Goguen, J.A. Mathematical representation of hierarchically organised systems. In E. Attinger (ed.), Global Systems Dynamics, S. Karger, 1971, 112-128.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....8, 10] and many others and is based on a dynamical systems theory vocabulary. ZS posits algebraic concepts between specification levels, behavior, states, and components. ZS relies on compositionally closed systems of components. The concepts form a natural tie in with categorytheoretic concepts [1, 3, 4, 5, 6, 11]. The history of science and engineering is replete with stories of great advances occurring after notions are examined by new methods for This work was partially supported by the Shodor Computational Science Fund funded by the Shodor Education Foundation, Inc. Durham, NC and NSF Grant ....

....have a complete mechanism for describing systems and then reasoning about them. The ZS approach strongly suggests category theory, with its emphasis on morphisms, as a natural formalization setting. Category theory also played a role in development of General Systems Theory, primarily by Goguen [3, 4, 5, 6]. The usefulness of the categorical approach is that it links ZS and system theory on the one hand to 2 abstract algebra, logic, and theoretical computer science on the other. 3.1. Basic Categories. A category consists of objects obC and arrows arC. There is no fixed notion of what the ....

J. A. Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, Basel, 1971.

....to make this technology have broader appeal to industry. We begin with some formal preliminaries and a brief discussion of Specware TM. 2. Formal Preliminaries The field of category theory [4,10,12] provides a foundational theory. This theory was applied to systems theory and systems engineering [3,6]. This theory was embodied in the software development tool Specware TM [14,16,17] 2.1. Category of Signatures A signature consists of the following: 1. A set S of sort symbols 2. A triple O = C, F, P of operators, where: C is a set of sorted constant symbols, F is a set of sorted ....

Goguen, J. A., Mathematical Representation of Hierarchically Organized Systems, in Global Systems Dynamics, ed. E. Attinger and S. Karger, 1970, pp. 112-128.

....laws of system modularisation and composition has been recognised since the early 70s when J. Goguen proposed the use of categorical techniques in General Systems Theory for unifying a variety of notions of system behaviour, including that of physical components, and their composition techniques [Goguen 71, 73, Goguen and Ginali 78] Similar principles have been used to formalise process models for concurrent systems [Sassone et al. 93] such as transition systems, synchronisation trees, event structures, etc. Based on similar categorical models, modularisation principles like those typical of ....

....11 4 Parallel composition One of the advantages of working in the proposed categorical framework is that mechanisms for building complex systems out of components can be formalised through universal constructs. A general principle is given by J. Goguen in his work on General Systems Theory [Goguen 71, 73, Goguen and Ginali 78] given a category of widgets, the operation of putting a system of widgets together to form a super widget corresponds to taking a colimit of the diagram of widgets that shows how to interconnect them . In this section, we investigate the applicability of these ....

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J.Goguen, "Mathematical Representation of Hierarchically Organised Systems", in E.Attinger (ed) Global Systems Dynamics, Krager 1971, 112-128.

....to the assumptions that are made on the environment in rely guarantee styles of specification. 1 Introduction In the early 70 s, J. Goguen proposed the use of categorical techniques in General Systems Theory for unifying a variety of notions of system behaviour and their composition techniques [Goguen 71, 73, Goguen and Ginali 78] His approach has been summarised in a very simple but far reaching principle: given a category of widgets, the operation of putting a system of widgets together to form a super widget corresponds to taking a colimit of the diagram of widgets that shows how to ....

J.Goguen, "Mathematical Representation of Hierarchically Organised Systems", in E.Attinger (ed) Global Systems Dynamics, Krager 1971, 112128.

....[ST88a] and in connection with the foundations of formal program development [BV85, ST88b] since then. The ideas in Sections 3 and 8 concerning building theories and logics in a structured fashion have their roots in CLEAR and are related to Goguen s earlier work on general systems theory [Gog71] ST88a] considers a language of structured specifications which is similar to but richer than the language of structured presentations introduced in Section 3. As discussed in [ST92] there is an essential difference between the view of structured presentations purely as theory presentations, ....

J.A. Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128, S. Karger, 1971.

....which models systems of concurrent, interacting objects by diagrams which assign an operational semantics to each object in a system. The behaviour of the whole system is given by a limit construction. In modelling behaviour by limits we follow earlier work by Goguen on Categorical Systems Theory [4, 5, 6]. This approach pays particular attention to the hierarchical structure of systems, and provides means of constructing systems from component parts in a way that captures both complex objects and parallel composition with synchronisation [16] The operational semantics of objects can be very ....

....g impose on the system. As this example illustrates, we can take the limit of a diagram of transition systems (i.e. a collection of transition systems connected by morphisms) to be the behaviour of the system. In doing so, we follow the Behaviour as Limit slogan of categorical systems theory [4, 5, 6]. An important property of the category of transition systems is that a limit behaviour always exists, even if, as in the example above, that behaviour is empty. Proposition 2 The category Tr is complete. A rather more intersesting model is given by labelled transition systems , in which a set of ....

Joseph Goguen. Mathematical representation of hierarchically organised systems. In E. O. Attinger, editor, Global Systems Dynamics, pages 111--129. S. Karger, 1970.

.... are under investigation in the form of bisimulations ( 3] Other semantic approaches to FOOPS in the presence of method combiners are based on recent work by Goguen ( 17] on sheaf semantics of concurrent systems, in turn inspired by early work by the same author on General Systems Theory ([9, 10, 23, 11]) 17] proposes a categorical approach to concurrent systems based on sheaf theory. The approach is model theoretic, in the sense that it provides complete sets of possible behaviours for a system, and constraint based, as it focuses on the relations holding between system components 10 so that ....

J. Goguen. Mathematical Representation of Hierarchically Organized Systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

....use limits of monoids to obtain commutativity properties for a sheaf theoretic model of concurrency. Our approach to interconnections of objects is similar in many respects to Monteiro and Pereira s sheaf theoretic approach, and also relies on work by Goguen on categorical systems theory (see [9, 11, 13]; applications of this work to object orientation can be found in Goguen [16] Ehrich et al. . 8] Wolfram and Goguen [31] and Cirstea [4] We consider objects to have a local, hidden state, aspects of which may be observed by means of attributes, and which may be updated by methods. Thus, an ....

....of the shared clock. The point illustrated by this example is that systems can be thought of as diagrams, and that the behaviour of a system is its limit. In developing this for interconnections of systems of objects we are following ideas and results from categorical systems theory (cf. Goguen [9, 16]) It is possible to contrive some rather curious examples of the behaviour of composite processes. Let A = fag B = fbg C = fa; bg D = fa; bg be four processes, arranged as in Figure 1 where the morphisms between the ffifl fflfi C j j j3 Q Q Qs ffifl fflfi A ffifl fflfi B ....

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Joseph Goguen. Mathematical representation of hierarchically organised systems. In E. O. Attinger, editor, Global Systems Dynamics, pages 111-- 129. S. Karger, 1970.

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Goguen, J.A. Mathematical representation of hierarchically organised systems. In E. Attinger (ed.), Global Systems Dynamics, S. Karger, 1971, 112-128.

....CafeOBJ [5] Maude [3] BOBJ [14] and the European languages casl [4] and acttwo. Semantics follows that of Clear [1] using a category of theories, with theory morphisms for views, and colimit for module composition, based on ideas from an early category theoretic general systems theory [8]. This semantics works over any logical system, using the formalism of institutions [13] Other languages in uenced by parameterized programming include Ada, ML, C , and Modula, none of which has views, so that actual parameter syntax must contain formal parameter syntax; only ML has higher ....

Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112-128. S. Karger, 1971.

....module) as well as its awkward treatment of sharing. The Clear module system semantics [1] uses the category of theories, where views are the natural notion of morphism for theories, and module composition is given by colimit, inspired by an earlier category theoretic approach to general systems [7]; this applies directly to OBJ3. Our semantics of higher order parameterized programming and review of other approaches is given in [15] ....

Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

....T v Figure 2: Application of a Parameterised Theory 2.4 Theory Instantiation The specification language Clear provides a semantics for applying (also called instantiating ) parameterised theories. This semantics uses colimits, and was inspsired by some earlier ideas from General Systems Theory [11]: Given a parameterised theory Phi : T T 0 and a view v : T A of an actual theory A as an instance of T , then the result of applying Phi to v is the pushout shown in Figure 2, whose pushout object is denoted T 0 [v] and whose morphism opposite to Phi may be denoted Phi 0 . The ....

Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

....was inspired by the Clear specification language [7, 8] In fact, OBJ can be regarded as an implementation of Clear. In particular, the notion of view was developed in collaboration with Rod Burstall for use in Clear. Clear s approach was in turn inspired by some ideas in general system theory [27]. A key idea is the use of colimits of diagrams of theories to determine the result of module expression evaluation. Although colimits are beyond the scope of this paper, they give a precise foundation for parameterized programming, and moreover, a foundation that is independent of the particular ....

Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

.... [50] for a general discussion of the relation between initial algebra semantics and compositionality in linguistics) It is also interesting to note that this approach is consistent with the General Systems Theory doctrine that the limit of a diagram solves an arbitrary system of constraints [12, 13, 16]. Also note that ambiguity of meaning can arise naturally in this setting, exactly because unification is not necessarily unitary. 7.9 Differential Equations Standard techniques of functional analysis, such as differential operators on function spaces (e.g. see [53] seem ideally suited for ....

Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

....and that we are not at all dependent upon interleaving. In particular, Lilius [34] has shown how to model Petri nets in our sheaf theoretic framework. This paper builds on a much earlier paper [15] which used sheaf theory as part of a research programme on Categorical General Systems Theory [13, 14, 18]. My interest in this area was revived by the desire to give a semantics for foops (a Functional Object Oriented Programming System) 22, 25] and for the Rewrite Rule Machine, a multi grain hierarchical massively parallel graph rewriting machine (see [19] and [17] The main points made in this ....

....systems, through diagrams and their limits, while the second concerns glueing together behaviour over domains. 3. 3 Interconnection The principles that objects are sheaves, systems are diagrams, and behaviour is limit are all taken from some earlier work in categorical General System Theory [13, 14, 18]. Another principle from this work is that interconnecting systems corresponds to taking colimits in the category of systems, where sharing is indicated by inclusion maps from shared parts into the systems that share them. The papers [13, 14, 18] develop some very general results in this setting, ....

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Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

....some physical (or conceptual) system, then the limit provides an object which (together with its projection morphisms) represents all possible behaviours of the system that are consistent with the given constraints. This intuition goes back to some work on General System Theory from 1969 74, [16, 27], and has many applications in computing science: 4.1 Products. An early achievement of category theory was to give a precise definition for the notion of product, which was previously known in many special cases, but only understood vaguely as a general concept. The definition is due to Mac ....

....say widgets, then the result of interconnecting a system of widgets to form a super widget corresponds to taking the colimit of the diagram of widgets in which the morphisms show how they are interconnected. At least for me, this intuition arose in the context of General Systems Theory [16, 27]. It may be interesting to note that the duality between the categorical definitions of limits and colimits suggests a similar duality between the intuitive notions of solution and interconnection. Now some examples: 6.1 Putting Theories together to make Specifications. Complexity is a fundamental ....

Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

.... for module expression evaluation is the use of colimits for putting together arbitrary structures, such as code, specification, and documentation; this idea has its origins in some work from the early 1970s on the interconnection of general systems using the category theoretic notion of colimit [9, 10]. 2.3 OBJ OBJ arose around 1976 as an executable formal notation for abstract data types with subsorts [11] Initial algebra semantics is executed by interpreting equations as rewrite rules [29, 12] In 1979, Joseph Tardo completed an inplementation, called OBJT, which included both subsorts and ....

Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

....in such a way as to include hierarchical whole part relationships. Systems were taken to be diagrams in a category, behaviors were given by their limits, and interconnections were given by colimits of diagrams; some very general laws about interconnection and behavior hold in this setting [35, 37, 78]. The most complete exposition is in [62] which has full proofs of all results. 3 The ADJ group, fGoguen, Thatcher, Wagner, Wrightg, was formed during my tenure as Research Fellow in the Mathematical Sciences at IBM Research, Yorktown Heights, initially to study the relationship between ....

Joseph Goguen. Mathematical representation of hierarchically organized systems. In E. Attinger, editor, Global Systems Dynamics, pages 112--128. S. Karger, 1971.

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Goguen, J. A., Mathematical Representation of Hierarchically Organized Systems, in Global Systems Dynamics, ed. E. Attinger and S. Karger, 1970, pp. 112-128.

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