On Representations of Abstract Systems with Partial Inputs and Outputs

Electron. Proc. Theor. Comput. Sci.

2017

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We consider the problem of proving that each point in a given set of states ("target set") can indeed be reached by a given nondeterministic continuous-time dynamical system from some initial state. We consider this problem for abstract continuous-time models that can be concretized as various kinds of continuous and hybrid dynamical systems.The approach to this problem proposed in this paper is based on finding a suitable superset S of the target set which has the property that each partial trajectory of the system which lies entirely in S either is defined as the initial time moment, or can be locally extended backward in time, or can be locally modified in such a way that the resulting trajectory can be locally extended back in time.This reformulation of the problem has a relatively simple logical expression and is convenient for applying various local existence theorems and local dynamics analysis methods to proving reachability which makes it suitable for reasoning about the behavior of continuous and hybrid dynamical systems in proof assistants such as Mizar, Isabelle, etc.

Section: Introductionmentioning

confidence: 99%

On the Underapproximation of Reach Sets of Abstract Continuous-Time Systems

Electron. Proc. Theor. Comput. Sci.

2017

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“…This formalization was proposed in [28][29][30][31][32] and inspired by the notion of a solution system from O. Hájek's Theory of processes [33,34]. In this formalization the global non-negative real time scale is assumed and continuous-time systems are modeled as sets of trajectories considered as functions on real time intervals which take values in an arbitrary fixed set of states.…”

Section: Introductionmentioning

confidence: 99%

“…In this formalization the global non-negative real time scale is assumed and continuous-time systems are modeled as sets of trajectories considered as functions on real time intervals which take values in an arbitrary fixed set of states. These sets must satisfy certain weak assumptions (more details are given in Section 2) [29]:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Local Characterization of Global Timed Bisimulation for Abstract Continuous-Time Systems

Coalgebraic Methods in Computer Science

2016

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We consider two notions of timed bisimulation on states of continuous-time dynamical systems: global and local timed bisimulation. By analogy with the notion of a bisimulation relation on states of a labeled transition system which requires the existence of matching transitions starting from states in such a relation, local timed bisimulation requires the existence of sufficiently short (locally defined) matching trajectories. Global timed bisimulation requires the existence of arbitrarily long matching trajectories. For continuous-time systems the notion of a global bisimulation is stronger than the notion of a local bisimulation and its definition has a non-local character. In this paper we give a local characterization of global timed bisimulation. More specifically, we consider a large class of abstract dynamical systems called Nondeterministic Complete Markovian Systems (NCMS) which covers various concrete continuous and discrete-continuous (hybrid) dynamical models and introduce the notion of an f +-timed bisimulation, where f + is a so called extensibility measure. This notion has a local character. We prove that it is equivalent to global timed bisimulation on states of a NCMS. In this way we give a local characterization of the notion of a global timed bisimulation.

“…Besides modeling data processed by programs, nominative data can be also applied to modeling data processed by signal processing systems in the context of the mathematical systems theory [4,6,7,5,3].…”

mentioning

confidence: 99%

Simple-Named Complex-Valued Nominative Data – Definition and Basic Operations

Nikitchenko

Kryvolap

et al. 2017

Formalized Mathematics

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Summary In this paper we give a formal definition of the notion of nominative data with simple names and complex values [15, 16, 19] and formal definitions of the basic operations on such data, including naming, denaming and overlapping, following the work [19]. The notion of nominative data plays an important role in the composition-nominative approach to program formalization [15, 16] which is a development of composition programming [18]. Both approaches are compared in [14]. The composition-nominative approach considers mathematical models of computer software and data on various levels of abstraction and generality and provides mathematical tools for reasoning about their properties. In particular, nominative data are mathematical models of data which are stored and processed in computer systems. The composition-nominative approach considers different types [14, 19] of nominative data, but all of them are based on the name-value relation. One powerful type of nominative data, which is suitable for representing many kinds of data commonly used in programming like lists, multidimensional arrays, trees, tables, etc. is the type of nominative data with simple (abstract) names and complex (structured) values. The set of nominative data of given type together with a number of basic operations on them like naming, denaming and overlapping [19] form an algebra which is called data algebra. In the composition-nominative approach computer programs which process data are modeled as partial functions which map nominative data from the carrier of a given data algebra (input data) to nominative data (output data). Such functions are also called binominative functions. Programs which evaluate conditions are modeled as partial predicates on nominative data (nominative predicates). Programming language constructs like sequential execution, branching, cycle, etc. which construct programs from the existing programs are modeled as operations which take binominative functions and predicates and produce binominative functions. Such operations are called compositions. A set of binominative functions and a set of predicates together with appropriate compositions form an algebra which is called program algebra. This algebra serves as a semantic model of a programming language. For functions over nominative data a special computability called abstract computability is introduces and complete classes of computable functions are specified [16]. For reasoning about properties of programs modeled as binominative functions a Floyd-Hoare style logic [1, 2] is introduced and applied [12, 13, 8, 11, 9, 10]. One advantage of this approach to reasoning about programs is that it naturally handles programs which process complex data structures (which can be quite straightforwardly represented as nominative data). Also, unlike classical Floyd-Hoare logic, the mentioned logic allows reasoning about assertions which include partial pre- and post-conditions [11]. Besides modeling data processed by programs, nominative data can be also applied to modeling data processed by signal processing systems in the context of the mathematical systems theory [4, 6, 7, 5, 3].