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.
Several works have defined system models. Lee and Sangiovanni-Vincentelli @cite_23 define , which intuitively requires that each output must be provoked by an input that occurred at least a @math -difference earlier. They show that fixed points exist, based on Banach's theorem. @cite_1 generalize this to a notion of superdense'' time where multiple events may occur simultaneously. @cite_5 , in the quantum scenario, describe a type of strict causality based on a that can be seen as a generalization of delta causality. Naundorf @cite_10 considers causality without any minimal time distance, and proves that fixed points still exist. Matsikoudis and Lee @cite_2 then show a fixed point theorem for the same notion, which they refer to as . They show that it is implied by a more natural notion of (strict) causality where outputs can be influenced only by inputs that occur strictly earlier, under the assumption that the ordering of inputs is well-founded A partial order on a set @math is if every nonempty subset of @math has one or more minimal elements. . We show in app:equivalence-causal that the strict causality notion of @cite_2 is essentially equivalent to the definition we introduce in this work.
@cite_1 @cite_23 @cite_2 @cite_5 @cite_10
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Deterministic timed systems can be modeled as fixed point problems (B. Roscoe and G. Reed, 1988), (R. K. Yates, 1993), (E. A. Lee, 1999). In particular, any connected network of timed systems can be modeled as a single system with feedback, and the system behavior is the fixed point of the corresponding system equation, when it exists. For delta-causal systems, we can use the Cantor metric to measure the distance between signals and the Banach fixed-point theorem to prove the existence and uniqueness of a system behavior. Moreover, the Banach fixed-point theorem is constructive: it provides a method to construct the unique fixed point through iteration. In this paper, we extend this result to systems modeled with the superdense model of time (O. , 1992), (Z. Manna and A. Pnueli, 1993) used in hybrid systems. We call the systems we consider eventually delta-causal, a strict generalization of delta-causal in which multiple events may be generated on a signal in zero time. With this model of time, we can use a generalized ultrametric (Generalized ultrametric spaces, I, 1996) instead of a metric to model the distance between signals. The existence and uniqueness of behaviors for such systems comes from the fixed-point theorem of (S. Priess-Crampe and P. Ribenboim, 1993), but this theorem gives no constructive method to compute the fixed point This leads us to define petrics, a generalization of metrics, which we use to generalize the Banach fixed-point theorem to provide a constructive fixed-point theorem. This new fixed-point theorem allows us to construct the unique behavior of eventually delta-causal systems. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. In this paper we recast the classical Darondeau---Degano's causal semantics of concurrency in a coalgebraic setting, where we derive a compact model. Our construction is inspired by the one of Montanari and Pistore yielding causal automata, but we show that it is instance of an existing categorical framework for modeling the semantics of nominal calculi, whose relevance is further demonstrated. The key idea is to represent events as names, and the occurrence of a new event as name generation. We model causal semantics as a coalgebra over a presheaf, along the lines of the Fiore---Turi approach to the semantics of nominal calculi. More specifically, we take a suitable category of finite posets, representing causal relations over events, and we equip it with an endofunctor that allocates new events and relates them to their causes. Presheaves over this category express the relationship between processes and causal relations among the processes' events. We use the allocation operator to define a category of well-behaved coalgebras: it models the occurrence of a new event along each transition. Then we turn the causal transition relation into a coalgebra in this category, where labels only exhibit maximal events with respect to the source states' poset, and we show that its bisimilarity is essentially Darondeau---Degano's strong causal bisimilarity. This coalgebra is still infinite-state, but we exploit the equivalence between coalgebras over a class of presheaves and History Dependent automata to derive a compact representation, where states only retain the poset of the most recent events for each atomic subprocess, and are isomorphic up to order-preserving permutations. Remarkably, this reduction of states is automatically performed along the equivalence. We address the problem of visually detecting causal events and fitting them together into a coherent story of the action witnessed by the camera. We show that this can be done by reasoning about the motions and collisions of surfaces, using high-level causal constraints derived from psychological studies of infant visual behavior. These constraints are naive forms of basic physical laws governing substantiality, contiguity, momentum, and acceleration. We describe two implementations. One system parses instructional videos, extracting plans of action and key frames suitable for storyboarding. Since learning will play a role in making such systems robust, we introduce a new framework for higher-order hidden Markov models and demonstrate its use in a second system that segments stereo video into actions in near real-time. Rather than attempt accurate low-level vision, both systems use high-level causal analysis to integrate fast but sloppy pixel-based representations over time. The output is suitable for summary, indexing, and automated editing.