Completeness and Complexity of Reasoning about Call-by-Value in Hoare Logic

We provide a sound and relatively complete Hoare logic for reasoning about partial correctness of recursive procedures in presence of local variables and the call-by-value parameter mechanism and in which the correctness proofs support contracts and are linear in the length of the program. We argue that in spite of the fact that Hoare logics for recursive procedures were intensively studied, no such logic has been proposed in the literature.

Utilization of the fractal dimension metric in training of dense Neural Networks

The process of training of Artificial Neural Networks essentially is optimization of the values of the weights $ w_{pq} $ associated with the arcs, connecting the nodes of the layers. This is a process of minimization of the Loss function (maximization of Accuracy function). During the training, the training data set recursively is utilized at subsequent stages, called \textit{Epochs}. The training continues until a satisfactory values of the Loss, Accuracy etc. parameters are reached. The matrices $ W^{UV} $ comprising the weights of the arcs connecting the layers $ U $ and $ V $, can be regarded as gray-scale images of a surface. Starting as random matrices, processed by recursive procedures, they gradually become fractal structures, characterized with respective fractal dimension $ D_f $. In the presented article we have made an attempt to utilize the correspondence of $ D_f $ with the Loss/Accuracy values, in order to forecast the optimal ending point of the NN training process. Similar conclusions were made for the correspondence between the number of layer’s nodes and $ D_f $. An attempt to apply statistically more rigorous approach in the determination of the slope of the regression line in Richardson-Mandelbrot plot, was made.

Symbolic execution formally explained

In this paper, we provide a formal explanation of symbolic execution in terms of a symbolic transition system and prove its correctness and completeness with respect to an operational semantics which models the execution on concrete values.We first introduce a formalmodel for a basic programming languagewith a statically fixed number of programming variables. This model is extended to a programming language with recursive procedures which are called by a call-by-value parameter mechanism. Finally, we present a more general formal framework for proving the soundness and completeness of the symbolic execution of a basic object-oriented language which features dynamically allocated variables.

APPLYING STATE SPACE MODELS TO STOCHASTIC CLAIMS RESERVING

The paper solves the loss reserving problem using Kalman recursions in linear statespace models. In particular, if one orders claims data from run-off triangles to time series with missing observations, then state space formulation can be applied for projections or interpolations of IBNR (Incurred But Not Reported) reserves. Namely, outputs of the corresponding Kalman recursion algorithms for missing or future observations can be taken as the IBNR projections. In particular, by means of such recursive procedures one can perform effectively simulations in order to estimate numerically the distribution of IBNR claims which may be very useful in terms of setting and/or monitoring of prudency level of loss reserves. Moreover, one can generalize this approach to the multivariate case of several dependent run-off triangles for correlated business lines and the outliers in claims data can be also treated effectively in this way. Results of a numerical study for several sets of claims data (univariate and multivariate ones) are presented.

Simulation of Elastic Tree-Like Dynamic Systems in Presence of External Holonomic Constraints

Modern scientific literature pays close attention to the problems of optimal modeling of elastic dynamic systems. The symbolic-recursive model of Newton --- Euler method with the provision of computational algorithms with a degree of complexity proportional to the dimension of these systems, i.e., O(n), has been adapted for dynamic systems with open kinematic chains, in particular, for elastic manipulators. If dynamic systems have closed kinematic chains, it is extremely difficult to propagate the strategy of numerical analysis without the mass matrix inversion. Consequently, the task of optimal modeling of tree-like dynamic systems is reduced to the search for combined strategies that use the procedures of strategies with and without inversion of mass matrices simultaneously. The paper introduces a method of numerical dynamic analysis of elastic tree-like multilink systems, the method combining the procedure of inverting the mass matrix with the procedure of effective kinematic calculation, borrowed from the generalized Newton --- Euler method. An approximate dynamic analysis technique is proposed that fully reproduces the recursive procedures of the generalized Newton --- Euler method. The technique is confirmed to the extent that the period of inversion of these systems is less than the time of their full functioning, and the range of displacements during one cycle is less than the complete revolution of the mechanism. The use of the approximate method for the dynamic analysis of elastic mechanisms is considered using the example of a numerical dynamic calculation of a slider-crank mechanism with an elastic connecting rod

GENERATING COMBINATORIAL OBJECTS USING RECURSIVE PROCEDURES AND FUNCTIONS

The article describes the methodology to solve problems of Olympiads in informatics generating combinatorial objects using recursion. Distance learning system DL.GSU.BY is the effective technical base for teaching.

Reachability Analysis of Asynchronous Dynamic Pushdown Networks Based on Tree Semantics Approach

<em>ADPN (Asynchronous Dynamic Pushdown Networks) are an abstract model for concurrent programs with recursive procedures and dynamic thread creation. Usually, asynchronous dynamic pushdown networks are described with interleaving semantics, in which the backward analysis is not effective. In order to improve interleaving semantics, tree semantics approach was introduced. This paper extends the tree semantics to ADPN. Because the reachability problem of ADPN is also undecidable, we address the context-bounded reachability problem and provide an algorithm for backward reachability analysis with tree-based semantics Approach.</em>

Systematic design of robust controller for electro-hydraulic system with model uncertainties

A systematic method is developed to design new robust tracking control for an electro-hydraulic system. In this article, the model dynamics is represented in strict feedback form to be linearly parameterized with regard to model parameters to consider its uncertainties and the separate error dynamics are derived for the tracking of states. The proposed method is intuitively similar to a previously developed integrator backstepping approach since it designs fictitious controls as desired states for recursive procedures. However, unlike previous works that have relied exclusively on Lyapunov analysis using the sum of the squares of state errors, this work utilizes a passivity approach with an energy function for stability. The advantage of the proposed method is that the fictitious control can be designed in a decoupled form and an L2-gain analysis can be applied to guarantee the robustness to model uncertainties. For the separate error dynamics, one part of the fictitious control is designed using stability analysis with a strict passivity formulation for error convergence; this leads to a separation of the other fictitious control design for robustness from the effects of other equations of dynamics. Hence, this method makes it possible to treat the terms caused by the model uncertainties as disturbance and to design the fictitious robust control using the L2-gain analysis for the disturbance attenuation, which does not require the bound function of the disturbance to be known. The validity of the proposed method is shown through simulations.