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Paul E. Schupp - One of the best experts on this subject based on the ideXlab platform.

  • GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY
    2012
    Co-Authors: Carl G Jockusch, Paul E. Schupp
    Abstract:

    Abstract. Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically Computable if there is a Partial Computable Function which agrees with the characteristic Function of A on its domain D, and furthermore D has density 1, i.e. limn→ ∞ |{k < n: k ∈ D}|/n = 1. A set A is called coarsely Computable if there is a Computable set R such that the symmetric difference of A and R has density 0. We prove that there is a c.e. set which is generically Computable but not coarsely Computable and vice versa. We show that every nonzero Turing degree contains a set which is neither generically Computable nor coarsely Computable. We prove that there is a c.e. set of density 1 which has no Computable subset of density 1. Finally, we define and study generic reducibility. 1

  • generic computability turing degrees and asymptotic density
    Journal of The London Mathematical Society-second Series, 2012
    Co-Authors: Carl G Jockusch, Paul E. Schupp
    Abstract:

    Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically Computable if there is a Partial Computable Function which agrees with the characteris- tic Function of A on its domain D, and furthermore D has density 1, i.e. limn!1 |{k < n : k 2 D}|/n = 1. A set A is called coarsely Computable if there is a Computable set R such that the symmetric difference of A and R has density 0. We prove that there is a c.e. set which is generi- cally Computable but not coarsely Computable and vice versa. We show that every nonzero Turing degree contains a set which is not generically Computable and also a set which is not coarsely Computable. We prove that there is a c.e. set of density 1 which has no Computable subset of density 1. Finally, we define and study generic reducibility.

  • generic computability turing degrees and asymptotic density
    Journal of The London Mathematical Society-second Series, 2012
    Co-Authors: Carl G Jockusch, Paul E. Schupp
    Abstract:

    Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically Computable if there is a Partial Computable Function that agrees with the characteristic Function of A on its domain D, and furthermore D has density 1, that is, lim n→∞ |{kComputable if there is a Computable set R such that the symmetric difference of A and R has density 0. We prove that there is a computably enumerable (c.e.) set that is generically Computable but not coarsely Computable and vice versa. We show that every nonzero Turing degree contains a set that is neither generically Computable nor coarsely Computable. We prove that there is a c.e. set of density 1 that has no Computable subset of density 1. Finally, we define and study generic reducibility.

  • generic computability turing reducibility and asymptotic density
    arXiv: Group Theory, 2010
    Co-Authors: Carl G Jockusch, Paul E. Schupp
    Abstract:

    Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically Computable if there is a Partial Computable Function f whose domain has density 1 and which agrees with the characteristic Function of A on its domain. A set A is coarsely Computable if there is a Computable set C such that the symmetric difference of A and C has density 0. We prove that there is a c.e. set which is generically Computable but not coarsely Computable and vice versa. We show that every nonzero Turing degree contains a set which is not coarsely Computable. We prove that there is a c.e. set of density 1 which has no Computable subset of density 1. As a corollary, there is a generically Computable set A such that no generic algorithm for A has Computable domain. We define a general notion of generic reducibility in the spirt of Turing reducibility and show that there is a natural order-preserving embedding of the Turing degrees into the generic degrees which is not surjective.

Patey Ludovic - One of the best experts on this subject based on the ideXlab platform.

  • Ramsey-type graph coloring and diagonal non-computability
    'Springer Science and Business Media LLC', 2015
    Co-Authors: Patey Ludovic
    Abstract:

    International audienceA Function is diagonally non-Computable (d.n.c.) if it diagonalizes against the universal Partial Computable Function. D.n.c. Functions play a central role in algorithmic ran-domness and reverse mathematics. Flood and Towsner asked for which Functions h, the principle stating the existence of an h-bounded d.n.c. Function (h-DNR) implies Ramsey-type weak König's lemma (RWKL). In this paper, we prove that for every Computable order h, there exists an ω-model of h-DNR which is not a not model of the Ramsey-type graph coloring principle for two colors (RCOLOR 2) and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a Computable non-reducibility into a separation over ω-models

  • Ramsey-type graph coloring and diagonal non-computability
    2014
    Co-Authors: Patey Ludovic
    Abstract:

    A Function is diagonally non-Computable (d.n.c.) if it diagonalizes against the universal Partial Computable Function. D.n.c. Functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which Functions h, the principle stating the existence of an h-bounded d.n.c. Function (DNR_h) implies the Ramsey-type K\"onig's lemma (RWKL). In this paper, we prove that for every Computable order h, there exists an~$\omega$-model of DNR_h which is not a not model of the Ramsey-type graph coloring principle for two colors (RCOLOR2) and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a Computable non-reducibility into a separation over omega-models.Comment: 18 page

Carl G Jockusch - One of the best experts on this subject based on the ideXlab platform.

  • GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY
    2012
    Co-Authors: Carl G Jockusch, Paul E. Schupp
    Abstract:

    Abstract. Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically Computable if there is a Partial Computable Function which agrees with the characteristic Function of A on its domain D, and furthermore D has density 1, i.e. limn→ ∞ |{k < n: k ∈ D}|/n = 1. A set A is called coarsely Computable if there is a Computable set R such that the symmetric difference of A and R has density 0. We prove that there is a c.e. set which is generically Computable but not coarsely Computable and vice versa. We show that every nonzero Turing degree contains a set which is neither generically Computable nor coarsely Computable. We prove that there is a c.e. set of density 1 which has no Computable subset of density 1. Finally, we define and study generic reducibility. 1

  • generic computability turing degrees and asymptotic density
    Journal of The London Mathematical Society-second Series, 2012
    Co-Authors: Carl G Jockusch, Paul E. Schupp
    Abstract:

    Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically Computable if there is a Partial Computable Function which agrees with the characteris- tic Function of A on its domain D, and furthermore D has density 1, i.e. limn!1 |{k < n : k 2 D}|/n = 1. A set A is called coarsely Computable if there is a Computable set R such that the symmetric difference of A and R has density 0. We prove that there is a c.e. set which is generi- cally Computable but not coarsely Computable and vice versa. We show that every nonzero Turing degree contains a set which is not generically Computable and also a set which is not coarsely Computable. We prove that there is a c.e. set of density 1 which has no Computable subset of density 1. Finally, we define and study generic reducibility.

  • generic computability turing degrees and asymptotic density
    Journal of The London Mathematical Society-second Series, 2012
    Co-Authors: Carl G Jockusch, Paul E. Schupp
    Abstract:

    Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically Computable if there is a Partial Computable Function that agrees with the characteristic Function of A on its domain D, and furthermore D has density 1, that is, lim n→∞ |{kComputable if there is a Computable set R such that the symmetric difference of A and R has density 0. We prove that there is a computably enumerable (c.e.) set that is generically Computable but not coarsely Computable and vice versa. We show that every nonzero Turing degree contains a set that is neither generically Computable nor coarsely Computable. We prove that there is a c.e. set of density 1 that has no Computable subset of density 1. Finally, we define and study generic reducibility.

  • generic computability turing reducibility and asymptotic density
    arXiv: Group Theory, 2010
    Co-Authors: Carl G Jockusch, Paul E. Schupp
    Abstract:

    Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically Computable if there is a Partial Computable Function f whose domain has density 1 and which agrees with the characteristic Function of A on its domain. A set A is coarsely Computable if there is a Computable set C such that the symmetric difference of A and C has density 0. We prove that there is a c.e. set which is generically Computable but not coarsely Computable and vice versa. We show that every nonzero Turing degree contains a set which is not coarsely Computable. We prove that there is a c.e. set of density 1 which has no Computable subset of density 1. As a corollary, there is a generically Computable set A such that no generic algorithm for A has Computable domain. We define a general notion of generic reducibility in the spirt of Turing reducibility and show that there is a natural order-preserving embedding of the Turing degrees into the generic degrees which is not surjective.

Drago Antonino - One of the best experts on this subject based on the ideXlab platform.

  • Turing-Church thesis, constructve mathematics and intuitionist logic
    2021
    Co-Authors: Drago Antonino
    Abstract:

    At a first glance the Theory of computation relies on potential infinity and an organization aimed at solving a problem. Under such aspect it is like Mendeleev theory of chemistry. Also its theoretical development reiterates that of this scientific theory: it makes use of doubly negated propositions and its reasoning proceeds through ad absurdum proofs; a final, universal predicate of equivalence of all definitions of a computations is translated into an equality one, and at the same time intuitionist logic into classical logic. Yet, the last step of this development of current theory includes both a misleading notion of thesis and intuitive notions (e.g. the Partial Computable Function, as stressed by some scholars). A program for a rational re-construction of the theory according to the theoretical development of the above mentioned theories is sketchy suggested.Comment: The very nature of Turing-Church's thesis. Sketch of a program for a rational re-formulation of the theory of computatio

Antonino Drago - One of the best experts on this subject based on the ideXlab platform.

  • turing church thesis constructve mathematics and intuitionist logic
    arXiv: Logic, 2021
    Co-Authors: Antonino Drago
    Abstract:

    At a first glance the Theory of computation relies on potential infinity and an organization aimed at solving a problem. Under such aspect it is like Mendeleev theory of chemistry. Also its theoretical development reiterates that of this scientific theory: it makes use of doubly negated propositions and its reasoning proceeds through ad absurdum proofs; a final, universal predicate of equivalence of all definitions of a computations is translated into an equality one, and at the same time intuitionist logic into classical logic. Yet, the last step of this development of current theory includes both a misleading notion of thesis and intuitive notions (e.g. the Partial Computable Function, as stressed by some scholars). A program for a rational re-construction of the theory according to the theoretical development of the above mentioned theories is sketchy suggested.