Abstract We propose the concept of a system algebra with a parallel composition operation and an interface connection operation, and formalize composition-order invariance, which postulates that the order of composing and connecting systems is irrelevant, a generalized form of associativity. Composition-order invariance explicitly captures a common property that is implicit in any context where one can draw a figure (hiding the drawing order) of several connected systems, which appears in many scientific contexts. This abstract algebra captures settings where one is interested in the behavior of a composed system in an environment and wants to abstract away anything internal not relevant for the behavior. This may include physical systems, electronic circuits, or interacting distributed systems. One specific such setting, of special interest in computer science, are functional system algebras, which capture, in the most general sense, any type of system that takes inputs and produces outputs depending on the inputs, and where the output of a system can be the input to another system. The behavior of such a system is uniquely determined by the function mapping inputs to outputs. We consider several instantiations of this very general concept. In particular, we show that Kahn networks form a functional system algebra and prove their composition-order invariance. Moreover, we define a functional system algebra of causal systems, characterized by the property that inputs can only influence future outputs, where an abstract partial order relation captures the notion of “later”. This system algebra is also shown to be composition-order invariant and appropriate instantiations thereof allow to model and analyze systems that depend on time.
Hardy has developed an abstract theory, in which composition-order invariance (there called order independence) plays an important role @cite_9 . That work, however, focuses on physical systems. Lee and Sangiovanni-Vincentelli @cite_23 also introduce an abstract system model, but it is specific to systems that consider some form of time and does not follow an algebraic approach.
@cite_9 @cite_23
1689232663 2004142419 1497573972 2039005177
We develop a theory for describing composite objects in physics. These can be static objects, such as tables, or things that happen in spacetime (such as a region of spacetime with fields on it regarded as being composed of smaller such regions joined together). We propose certain fundamental axioms which, it seems, should be satisfied in any theory of composition. A key axiom is the order independence axiom which says we can describe the composition of a composite object in any order. Then we provide a notation for describing composite objects that naturally leads to these axioms being satisfied. In any given physical context we are interested in the value of certain properties for the objects (such as whether the object is possible, what probability it has, how wide it is, and so on). We associate a generalized state with an object. This can be used to calculate the value of those properties we are interested in for for this object. We then propose a certain principle, the composition principle, which says that we can determine the generalized state of a composite object from the generalized states for the components by means of a calculation having the same structure as the description of the generalized state. The composition principle provides a link between description and prediction. Abstract Two conditions on a collection of simple orders - unimodality and straightness - are necessary but not jointly sufficient for unidimensional unfolding representations. From the analysis of these conditions, a polynomial time algorithm is derived for the testing of unidimensionality and for the construction of a representation when one exists. In this paper the abstraction-refinement paradigm based on 3-valued logics is extended to the setting of probabilistic systems. We define a notion of abstraction for Markov chains. To be able to relate the behavior of abstract and concrete systems, we equip the notion of abstraction with the concept of simulation. Furthermore, we present model checking for abstract probabilistic systems (abstract Markov chains) with respect to specifications in probabilistic temporal logics, interpreted over a 3-valued domain. More specifically, we introduce a 3-valued version of probabilistic computation-tree logic (PCTL) and give a model checking algorithm w.r.t. abstract Markov chains. Abstract To support our reasoning abilities perception must recover environmental regularities—e.g., rigidity, “objectness,” axes of symmetry—for later use by cognition. To create a theory of how our perceptual apparatus can produce meaningful cognitive primitives from an array of image intensities we require a representation whose elements may be lawfully related to important physical regularities, and that correctly describes the perceptual organization people impose on the stimulus. Unfortunately, the representations that are currently available were originally developed for other purposes (e.g., physics, engineering) and have so far proven unsuitable for the problems of perception or common-sense reasoning. In answer to this problem we present a representation that has proven competent to accurately describe an extensive variety of natural forms (e.g., people, mountains, clouds, trees), as well as man-made forms, in a succinct and natural manner. The approach taken in this representational system is to describe scene structure at a scale that is similar to our naive perceptual notion of “a part,” by use of descriptions that reflect a possible formative history of the object, e.g., how the object might have been constructed from lumps of clay. For this representation to be useful it must be possible to recover such descriptions from image data; we show that the primitive elements of such descriptions may be recovered in an overconstrained and therefore reliable manner. We believe that this descriptive system makes an important contribution towards solving current problems in perceiving and reasoning about natural forms by allowing us to construct accurate descriptions that are extremely compact and that capture people's intuitive notions about the part structure of three-dimensional forms.