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13
Interconnection of Object Specifications
 Formal Methods and Object Technology
, 1996
"... ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and ap ..."
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Cited by 8 (2 self)
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ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and applying no methods corresponds to the identity of the monoid. A monoid is a set M with an associative binary operation ffl M : M \ThetaM ! M , usually referred to as `multiplication', which has an identity element e M 2 M . If M = (M; ffl M ; e M ) is a monoid, we often write just M for M, and e for e M ; moreover for m;m 0 2 M , we usually write mm 0 instead of m ffl M m 0 . For example, A , the set of lists containing elements of A, together with concatenation ++ : A \ThetaA ! A and the empty list [ ] 2 A , is a monoid. This example is especially important for the material in later sections. A monoid homomorphism is a structure preserving map between the carriers of ...
Compositional abstractions of hybrid control systems
 In Proceedings of the 40th IEEE Conference on Decision and Control
, 2001
"... Abstract. Abstraction is a natural way to hierarchically decompose the analysis and design of hybrid systems. Given a hybrid control system and some desired properties, one extracts an abstracted system while preserving the properties of interest. Abstractions of purely discrete systems is a mature ..."
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Abstract. Abstraction is a natural way to hierarchically decompose the analysis and design of hybrid systems. Given a hybrid control system and some desired properties, one extracts an abstracted system while preserving the properties of interest. Abstractions of purely discrete systems is a mature area, whereas abstractions of continuous systems is a recent activity. In this paper we present a framework for abstraction that applies to discrete, continuous, and hybrid systems. We introduce a composition operator that allows to build complex hybrid systems from simpler ones and show compatibility between abstractions and this compositional operator. Besides unifying the existing methodologies we also propose constructions to obtain abstractions of hybrid control systems.
Quotients of fully nonlinear control systems
 SIAM Journal on Control and Optimization
, 2001
"... Abstract. In this paper, we introduce and study quotients of fully nonlinear control systems. Our definition is inspired by categorical definitions of quotients as well as recent work on abstractions of affine control systems. We show that quotients exist under mild regularity assumptions and charac ..."
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Cited by 7 (5 self)
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Abstract. In this paper, we introduce and study quotients of fully nonlinear control systems. Our definition is inspired by categorical definitions of quotients as well as recent work on abstractions of affine control systems. We show that quotients exist under mild regularity assumptions and characterize the structure of the quotient state/input space. This allows one to understand how states and inputs of the quotient system are related to states and inputs of the original system. We also introduce a notion of projectability which turns out to be equivalent to controlled invariance. This allows one to regard previous work on symmetries, partial symmetries, and controlled invariance as leading to special types of quotients. We also show the existence of quotients that are not induced by symmetries or controlled invariance. Such decompositions have a potential use in a theory of hierarchical control based on quotients.
D.: Representing systems with hidden state
 In: The TwentyFirst National Conference on Artificial Intelligence (AAAI
, 2006
"... We discuss the problem of finding a good state representation in stochastic systems with observations. We develop a duality theory that generalizes existing work in predictive state representations as well as automata theory. We discuss how this theoretical framework can be used to build learning al ..."
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Cited by 5 (3 self)
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We discuss the problem of finding a good state representation in stochastic systems with observations. We develop a duality theory that generalizes existing work in predictive state representations as well as automata theory. We discuss how this theoretical framework can be used to build learning algorithms, approximate planning algorithms as well as to deal with continuous observations.
INTERCONNECTION OF PROBABILISTIC SYSTEMS
, 2000
"... There is a growing interest in models for probabilistic systems. This fact is motivated by engineering applications, namely in problems concerning the evaluation of the performance of systems. It is of ..."
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Cited by 3 (3 self)
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There is a growing interest in models for probabilistic systems. This fact is motivated by engineering applications, namely in problems concerning the evaluation of the performance of systems. It is of
Symbolic models for control systems
, 2007
"... In this paper we provide a bridge between the infinite state models used in control theory to describe the evolution of continuous physical processes and the finite state models used in computer science to describe software. We identify classes of control systems for which it is possible to constru ..."
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Cited by 3 (2 self)
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In this paper we provide a bridge between the infinite state models used in control theory to describe the evolution of continuous physical processes and the finite state models used in computer science to describe software. We identify classes of control systems for which it is possible to construct equivalent (bisimilar) finite state models. These constructions are based on finite, but otherwise arbitrary, partitions of the set of inputs or outputs of a control system.
A AlgebraCoalgebra Duality in Brzozowski’s Minimization Algorithm
"... We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simpl ..."
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We give a new presentation of Brzozowski’s algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. Notably, we derive algorithms to obtain minimal, language equivalent automata from Moore, nondeterministic and weighted automata.
Paracategories II: Adjunctions, fibrations and examples from probabilistic automata theory
, 2002
"... In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We setup the relevant notion of adjunction betwee ..."
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In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We setup the relevant notion of adjunction between paracategories and apply it to define (co)completeness and cartesian closure, exemplified by the paracategory of bivariant functors and dinatural transformations. We introduce partial multicategories to account for partial tensor products. We also consider fibrations for paracategories and their indexedparacategory version. Finally, we instantiate all these concepts in the context of probabilistic automata.
Higher Dimensional Trees, Algebraically
"... Abstract. In formal language theory, James Rogers published a series of innovative papers generalising strings and trees to higher dimensions.Motivated by applications in linguistics, his goal was to smoothly extend the core theory of the formal languages of strings and trees to these higher dimensi ..."
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Abstract. In formal language theory, James Rogers published a series of innovative papers generalising strings and trees to higher dimensions.Motivated by applications in linguistics, his goal was to smoothly extend the core theory of the formal languages of strings and trees to these higher dimensions. Rogers ’ definitions focussed on a specific representation of higher dimensional trees. This paper presents an alternative approach which focusses more on their universal properties and is based upon category theory, algebras, coalgebras and containers. Our approach reveals that Rogers ’ trees are canonical constructions which are also particularly beautiful. We also provide new theoretical results concerning higher dimensional trees. Finally, we provide evidence for our devout conviction that clean mathematical theories provide the basis for clean implementations by showing how our abstract presentation makes computing with higher dimensional trees easier. 1
The Duality of State and Observation in Probabilistic Transition Systems
"... Abstract. In this paper we consider the problem of representing and reasoning about systems, especially probabilistic systems, with hidden state. We consider transition systems where the state is not completely visible to an outside observer. Instead, there are observables that partly identify the s ..."
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Abstract. In this paper we consider the problem of representing and reasoning about systems, especially probabilistic systems, with hidden state. We consider transition systems where the state is not completely visible to an outside observer. Instead, there are observables that partly identify the state. We show that one can interchange the notions of state and observation and obtain what we call a dual system. In the case of deterministic systems, the double dual gives a minimal representation of the behaviour of the original system. We extend these ideas to probabilistic transition systems and to partially observable Markov decision processes (POMDPs). 1