This paper describes the first fully automated conditioned slicing system, C##SIT, detailing the theory that underlies it, its architecture and the way it combines symbolic execution, theorem proving and slicing technologies. The use of C##SIT is illustrated with respect to the applications of testing and comprehension. ### #####: Conditioned Slicing, program conditioning, slicing, symbolic execution, path analysis

. Characteristic properties of majorant-computable real-valued functions are studied. A formal theory of computability over the reals which satisfies the requirements of numerical analysis used in Computer Science is constructed on the base of the definition of majorant-computability proposed in [13]. A model-theoretical characterization of majorantcomputability real-valued functions and their domains is investigated. A theorem which connects the graph of a majorant-computable function with validity of a finite formula on the set of hereditarily finite sets on IR, HF( IR) (where IR is a proper elementary enlargement of the standard reals) is proven. A comparative analysis of the definition of majorantcomputability and the notions of computability earlier proposed by Blum et al., Edalat, Sunderhauf, Pour-El and Richards, Stoltenberg-Hansen and Tucker is given. Examples of majorant-computable real-valued functions are presented. 1 Introduction In the recent time, attention to the prob...

Program slicing is a technique for program simplification based upon the deletion of statements which cannot affect the values of a chosen set of variables. Because slicing extracts a subcomponent of the program concerned with some specific computation on a set of variables, it can be used to assist program comprehension, allowing a programmer to remodularise a program according to arbitrarily selected slicing criteria. In this paper it is shown that the simplification power of slicing can be improved if the syntactic restriction to statement deletion is removed, allowing slices to be constructed using any simplifying transformation which preserves the effect of the original program upon the set of variables of interest. It is also shown that quasi static slicing, first proposed by Venkatesh (and defined here in a slightly more general form), is the most suitable slicing paradigm for program comprehension. The various forms of slice are formally defined, an algorithm, based upon transf...

Based on an eective theory of continuous domains, notions of computability for operators and real-valued functionals dened on the class of continuous functions are introduced. Denability and semantic characterisation of computable functionals are given. Also we propose a recursion scheme which is a suitable tool for formalisation of complex systems, such as hybrid systems. In this framework the trajectories of continuous parts of hybrid systems can be represented by computable functionals. 1

We develop a unified framework for dealing with constructibility and absoluteness in set theory, decidability of relations in effective structures (like the natural numbers), and domain independence of queries in database theory. Our framework and results suggest that domain-independence and absoluteness might be the key notions in a general theory of constructibility, predicativity, and computability. 1

We propose semantic characterisations of second-order computability over the reals based on -denability theory. Notions of computability for operators and real-valued functionals dened on the class of continuous functions are introduced via domain theory. We consider the reals with and without equality and prove theorems which connect computable operators and real-valued functionals with validity of nite -formulas. 1

We investigate the concept of generalised computability of operators and functionals dened on the set of continuous functions, rstly introduced in [9]. By working in the reals, with equality and without equality, we study properties of generalised computable operators and functionals. Also we propose an interesting application to formalisation of hybrid systems. We obtain some class of hybrid systems, which trajectories are computable in the sense of computable analysis.

This work is the next step in a series of papers [3, 4] using arguments from definability theory to logically characterise computable continuous data. In order to do this we have proposed the notion of majorant-computability and developed logical approach to computability over continuous data. This approach is based on representations of continuous data by suitable structures without the equality test and Σ-definability in extensions of the structures by hereditarily finite sets. One of the main features of the notion of majorant-computability is that on the one side it is independent from concrete representations of the elements of structures on the other side it is flexible, i.e. we can change the language of Σ-formulas to express appropriate computability properties. In this talk we introduce and study the language of ΣK-formulas which is an extension of the language of Σ-formulas. This language simplifies reasoning about computability of higher type continuous data, and admits elimination of universal quantifiers bounded by computable compact sets. In order to show these properties we prove Uniformity principle for Σ-definability over the real

We introduce majorant computability of functions on reals. A structural theorem is proved, which connects the graph of a majorant--computable function with validity of a finite formula on the set of the hereditarily finite set HF( ¯ R) (where ¯ R is an elementary proper extension of standard real numbers). The class of majorant--computable functions in our approach include an interesting class of real total functions having meromorphic extension on C. This class in particular contains functions which are solutions of known differential equations. The notion of a majorant--computable functional on the set of total majorant--computable real functions is defined. As an example of a majorant--computable functional the Riemann integral is proposed. 1 Section 1. Introduction Section 1 Introduction The semiring of natural numbers is the classical structure for which the concept of computability has been defined and studied rather well. Though several definitions were independently proposed,...

The concept of majorant-computability over the reals which integrates methods and ideas of continuous mathematics and the modern mathematical logic is introduced. Properties of majorant-computable real-valued functions are studied. Definability of majorant-computable real-valued functions and majorant-enumerable subsets of the reals is investigated in this framework. A theorem which connects the graph of a majorant-computable function with validity of a finite formula on the set of hereditarily finite sets on R, HF( R) (where R is an elementary proper extension of the standard reals) is proved. The Mandelbrot set and Julia sets are given as examples od sets that are majorantenumerable. Keywords: majorant-computabile function, majorant-enumerable set, computational process, approximation, definability. 1 Introduction We introduce the concept of majorant-computability over the reals which integrates methods and ideas of continuous mathematics and the modern mathematical logic, in pa...