Feedback is an essential composition operator in many classes of reactive and other systems. This paper studies feedback in the context of compositional theories with refinement. Such theories allow to reason about systems on a component-by-component basis, and to characterize substitutability as a refinement relation. Although compositional theories of feedback do exist, they are limited either to deterministic systems (functions) or input-receptive systems (total relations). In this work we propose a compositional theory of feedback which applies to non-deterministic and non-input-receptive systems (e.g., partial relations). To achieve this, we use the semantic frameworks of predicate and property transformers, and relations with fail and unknown values. We show how to define instantaneous feedback for stateless systems and feedback with unit delay for stateful systems. Both operations preserve the refinement relation, and both can be applied to non-deterministic and non-input-receptive systems.
Our idea to use relations with an unknown value is inspired from fix-point based methods such as , which are used to reason about instantaneous feedback in cyclic combinational circuits and synchronous languages @cite_8 @cite_11 @cite_2 . These methods deal with monotonic functions over the flat CPO with unknown as the least element. Because they assume total functions, they are limited to deterministic and input-receptive systems. Refinement is also absent from these frameworks.
@cite_11 @cite_2 @cite_8
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Received 28 May 2002; revised 18 September 2002; accepted 24 September 2002; published 14 February 2003. [1] When an inverse problem is solved to estimate an unknown function such as the hydraulic conductivity in an aquifer or the contamination history at a site, one constraint is that the unknown function is known to be everywhere nonnegative. In this work, we develop a statistically rigorous method for enforcing function nonnegativity in Bayesian inverse problems. The proposed method behaves similarly to a Gaussian process with a linear variogram (i.e., unrestricted Brownian motion) for parameter values significantly greater than zero. The method uses the method of images to define a prior probability density function based on reflected Brownian motion that implicitly enforces nonnegativity. This work focuses on problems where the unknown is a function of a single variable (e.g., time). A Markov chain Monte Carlo (MCMC) method, specifically, a highly efficient Gibbs sampler, is implemented to generate conditional realizations of the unknown function. The new method is applied to the estimation of the trichloroethylene (TCE) and perchloroethylene (PCE) contamination history in an aquifer at Dover Air Force Base, Delaware, based on concentration profiles obtained from an underlying aquitard. INDEX TERMS: 1831 Hydrology: Groundwater quality; 1869 Hydrology: Stochastic processes; 3260 Mathematical Geophysics: Inverse theory; KEYWORDS: stochastic inverse modeling, contaminant source identification, inference under constraints, Markov chain Monte Carlo (MCMC), Gibbs sampling, Bayesian inference In this paper we present an algebraic construction of the category of monotonic predicate transformers from the category of relations which is similar to the standard algebraic construction of the integers from the natural numbers. The same construction yields the category of relations from the category of total functions. This provides a mechanism through which the rich type structure of the category of total functions can be promoted to successively weaker ones in the categories of relations and predicate transformers. In addition, it has exposed two complete rules for the refinement and composition of specifications in Morgan's refinement calculus. In sparse approximation theory, the fundamental problem is to reconstruct a signal A∈ℝn from linear measurements 〈Aψi〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [9] where the reconstruction can be done with very few—O(k logn)—linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [9, 4, 23] prove that there exists a single O(k logn) ×n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, logn, 1 e, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + e and improves the reconstruction time from poly(n) to poly(k logn) Our second result is a randomized construction of O(kpolylog (n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [20, 21], Streaming algorithms [11, 12, 6] and Complexity Theory [1] for this case Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement Abstract Incomplete relations are relations which contain null values, whose meaning is “value is at present unknown”. Such relations give rise to two types of functional dependency (FD). The first type, called the strong FD (SFD), is satisfied in an incomplete relation if for all possible worlds of this relation the FD is satisfied in the standard way. The second type, called the weak FD (WFD), is satisfied in an incomplete relation if there exists a possible world of this relation in which the FD is satisfied in the standard way. We exhibit a sound and complete axiom system for both strong and weak FDs, which takes into account the interaction between SFDs and WFDs. An interesting feature of the combined axiom system is that it is not k -ary for any natural number k ⩾ 0. We show that the combined implication problem for SFDs and WFDs can be solved in time polynomial in the size of the input set of FDs. Finally, we show that Armstrong relations exist for SFDs and WFDs.