## Object-Oriented Hybrid Systems of Coalgebras plus Monoid Actions (1996)

Venue: | Algebraic Methodology and Software Technology (AMAST |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Jacobs96object-orientedhybrid,

author = {Bart Jacobs},

title = {Object-Oriented Hybrid Systems of Coalgebras plus Monoid Actions},

journal = {Algebraic Methodology and Software Technology (AMAST},

year = {1996},

volume = {239},

pages = {520--535}

}

### OpenURL

### Abstract

. Hybrid systems combine discrete and continuous dynamics. We introduce a semantics for such systems consisting of a coalgebra together with a monoid action. The coalgebra captures the (discrete) operations on a state space that can be used by a client (like in the semantics of ordinary (non-temporal) object-oriented systems). The monoid action captures the influence of time on the state space, where the monoids that we consider are the natural numbers monoid (N; 0; +) of discrete time, and the positive reals monoid (R0 ; 0; +) of real time. Based on this semantics we develop a hybrid specification formalism with timed method applications: it involves expressions like s:meth@ff, with the following meaning: in state s let the state evolve for ff units of time (according to the monoid action), and then apply the (coalgebraic) method meth. In this formalism we specify various (elementary) hybrid systems, investigate their correctness, and display their behaviour in simulations. We furthe...

### Citations

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Citation Context ...0 ) = 0. The three (interpretations of the) methods can be combined into a single function U ! f0; 1g \Theta U \Theta U , giving us a coalgebra on U of the functor X 7! f0; 1g \Theta X \Theta X , see =-=[21, 34, 35] for more -=-(categorical) background information on coalgebras. 2 One could also write "equations" with bisimilarity $ instead of equality = between terms inhabiting states, see e.g. [21, 20], but we sh... |

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- 1996
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Citation Context ...f a class is represented by several "hidden" sorts which are only indirectly accessible via the "visible" sorts. The (product of the) hidden sorts corresponds to our (coalgebraic) =-=state space X , see [24]-=- for a comparison. Here we extend the coalgebraic semantics and specification with temporal aspects. ffl At the semantic level we continue to use coalgebras to accommodate for methods (objectoriented ... |

10 |
System theory: a unified state-space approach to continuous and discrete systems
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Citation Context ...imilarity, refinement, inheritance) extend to hybrid systems, via similar definitions and results. It thus establishes a conceptual unity, at a mathematical level. As such it follows the tradition of =-=[4, 22, 32, 11]. A -=-next step is to transfer these notions and result to the programming level. This involves including monoid actions in actual classes (in an object-oriented hybrid language). Such an action ��, wri... |

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Citation Context ...eritance : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41 6 Non-homogeneous specifications and models 45 [This is the full version of the AMAST 1996 conference paper =-=[18]-=-. The latter only contains the essentials from the present sections 3 and 4. The work presented here was carried out while the author was employed at CWI in Amsterdam.] 1 Introduction Hybrid systems c... |

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Citation Context ...et ffi j s:val@(ff + fi) = 1g is an upwardly open interval [0; fl) ` R0 . We conclude this subsection on hybrid timers with a more complicated example, which is well-known in the literature, see e.g. =-=[23, 39]). It invo-=-lves a "watch-dog" surveying a number of processes and expecting signals from these processes that everything is all-right at regular intervals, see Figure 3. For convenience, our specificat... |

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Citation Context ... 8 The statements that we use in this paper only involve actual equations between observable output values. One may also wish to use "equations" between states, in the form of bisimilarity $=-= (like in [21, 20]-=-). This gives additional statements of the form s: \Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma! proc(b i )@fi i $ s: \Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma! proc(d j )@ffi j . The appro... |

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Citation Context ...imilarity, refinement, inheritance) extend to hybrid systems, via similar definitions and results. It thus establishes a conceptual unity, at a mathematical level. As such it follows the tradition of =-=[4, 22, 32, 11]. A -=-next step is to transfer these notions and result to the programming level. This involves including monoid actions in actual classes (in an object-oriented hybrid language). Such an action ��, wri... |

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Citation Context ... of such processes via discrete actions like opening or closing of valves, and (instantaneous) addition of extra substances, may be described via procedures acting on the state space (see for example =-=[31]-=-). In this subsection we shall consider two such simple situations, involving chemical substances in a reaction container. The second example specification in Figure 5 will be illustrated with a simul... |