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Papers by Alexander Yakhnis

We consider the problem of constructing a controller for a hybrid system which will solve the via... more We consider the problem of constructing a controller for a hybrid system which will solve the viability problem that all points of plant trajectories stay inside a given viability set. Here, a controller is a network of three successive devices, a digital to analog converter, a digital program (a computer together with its control software), and an analog to digital converter. We model a controller as an input-output automaton, which we call a control automaton. We give a necessary and sufficient condition that must be satisfied in order that a finite state control automaton solves a viability problem. Our results apply to plants modelled by vector differential equations with control and disturbance parameters, or to plants modelled by differential inclusions with a control parameter. We represent imprecise sensing of plant state. The main restrictions on the range of applicability of our results are that the set of admissible control laws is finite, and that we can neglect delays when control laws are reset. The latter restriction seems to be inessential.

Annals of Pure and Applied Logic, 1993

S., A. Yakhnis and V. Yakhnis, Some effectively infinite classes of enumerations, Annals of Pure ... more S., A. Yakhnis and V. Yakhnis, Some effectively infinite classes of enumerations, Annals of Pure and Applied Logic 60 (1993) 207-235.

The authors propose a game framework for analyzing, extracting, and verifying digital control pro... more The authors propose a game framework for analyzing, extracting, and verifying digital control programs for continuous plants by regarding such programs as finite-state winning strategies in associated games. They call such interacting systems of digital control programs and continuous plants `hybrid systems' and model them as networks of interacting concurrent digital programs or automata, following the approach of Nerode and Remmel (1990). This extends to hybrid systems the paradigm introduced by A. Nerode et al. (1992) for analyzing concurrent digital programs meeting program specifications as winning finite-state strategies in associated two-person games. This formulation is intended to facilitate the transfer of recent tools from logic and concurrency and dynamical systems to extraction and verification of digital control programs for continuous systems

Theoretical Computer Science, 1995

Hybrid systems are interacting systems of digital automata and continuous plants subject to distu... more Hybrid systems are interacting systems of digital automata and continuous plants subject to disturbances. The digital automata are used to force the state trajectory of the continuous plant to obey a performance specification. For the basic concepts and notation for hybrid systems, see , and other papers in the same volume. Here we introduce tools for analyzing enforcing viability of all possible plant state trajectories of a hybrid system by suitable choices of finite state control automata. Thus, the performance specification considered here is that the state of the plant remain in a prescribed viability set of states at all times (Aubin, 1991). The tools introduced are local viability graphs and viability graphs for hybrid systems. We construct control automata which guarantee viability as the fixpoints of certain operators on graphs. When control and state spaces are compact, the viability set is closed, and a non-empty closed subset of a viability graph is given with a sturdiness property, one can extract finite state automata guaranteeing viable trajectories. This paper is a sequel to , especially Appendix II.

We consider the problem of constructing a “controller” for a hybrid system which will solve the v... more We consider the problem of constructing a “controller” for a hybrid system which will solve the viability problem that all points of plant trajectories stay inside a given “viability set”. Here, a “controller” is a network of three successive devices, a digital to analog converter, a digital program (a computer together with its control software), and an analog to digital converter. We model a controller as an input-output automaton, which we call a “control automaton”. We give a necessary and sufficient condition that must be satisfied in order that a finite state control automaton solves a viability problem. Our results apply to plants modelled by vector differential equations with control and disturbance parameters, or to plants modelled by differential inclusions with a control parameter. We represent imprecise sensing of plant state. The main restrictions on the range of applicability of our results are that the set of admissible control laws is finite, and that we can neglect delays when control laws are reset. The latter restriction seems to be inessential.

Annals of Pure and Applied Logic, 1993

Annals of Pure and Applied Logic, 1996

We begin by introducing games and strategies on trees (Section 2), define computational games (Se... more We begin by introducing games and strategies on trees (Section 2), define computational games (Section 3), give the connection with automata (Section 4), introduce computational state models (Section 5), introduce strategy models (Section 6), look at operations on ...

Mathematical Logic Quarterly, 1998

We define a new type of two player game occurring on a tree. The tree may have no root and may ha... more We define a new type of two player game occurring on a tree. The tree may have no root and may have arbitrary degrees of nodes. These games extend the class of games considered by Gurevich-Harrington in [5]. We prove that in the game one of the players has a winning strategy which depends on finite bounded information about the past part of a play and on future of each play that is isomorphism types of tree nodes. This result extends further the Gurevich-Harrington determinacy theorem from [5].

As our society becomes technologically more complex, computers are being used in greater and grea... more As our society becomes technologically more complex, computers are being used in greater and greater numbers of high consequence systems. Giving a machine control over the lives of humans can be disturbing, especially if the software that is run on such a machine has bugs. Formal reasoning is one of the most powerful techniques available to demonstrate the correctness of a piece of software. When reasoning about software and its development, one frequently encounters expressions that contain partial functions. As might be expected, the presence of partial functions introduces an additional dimension of difficulty to the reasoning framework. This difficulty produces an especially strong impact in the case of high consequence systems. An ability to use formal methods for constructing software is essential if we want to obtain greater confidence in such systems through formal reasoning. This is only reasonable under automation of software development and verification. However, the ubiquitous presence of partial functions prevents a uniform application to software of any tools not specifically accounting for partial functions. In this paper we will describe a framework for reasoning about software, based on the nonstrict explicit domain approach, that is applicable to a large class of software/hardware systems. In this framework the Hoare triples containing partial functions can be reasoned about automatically in a well-defined and uniform manner.

Annals of Pure and Applied Logic, 1990

We consider the problem of constructing a controller for a hybrid system which will solve the via... more We consider the problem of constructing a controller for a hybrid system which will solve the viability problem that all points of plant trajectories stay inside a given viability set. Here, a controller is a network of three successive devices, a digital to analog converter, a digital program (a computer together with its control software), and an analog to digital converter. We model a controller as an input-output automaton, which we call a control automaton. We give a necessary and sufficient condition that must be satisfied in order that a finite state control automaton solves a viability problem. Our results apply to plants modelled by vector differential equations with control and disturbance parameters, or to plants modelled by differential inclusions with a control parameter. We represent imprecise sensing of plant state. The main restrictions on the range of applicability of our results are that the set of admissible control laws is finite, and that we can neglect delays when control laws are reset. The latter restriction seems to be inessential.

Annals of Pure and Applied Logic, 1993

S., A. Yakhnis and V. Yakhnis, Some effectively infinite classes of enumerations, Annals of Pure ... more S., A. Yakhnis and V. Yakhnis, Some effectively infinite classes of enumerations, Annals of Pure and Applied Logic 60 (1993) 207-235.

The authors propose a game framework for analyzing, extracting, and verifying digital control pro... more The authors propose a game framework for analyzing, extracting, and verifying digital control programs for continuous plants by regarding such programs as finite-state winning strategies in associated games. They call such interacting systems of digital control programs and continuous plants `hybrid systems' and model them as networks of interacting concurrent digital programs or automata, following the approach of Nerode and Remmel (1990). This extends to hybrid systems the paradigm introduced by A. Nerode et al. (1992) for analyzing concurrent digital programs meeting program specifications as winning finite-state strategies in associated two-person games. This formulation is intended to facilitate the transfer of recent tools from logic and concurrency and dynamical systems to extraction and verification of digital control programs for continuous systems

Theoretical Computer Science, 1995

Hybrid systems are interacting systems of digital automata and continuous plants subject to distu... more Hybrid systems are interacting systems of digital automata and continuous plants subject to disturbances. The digital automata are used to force the state trajectory of the continuous plant to obey a performance specification. For the basic concepts and notation for hybrid systems, see , and other papers in the same volume. Here we introduce tools for analyzing enforcing viability of all possible plant state trajectories of a hybrid system by suitable choices of finite state control automata. Thus, the performance specification considered here is that the state of the plant remain in a prescribed viability set of states at all times (Aubin, 1991). The tools introduced are local viability graphs and viability graphs for hybrid systems. We construct control automata which guarantee viability as the fixpoints of certain operators on graphs. When control and state spaces are compact, the viability set is closed, and a non-empty closed subset of a viability graph is given with a sturdiness property, one can extract finite state automata guaranteeing viable trajectories. This paper is a sequel to , especially Appendix II.

We consider the problem of constructing a “controller” for a hybrid system which will solve the v... more We consider the problem of constructing a “controller” for a hybrid system which will solve the viability problem that all points of plant trajectories stay inside a given “viability set”. Here, a “controller” is a network of three successive devices, a digital to analog converter, a digital program (a computer together with its control software), and an analog to digital converter. We model a controller as an input-output automaton, which we call a “control automaton”. We give a necessary and sufficient condition that must be satisfied in order that a finite state control automaton solves a viability problem. Our results apply to plants modelled by vector differential equations with control and disturbance parameters, or to plants modelled by differential inclusions with a control parameter. We represent imprecise sensing of plant state. The main restrictions on the range of applicability of our results are that the set of admissible control laws is finite, and that we can neglect delays when control laws are reset. The latter restriction seems to be inessential.

Annals of Pure and Applied Logic, 1993

Annals of Pure and Applied Logic, 1996

We begin by introducing games and strategies on trees (Section 2), define computational games (Se... more We begin by introducing games and strategies on trees (Section 2), define computational games (Section 3), give the connection with automata (Section 4), introduce computational state models (Section 5), introduce strategy models (Section 6), look at operations on ...

Mathematical Logic Quarterly, 1998

We define a new type of two player game occurring on a tree. The tree may have no root and may ha... more We define a new type of two player game occurring on a tree. The tree may have no root and may have arbitrary degrees of nodes. These games extend the class of games considered by Gurevich-Harrington in [5]. We prove that in the game one of the players has a winning strategy which depends on finite bounded information about the past part of a play and on future of each play that is isomorphism types of tree nodes. This result extends further the Gurevich-Harrington determinacy theorem from [5].

As our society becomes technologically more complex, computers are being used in greater and grea... more As our society becomes technologically more complex, computers are being used in greater and greater numbers of high consequence systems. Giving a machine control over the lives of humans can be disturbing, especially if the software that is run on such a machine has bugs. Formal reasoning is one of the most powerful techniques available to demonstrate the correctness of a piece of software. When reasoning about software and its development, one frequently encounters expressions that contain partial functions. As might be expected, the presence of partial functions introduces an additional dimension of difficulty to the reasoning framework. This difficulty produces an especially strong impact in the case of high consequence systems. An ability to use formal methods for constructing software is essential if we want to obtain greater confidence in such systems through formal reasoning. This is only reasonable under automation of software development and verification. However, the ubiquitous presence of partial functions prevents a uniform application to software of any tools not specifically accounting for partial functions. In this paper we will describe a framework for reasoning about software, based on the nonstrict explicit domain approach, that is applicable to a large class of software/hardware systems. In this framework the Hoare triples containing partial functions can be reasoned about automatically in a well-defined and uniform manner.

Annals of Pure and Applied Logic, 1990