≤T) however, the positive and negative parts of this relation are separate. A basic classification of ≤se in terms of standard reducibilities is carried out and it is shown that the natural...
I will discuss some recent results in the analysis of the computability-theoretic and proof-theoretic content of Vaughtian model theory, that is, the study of special models such as prime, saturated, and homogeneous models, and associated results such as the omitting types theorem. This is a research program dating back to the 1970’s (see [4]), but which has recently picked up steam with the application...
Finite trees are given a well ordering in such a way that there is a 1-1 correspondence between finite trees and an initial segment of the ordinals. The ordinal ε0 is the supremum of all binary trees. We get the (fixpoint free) n-ary Veblen hierarchy as tree functions and the supremum of all trees is the small Veblen ordinal φΩω(0).
Informally, the enumeration degree dege(A) (or e-degree) of a set A of natural numbers is a class of sets which have the same enumeration difficulty (see [2] for the exact definition).
We define the notion of ordinal computability by generalizing standard TURING computability on tapes of length ω to computations on tapes of arbitrary ordinal length. The generalized TURING machine is able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets...
In recent years (though much influenced by writings of G. Kreisel going back to the 50’s as well as subsequent work by H. Luckhardt and others) an applied form of proof theory systematically evolved which is also called ‘Proof Mining’ ([10], see also [1]). It is concerned with transformations of prima facie ineffective proofs into proofs from which certain quantitative computational information as...
This paper extends the logical approach to computable analysis via Σ–definability to higher type continuous data such as functionals and operators. We employ definability theory to introduce computability of functionals from arbitrary domain to the real numbers. We show how this concept works in particular cases.
We continue the study of P systems with mobile membranes introduced in [6], which is a variant of P systems with active membranes having none of the features like polarizations, label change and division of non-elementary membranes. This variant was shown to be universal using only the simple operations of endocytosis and exocytosis; moreover, if elementary membrane division is allowed, it is capable...
The class of rudimentary relations and the small relational Grzegorczyk classes attracted fairly much attention during the latter half of the previous century, e.g. Gandy [6], Paris-Wilkie [20], and numerous others.
Let C be a program written in a formal language in order to be executed by some kind of machinery. A statement about C might be true or false and has the form C:M. For the time being, just consider the statement C:M as a collection of data yielding information about the resources required to execute C; and if we know that C:M is true (or false), we know something useful when it comes to...
Let us say that a c.e. operator E is degree invariant on any given Turing degree a if X,Y ∈ a → E(X) ≡ TE(Y) . In [4] we construct a c.e. operator E such that ∀X [ X < TE(X) < TX′] . While we are unable to produce degree invariance everywhere, we are able to ensure that for every degree...
It is shown that for any 2-computably enumerable Turing degrees a, l, if l ′ = 0′, and l < a, then there are 2-computably enumerable Turing degrees x0, x1 such that both l ≤ x0, x1 < a and x0 ∨ x1 = a hold, extending the Robinson low splitting theorem for the computably enumerable degrees to the difference hierarchy.
A significant proportion of algorithms for solid modelling and, more generally, algorithms based on geometric representations suffer of robustness problems. In fact, the impacted algorithms reveal to have numerical computations deeply nested with combinatorial computations. Traditionally, the numerical part of the computation is implemented in double precision floating-point numbers and generally...
Recent developments in the theory of computing give a canonical way of assigning a dimension to each point of n-dimensional Euclidean space. Computable points have dimension 0, random points have dimension n, and every real number in [0,n] is the dimension of uncountably many points. If X is a reasonably simple subset of n-dimensional Euclidean space (a union of computably closed sets), then the classical...
We present linear time solutions to two NP-complete problems, namely SAT and the directed Hamiltonian Path Problem (HPP), based on accepting networks of splicing processors (ANSP) having all resources (size, number of rules and symbols) linearly bounded by the size of the given instance. The underlying structure of these ANSPs does not depend on the number of clauses, in the case of SAT, and the number...
This is a survey of a century long history of interplay between Hilbert’s tenth problem (about solvability of Diophantine equations) and different notions and ideas from the Computability Theory.
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