Algebraic independence of certain Mahler numbers

In this note we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue-Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method.Comment: 8 page

On Pad\'e approximations and global relations of some Euler-type series

We shall consider some special generalizations of Euler's factorial series. First we construct Pad\'e approximations of the second kind for these series. Then these approximations are applied to study global relations of certain p-adic values of the series

On a result of Fel'dman on linear forms in the values of some E-functions

We shall consider a result of Fel'dman, where a sharp Baker-type lower bound is obtained for linear forms in the values of some E-functions. Fel'dman's proof is based on an explicit construction of Pad\'e approximations of the first kind for these functions. In the present paper we introduce Pad\'e approximations of the second kind for the same functions and use these to obtain a slightly improved version of Fel'dman's result

The Logic of Approximate Dependence

We extend the treatment of functional dependence, the basic concept of dependence logic, to include the possibility of dependence with a limited number of exceptions. We call this approximate dependence. The main result of the paper is a Completeness Theorem for approximate dependence atoms. We point out some problematic features of this which suggests that we should consider multi-teams, not just teams

A Note on Extensions of Infinitary Logic

We show that a strong form of the so called Lindstrom's Theorem fails to generalize to extensions of L_{kappa,omega} and L_{kappa,kappa}: For weakly compact kappa there is no strongest extension of L_{kappa,omega} with the (kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to kappa. With an additional set-theoretic assumption, there is no strongest extension of L_{kappa,kappa} with the (kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to <kappa

The size of a formula as a measure of complexity

We introduce a refinement of the usual Ehrenfeucht-Fra\"{\i}ss\'e game. The new game will help us make finer distinctions than the traditional one. In particular, it can be used to measure the size formulas needed for expressing a given property. We will give two versions of the game: the first version characterizes the size of formulas in propositional logic, and the second version works for first-order predicate logic.Comment: 25 page

Stationary sets and infinitary logic

Let K^0_lambda be the class of structures , where A subseteq lambda is disjoint from a club, and let K^1_lambda be the class of structures < lambda,, where A subseteq lambda contains a club. We prove that if lambda = lambda^{< kappa} is regular, then no sentence of L_{lambda^+ kappa} separates K^0_lambda and K^1_lambda. On the other hand, we prove that if lambda = mu^+, mu = mu^{< mu}, and a forcing axiom holds (and aleph_1^L= aleph_1 if mu = aleph_0), then there is a sentence of L_{lambda lambda} which separates K^0_lambda and K^1_lambda

An Ehrenfeucht-Fra\"{i}ss\'{e} Game for Lω1ωL_{\omega_1\omega}

Ehrenfeucht-Fraisse games are very useful in studying separation and equivalence results in logic. The standard finite Ehrenfeucht-Fraisse game characterizes equivalence in first order logic. The standard Ehrenfeucht-Fraisse game in infinitary logic characterizes equivalence in $L_{\infty\omega}$. The logic $L_{\omega_1\omega}$ is the extension of first order logic with countable conjunctions and disjunctions. There was no Ehrenfeucht-Fraisse game for $L_{\omega_1\omega}$ in the literature. In this paper we develop an Ehrenfeucht-Fraisse Game for $L_{\omega_1\omega}$. This game is based on a game for propositional and first order logic introduced by Hella and Vaananen. Unlike the standard Ehrenfeucht-Fraisse games which are modeled solely after the behavior of quantifiers, this new game also takes into account the behavior of connectives in logic. We prove the adequacy theorem for this game. We also apply the new game to prove complexity results about infinite binary strings.Comment: 22 pages, 1 figur

Algebraic independence of reciprocal sums of certain Fibonacci-type numbers

The paper studies algebraic independence of certain reciprocal sums of Fibonacci and Lucas sequences. Also more general binary recurrences are considered. The main tool is Mahler's method reducing the investigation of the algebraic independence of function values to the one of functions if these satisfy certain types of functional equations.Comment: 16 page

On simultaneous approximation of the values of certain Mahler functions

In this paper, we estimate the simultaneous approximation exponents of the values of certain Mahler functions. For this we construct Hermite-Pad\'{e} approximations of the functions under consideration, then apply the functional equations to get an infinite sequence of approximations and use the numerical approximations obtained from this sequence.Comment: 16 pages, added Corollary 1, corrected Appendix C, improved the proof of Theorem 3, added and updated reference