The Complexity and Expressive Power of Limit Datalog

Motivated by applications in declarative data analysis, in this article, we study Datalog Z —an extension of Datalog with stratified negation and arithmetic functions over integers. This language is known to be undecidable, so we present the fragment of limit Datalog Z programs, which is powerful enough to naturally capture many important data analysis tasks. In limit Datalog Z , all intensional predicates with a numeric argument are limit predicates that keep maximal or minimal bounds on numeric values. We show that reasoning in limit Datalog Z is decidable if a linearity condition restricting the use of multiplication is satisfied. In particular, limit-linear Datalog Z is complete for Δ 2 EXP and captures Δ 2 P over ordered datasets in the sense of descriptive complexity. We also provide a comprehensive study of several fragments of limit-linear Datalog Z . We show that semi-positive limit-linear programs (i.e., programs where negation is allowed only in front of extensional atoms) capture coNP over ordered datasets; furthermore, reasoning becomes coNEXP-complete in combined and coNP-complete in data complexity, where the lower bounds hold already for negation-free programs. In order to satisfy the requirements of data-intensive applications, we also propose an additional stability requirement, which causes the complexity of reasoning to drop to EXP in combined and to P in data complexity, thus obtaining the same bounds as for usual Datalog. Finally, we compare our formalisms with the languages underpinning existing Datalog-based approaches for data analysis and show that core fragments of these languages can be encoded as limit programs; this allows us to transfer decidability and complexity upper bounds from limit programs to other formalisms. Therefore, our article provides a unified logical framework for declarative data analysis which can be used as a basis for understanding the impact on expressive power and computational complexity of the key constructs available in existing languages.

On Robin’s criterion for the Riemann Hypothesis

Robin’s criterion says that the Riemann Hypothesis is equivalent to \[\forall n\geq 5041, \ \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n,\] where \sigma(n) is the sum of the divisors of n, \gamma represents the Euler–Mascheroni constant, and \log_i denotes the i-fold iterated logarithm. In this note we get the following better effective estimates: \begin{equation*} \forall n\geq3, \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n+\frac{0.3741}{\log_2^2n}. \end{equation*} The idea employed will lead us to a possible new reformulation of the Riemann Hypothesis in terms of arithmetic functions.

Fuzzy logical conclusions and conclusions in expert systems of medical diagnostics

The main problems in making a correct diagnosis are: subjectivity and insufficient qualifications of the doctor, difficulties in correctly assessing the patient’s complaints, signs and symptoms of the disease observed in the patient, as well as individual manifestations of the symptoms of the disease. In publications on the use of expert systems for medical diagnostics using fuzzy logic, the main attention was paid to the medical features of the problem. In this work, for the first time, general methodological aspects of building such systems, creating databases, representing by fuzzy sets of real numbers, digital scales, linguistic and Boolean data of symptom values are formulated. The types of membership functions that are advisable to use to represent the symptoms of diseases are proposed. In fuzzy-logical conclusions, not only the values of the characteristic functions of the logical terms of individual symptoms, but also complex arithmetic functions of their values are used.

A short remark on an arithmetic function

An explicit form of A. Shannon’s arithmetic function δ is given. A possible application of it is discussed for representation of the well-known arithmetic functions ω and Kronecker’s delta-function δ_{m,s}.

On a basic mean value theorem with explicit exponents

We follow a paper by Sedunova regarding Vaughan’s basic mean value Theorem to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by Cojocaru and Murty. As an application we derive a basic mean value theorem for the von Mangoldt generalized functions.

A generalized Davenport expansion

We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.