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E. A. Palyutin - One of the best experts on this subject based on the ideXlab platform.

  • p-Spectra of Abelian Groups
    Algebra and Logic, 2014
    Co-Authors: E. A. Palyutin
    Abstract:

    We consider four types of subgroups of Abelian groups: arbitrary subgroups (s-subgroups), algebraically closed subgroups (a-subgroups), pure subgroups (p-subgroups), and elementary subgroups (e-subgroups). A language L(X) is an extension of a language L by a set X of constants. A language LP is an extension of L by one Unary Predicate Symbol P. For i ∈ {s, a, p, e}, let Δ_i consist of sentences in LP, where L is the language of Abelian groups, expressing the fact that a Predicate P defines a subgroup of type i. For a complete theory T of Abelian groups and for i ∈ {s, a, p, e}, a cardinal function assigning a cardinal λ the supremum of the number of completions of sets (T^∗∪{P(a) | a ∈ X}∪Δ_i) in the language (L(X))P for complete extensions T∗ of T in the language L(X) for sets X of cardinality λ is called the (P, i)-spectrum of the theory T. For each i ∈ {s, a, p, e}, we describe all possible (P, i)-spectra of complete theories of Abelian groups.

  • The number of P-expansions of Abelian groups
    Algebra and Logic, 2013
    Co-Authors: E. A. Palyutin
    Abstract:

    In [1], there is a theorem which says that under the assumption of the generalized continuum hypothesis, a theory of any Abelian group is P -superstable if P defines an elementary subsystem. There, also, for this type of subgroups, a list of possible P -spectra of complete theories T of Abelian groups is given on the assumption that T is superstable. In [2], it was proved that a theory of every torsion-free Abelian group is P -stable if P defines an algebraically closed subgroup. A complete description of Abelian groups whose theories are P -stable if P defines an algebraically closed subgroup is contained in [3]. Below P -spectra of Abelian groups will be completely described for subgroups P of the following types: pure subgroups, elementary subsystems, algebraically closed subgroups, and arbitrary subgroups. In particular, we generalize the above-mentioned results in [1, 2]. A substructure B of a structure A is said to be algebraically closed if B contains each finite set X ⊆ A definable in A by a formula Φ(x) with parameters in B. By L(X) we denote a language obtained from L by adding X as a set of new constants. For a complete theory T in a language L, T (X) denotes an arbitrary completion of T in a language L(X). We say that X is a set in a theory T , bearing in mind that some completion T (X) has been fixed. Let LP be a language obtained from L by adding a new Unary Predicate Symbol P . Definition. Let T be a complete L-theory, Δ a set of LP -sentences, and X a set in T . Denote by CTΔ(X) the cardinality of a set of completions in a language (L(X))P of the set

Villegas Silva - One of the best experts on this subject based on the ideXlab platform.

Luis Miguel - One of the best experts on this subject based on the ideXlab platform.

Samuel Alexander - One of the best experts on this subject based on the ideXlab platform.

  • This sentence does not contain the Symbol X
    2013
    Co-Authors: Samuel Alexander
    Abstract:

    To appear in The Reasoner. In order to formalize the Liar’s Paradox, one approach is as follows. Work in the language of Peano arithmetic extended by a Unary Predicate Symbol T , and use Godel’s diagonal lemma to produce a sentence λ such that Peano arithmetic proves λ ↔ ¬T (pλq). One then refers to λ as a liar sentence, glossing it as “This sentence is not true.” A suprise may occur if we use a similar strategy to formalize

Daphne Koller - One of the best experts on this subject based on the ideXlab platform.

  • Asymptotic conditional probabilities: The non-Unary case
    Journal of Symbolic Logic, 1996
    Co-Authors: Adam J. Grove, Joseph Y. Halpern, Daphne Koller
    Abstract:

    AbstractMotivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1, …, N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kiĭ [24], if there is a non-Unary Predicate Symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.