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footing. Rather, one observes pockets of dense connections (direct interactions) among small subsets of components, but only loose connections between these densely connected subsets. There is thus a notion of "causal proximity" in physical systems under which variables naturally precipitate into small clusters. Identifying these clusters and using them to represent the joint provides the basis for great efficiency of storage (relative to the full joint distribution) as well as for potent inference algorithms.
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This process of decomposition may be undertaken to gain insight into the identity of the constituent components, which may reflect individual physical processes of interest. Also, functional decomposition may result in a compressed representation of the global function, a task which is feasible only
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Functional
Decomposition is a design method intending to produce a non-implementation, architectural description of a computer program. The software architect first establishes a series of functions and types that accomplishes the main processing problem of the computer program, decomposes each to
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on system variables should provide evidence of this hierarchical structure. The task of an observer who seeks to understand the system is then to infer the hierarchical structure from observations of these variables. This is the notion behind the hierarchical decomposition of a joint distribution,
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methods attempt to decompose a joint distribution along its causal fault lines, thus "cutting nature at its seams". The essential motivation behind these methods is again that within most systems (natural or artificial), relatively few components/events interact with one another directly on equal
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In practical scientific applications, it is almost never possible to achieve perfect functional decomposition because of the incredible complexity of the systems under study. This complexity is manifested in the presence of "noise," which is just a designation for all the unwanted and untraceable
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Decomposition of a function into non-interacting components generally permits more economical representations of the function. Intuitively, this reduction in representation size is achieved simply because each variable depends only on a subset of the other variables. Thus, variable
1618:. When a system is designed as pure functions, they can be reused, or replaced. A usual side effect is that the interfaces between blocks become simple and generic. Since the interfaces usually become simple, it is easier to replace a pure function with a related, similar function.
510:, presidential motorcade, etc.) all these other secondary variables are not directly relevant to the West Side Highway traffic. All we need (hypothetically) in order to predict the West Side Highway traffic is the weather and the GW Bridge traffic, because these two variables
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refers to the process of defining a system in functional terms, then defining lower-level functions and sequencing relationships from these higher level systems functions. The basic idea is to try to divide a system in such a way that each block of a
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However, while perfect functional decomposition is usually impossible, the spirit lives on in a large number of statistical methods that are equipped to deal with noisy systems. When a natural or artificial system is intrinsically hierarchical, the
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traffic" with values {"10mph", "5mph", "1mph"}. The point here is that while there are certainly many secondary variables that affect the weather variable (e.g., low pressure system over Canada,
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In other words, the system can be seen as acting separately on each of the components of the input signal. Commonly used examples of this type of decomposition are the
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methods can be thought of as implementing a function decomposition process in the presence of noise; that is, where functional dependencies are only expected to hold
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are constants. This decomposition aids in analysis, because now the output of the system can be expressed in terms of the components of the input. If we let
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relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts.
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from the rest of the world. Practical examples of this phenomenon surround us. Consider the particular case of "northbound traffic on the
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1902:
Tonge, Fred M. (1969), "Hierarchical aspects of computer languages", in Whyte, Lancelot Law; Wilson, Albert G.; Wilson, Donna (eds.),
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Simon, Herbert A. (1963), "Causal
Ordering and Identifiability", in Ando, Albert; Fisher, Franklin M.; Simon, Herbert A. (eds.),
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469:) takes on three possible values of {"moving slow", "moving deadly slow", "not moving at all"}. Now, let's say the variable
1568:{\displaystyle =a_{1}\cdot T\{g_{1}(t)\}+a_{2}\cdot T\{g_{2}(t)\}+a_{3}\cdot T\{g_{3}(t)\}+\dots +a_{n}\cdot T\{g_{n}(t)\}}
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the attempt to recover something of the intrinsic hierarchical structure which generated that joint distribution.
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between the components are critical to the function of the collection. All interactions may not be
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that states a
Knowledge editor's personal feelings or presents an original argument about a topic.
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West Side
Highway traffic from all other potential influences. That is, all other influences act
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Koestler, Athur (1973), "The tree and the candle", in Gray, William; Rizzo, Nicholas D. (eds.),
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Simon, Herbert A. (1973), "The organization of complex systems", in Pattee, Howard H. (ed.),
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1756:(Report). Fort Belvoir, VA: Defense Acquisition University Press. January 2001. p. 45.
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A review of other applications and function decomposition. Also presents methods based on
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can be decomposed into a linear combination of other functions, called component signals:
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depends on two other variables, "weather" with values of {"sun", "rain", "snow"}, and "
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Processes related to functional decomposition are prevalent throughout the fields of
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reveal common functions and types, and finally derives
Modules from this activity.
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Simon, Herbert A. (1996), "The architecture of complexity: Hierarchic systems",
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Causal influences on West Side
Highway traffic. Weather and GW Bridge traffic
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and the recently popular methods referred to as "causal decompositions" or
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Proceedings of the
Fourteenth International Conference on Machine Learning
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Zupan, Blaž; Bohanec, Marko; Bratko, Ivan; Demšar, Janez (July 1997).
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in Japan, etc.) and the Bridge traffic variable (e.g., an accident on
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678:. The input signal to an LTI system can be expressed as a function,
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Unity
Through Diversity: A Festschrift for Ludwig von Bertalanffy
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and a front panel. Later, when a different model needs an audio
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Practical applications of functional decomposition are found in
300:, synthesis, validation and verification of composite behavior.
1700:. San Francisco: Morgan Kaufmann Publishers. pp. 421–429.
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can be described without an "and" or "or" in the description.
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represent the effect of the system, then the output signal is
1028:{\displaystyle \{g_{1}(t),g_{2}(t),g_{3}(t),\dots ,g_{n}(t)\}}
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This exercise forces each part of the system to have a pure
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Expression of a function as the composition of two functions
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personal reflection, personal essay, or argumentative essay
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when the constituent processes possess a certain level of
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Functional decomposition is used in the analysis of many
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55:Learn how and when to remove these messages
1621:For example, say that one needs to make a
294:, but possibly deduced through repetitive
1868:, Massachusetts: MIT Press, pp. 5–31
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379:of variables. We would say that variable
282:(i.e., independence or non-interaction).
250:Learn how and when to remove this message
232:Learn how and when to remove this message
170:Learn how and when to remove this message
108:Learn how and when to remove this message
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195:This article includes a list of general
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1910:: American Elsevier, pp. 233–251
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1882:: George Braziller, pp. 3–27
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1751:Systems Engineering Fundamentals
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1808:Leyton, Michael (1992),
1788:The Ghost in the Machine
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1676:Knowledge representation
589:. Among such models are
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546:Knowledge representation
268:functional decomposition
1904:Hierarchical Structures
1851:The Illusion of Reality
572:hierarchical clustering
491:{\displaystyle {x_{1}}}
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1777:The Modularity of Mind
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1773:Fodor, Jerry
1745:
1738:Simon (1963)
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1718:graph theory
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407:screens off
264:engineering
214:introducing
1921:Categories
1765:References
512:screen off
377:entire set
314:screen off
297:perception
291:observable
280:modularity
272:functional
197:references
144:improve it
41:improve it
1894:Cambridge
1866:Cambridge
1841:170454227
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160:May 2023
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