Fixpoint Characterization Research Articles

Distribution-based limited fuzzy bisimulations for nondeterministic fuzzy transition systems

Bisimulation techniques have been adopted and developed in fuzzy transition systems (FTSs) to compare their behavior. Recently, distribution-based bisimulation has been introduced to nondeterministic fuzzy transition systems (NFTSs) to measure the similarity between two distributions which can be regarded as generalized states. However, the new bisimulation does not consider the similarity of states. In terms of limited approximate bisimilarity, in this article, a notion of distribution-based k-limited bisimulation is proposed for NFTSs, which can measure the similarity between two distributions in the neighboring subgraphs (subsystems) and take the similarity of states into account, where k is a natural number. This article discusses some properties of limited bisimulation and the nonexpansiveness of typical parallel composition operators with respect to the limited bisimulation which makes compositional verification possible. Then a fixed point characterization of the limited bisimulation is investigated and an algorithm is given to compute the degree of k-limited similarity between two distributions. Moreover, a fuzzy modal logic interpreted on distributions is adopted to characterize the limited bisimulation.

Jackknife empirical likelihood ratio test for logistic distribution using fixed point characterization

Tests for the goodness of fit of logistic distribution have received much attention in the literature as it is widely used in many studies related to bio-assay, population growth, etc. In this paper, we propose a method to test whether the data comes from a logistic distribution or not. We derive the asymptotic distribution of the test statistic. Also, we develop a test based on Jackknife Empirical Likelihood (JEL). The finite sample performance of the JEL ratio test is evaluated through Monte Carlo simulation. We illustrate the proposed method using a real data set.

A new goodness of fit test for gamma distribution with censored observations

In the present paper, we develop a new goodness of fit test for gamma distribution based on the fixed point characterization. We use U-Statistics theory to derive the test statistic. We discuss how the right censored observations are incorporated into the proposed test. The asymptotic properties of the test statistics in censored and uncensored cases are studied. We carry out a Monte Carlo simulation study to validate the finite sample performance of the proposed tests. We also illustrate the test procedures using real data sets.

Fuzzy Bisimulations for Nondeterministic Fuzzy Transition Systems

Bisimulations are established forms of behavioral equivalences for discrete event systems. Recently, bisimulations have been introduced to fuzzy transition systems (FTSs) and developed quickly. Relational lifting by using weight functions is one of the most used methods in bisimulation research, which lifts a crisp relation on the state spaces to a crisp relation on the distributions of state spaces of FTSs. This paper proposes a method, called fuzzy relational lifting method, via fuzzy similarity measures induced by residua (implications) in complete residuated lattices. In contrast with the existing method, fuzzy relational lifting method relaxes the conditions of lifting relation by using weight functions and is a generalization of the method. Some properties of fuzzy lifting operation are presented. Based on the fuzzy lifting operation, a fuzzy bisimulation for nondeterministic fuzzy transition systems (NFTSs) is introduced, which is much more natural and robust than some behavioral measures, and can measure the degree of similarity between any two states which are deemed to be equivalent intuitively but simply distinguished by other behavioral measures. Then some properties of fuzzy bisimulations are provided. Moreover, a fixedpoint characterization of fuzzy bisimulations is given and the greatest fuzzy bisimulation is computed. Finally, a real-valued logic is introduced to characterize fuzzy bisimulations over Godel algebra.

Approximate bisimulations for fuzzy-transition systems

Bisimulation techniques have been recently introduced to fuzzy transition systems to compare their behavior. When studying bisimulation, one of the most popular methods is relational lifting. In this paper, we first realize the relational lifting through a function S˜. The new relational lifting method relaxes the condition of lifting relation from using weight functions to function S˜ defined by residua (implications) in complete residuated lattices. We first discuss some properties of the lifting operation, and then define an α-bisimulation under S˜, where α∈[0,1], for nondeterministic fuzzy transition systems (NFTSs). The new bisimulation can handle the case that two states are equivalent intuitively but are simply treated as distinguishable by some behavioral measures, and so it is much more natural and robust. Then we provide some properties of α-bisimulation under S˜. The parallel composition operator of NFTSs is addressed and we show that α-bisimilarity under S˜ is nonexpansive with respect to the operator, which makes compositional verification possible. In addition, we give a fixed-point characterization of α-bisimulation under S˜ and compute α-bisimilarity under S˜. Finally, we introduce a real-valued logic to characterize α-bisimilarity under S˜.

The Temporal Logic of Coalitional Goal Assignments in Concurrent Multiplayer Games

We introduce and study a natural extension of the Alternating time temporal logic ATL , called Temporal Logic of Coalitional Goal Assignments (TLCGA). It features one new and quite expressive coalitional strategic operator, called the coalitional goal assignment operator ⦉ γ ⦊, where γ is a mapping assigning to each set of players in the game its coalitional goal , formalised by a path formula of the language of TLCGA, i.e., a formula prefixed with a temporal operator X , U , or G , representing a temporalised objective for the respective coalition, describing the property of the plays on which that objective is satisfied. Then, the formula ⦉ γ ⦊ intuitively says that there is a strategy profile Σ for the grand coalition Agt such that for each coalition C , the restriction Σ | C of Σ to C is a collective strategy of C that enforces the satisfaction of its objective γ (C) in all outcome plays enabled by Σ | C . We establish fixpoint characterizations of the temporal goal assignments in a μ-calculus extension of TLCGA, discuss its expressiveness and illustrate it with some examples, prove bisimulation invariance and Hennessy–Milner property for it with respect to a suitably defined notion of bisimulation, construct a sound and complete axiomatic system for TLCGA, and obtain its decidability via finite model property.

A Fixpoint Characterization of Three-Valued Disjunctive Hybrid MKNF Knowledge Bases

The logic of hybrid MKNF (minimal knowledge and negation as failure) is a powerful knowledge representation language that elegantly pairs ASP (answer set programming) with ontologies. Disjunctive rules are a desirable extension to normal rule-based reasoning and typically semantic frameworks designed for normal knowledge bases need substantial restructuring to support disjunctive rules. Alternatively, one may lift characterizations of normal rules to support disjunctive rules by inducing a collection of normal knowledge bases, each with the same body and a single atom in its head. In this work, we refer to a set of such normal knowledge bases as a head-cut of a disjunctive knowledge base. The question arises as to whether the semantics of disjunctive hybrid MKNF knowledge bases can be characterized using fixpoint constructions with head-cuts. Earlier, we have shown that head-cuts can be paired with fixpoint operators to capture the two-valued MKNF models of disjunctive hybrid MKNF knowledge bases. Three-valued semantics extends two-valued semantics with the ability to express partial information. In this work, we present a fixpoint construction that leverages head-cuts using an operator that iteratively captures three-valued models of hybrid MKNF knowledge bases with disjunctive rules. This characterization also captures partial stable models of disjunctive logic programs since a program can be expressed as a disjunctive hybrid MKNF knowledge base with an empty ontology. We elaborate on a relationship between this characterization and approximators in AFT (approximation fixpoint theory) for normal hybrid MKNF knowledge bases.

Data-Driven Optimal Control of Affine Systems: A Linear Programming Perspective

In this letter, we discuss the problem of optimal control for affine systems in the context of data-driven linear programming. First, we introduce a unified framework for the fixed point characterization of the value function, Q-function and relaxed Bellman operators. Then, in a model-free setting, we show how to synthesize and estimate Bellman inequalities from a small but sufficiently rich dataset. To guarantee exploration richness, we complete the extension of Willems’ fundamental lemma to affine systems.

Yablo’s Paradox, the Liar, and Referential Contradictions from a Graph Theory Point of View

F -systems are useful digraphs to model sentences that predicate the falsity of other sentences. Paradoxes like the Liar and the one of Yablo can be analyzed with that tool to find graph-theoretic patterns. In this paper we studied this general model consisting of a set of sentences and the binary relation ‘... affirms the falsity of...’ among them. The possible existence of non-referential sentences was also considered. To model the sets of all the sentences that can jointly be valued as true we introduced the notion of conglomerate, the existence of which guarantees the absence of paradox. Conglomerates also enabled us to characterize referential contradictions, i.e., sentences that can only be false under a classical valuation due to the interactions with other sentences in the model. A Kripke-style fixed-point characterization of groundedness was offered, and complete (meaning that every sentence is deemed either true or false) and consistent (meaning that no sentence is deemed true and false) fixed points were put in correspondence with conglomerates. Furthermore, argumentation frameworks are special cases of F -systems. We showed the relation between local conglomerates and admissible sets of arguments and argued about the usefulness of the concept for the argumentation theory.

Complete Abstractions for Checking Language Inclusion

We study the language inclusion problem L 1 ⊆ L 2 , where L 1 is regular or context-free. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of L 1 , obtained by approximating the Kleene iterates of its least fixpoint characterization, is included in L 2 . We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e., its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e., it guarantees termination of least fixpoint computations). This overapproximating abstraction of languages can be defined using quasiorder relations on words, where the abstraction gives the language of all the words “greater than or equal to” a given input word for that quasiorder. We put forward a range of such quasiorders that allow us to systematically design decision procedures for different language inclusion problems, such as regular languages into regular languages or into trace sets of one-counter nets, and context-free languages into regular languages. In the case of inclusion between regular languages, some of the induced inclusion checking procedures correspond to well-known state-of-the-art algorithms, like the so-called antichain algorithms. Finally, we provide an equivalent language inclusion checking algorithm based on a greatest fixpoint computation that relies on quotients of languages and, to the best of our knowledge, was not previously known.

Entropy-regularized 2-Wasserstein distance between Gaussian measures

Gaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. They furthermore stand as important special cases for frameworks providing geometries for probability measures, as the resulting geometry on Gaussians is often expressible in closed-form under the frameworks. In this work, we study the Gaussian geometry under the entropy-regularized 2-Wasserstein distance, by providing closed-form solutions for the distance and interpolations between elements. Furthermore, we provide a fixed-point characterization of a population barycenter when restricted to the manifold of Gaussians, which allows computations through the fixed-point iteration algorithm. As a consequence, the results yield closed-form expressions for the 2-Sinkhorn divergence. As the geometries change by varying the regularization magnitude, we study the limiting cases of vanishing and infinite magnitudes, reconfirming well-known results on the limits of the Sinkhorn divergence. Finally, we illustrate the resulting geometries with a numerical study.

Variational interior and interface transmission conditions: multidomain mixed Darcy/Stokes control problems

Multidomain mixed pairs of coupled stationary constrained boundary value problems, are formulated variationally in frameworks of reflexive Banach spaces as macro-hybrid systems, and analyzed. Interior synchronizing and interface coupling transmission constraints are modeled in terms of Lagrange multipliers as solutions of dual variational subpotential maximal monotone inclusions. Existence and uniqueness results are established via resolvent fixed-point characterizations of the corresponding primal and dual macro-hybridized problems. Multimedia mixed mechanical subsurface-surface Darcy/Stokes incompressible flow coupled pairs with intrinsic distributed control constraints exemplify the general macro-hybrid mixed systems. Coupling dual-primal and primal-dual interface macro-hybrid mixed variational conditions are lastly developed and applied to the Darcy/Stokes multimedia pair model.

A Fixed-Point Characterization of the Optimal Costate in Finite-Horizon Optimal Control Problems

A fixed-point characterization of the optimal costate in finite-horizon optimal control problems for nonlinear systems is presented. It is shown that the optimal initial condition of the costate variable must be a fixed-point, for any time, of the composition of the forward and backward flows of the underlying Hamiltonian dynamics. Such an abstract property is then translated into a constructive condition by relying on a sequence of repeated Lie brackets involving the Hamiltonian dynamics and evaluated at a single point in the state space. This leads to a system of algebraic equations in the unknown initial optimal costate that allows achieving a desired degree of accuracy of the approximation while always consisting of a number of equations equal to the dimension of the state of the underlying system, regardless of the achieved accuracy. A dual characterization of the optimal terminal value of the state is also discussed, together with a few computational aspects of the proposed strategy.

A fixed-point characterization of weakly compact sets in L1(μ) spaces

Let (Ω,Σ,μ) be a σ-finite measure space and consider the Lebesgue function space L1(μ) endowed with its standard norm. We obtain a characterization of weak compactness for closed bounded convex subsets of L1(μ) in terms of the existence of fixed points for certain classes of eventually affine, uniformly Lipschitzian mappings.

Robustly Complete Synthesis of Memoryless Controllers for Nonlinear Systems With Reach-and-Stay Specifications

This paper proposes a finitely terminating algorithm to solve reach-and-stay control problems for nonlinear systems. The algorithm is guaranteed to return a control strategy if the specification is robustly realizable. Such a feature is desirable as the commonly used abstraction-based methods are sound but not complete for systems that are not incrementally stable. Fundamental to the proposed method is a fixed-point characterization of the winning set of the system with respect to a given specification, i.e., the initial states that can be controlled to satisfy the specification. The use of an adaptive partitioning scheme not only guarantees the approximation precision of the winning set but also reduces computational time. The effectiveness and efficiency are illustrated by several benchmarking examples.

A Total Fractional-Order Variation Regularized Reconstruction Method for CT

The total variation (TV) regularized reconstruction methods for computed tomography (CT) may lead to staircase effects in the reconstructed images because of using the TV regularization. This paper develops a total fractional-order variation regularized CT reconstruction method, aiming at overcoming the weakness of the reconstruction methods based on the TV. Specifically, we propose an optimization model for CT reconstruction, including a fidelity term, a regularization term, and a constraint term. Here, the regularization is a total fractional-order variation arising from the fractional derivative of the underlying solution. To address the nondifferentiability of the resulting model, we introduce a fixed-point characterization for its solution through the proximity operators of the nondifferentiable functions. Based on the characterization, we further develop a fixed-point iterative scheme to solve the resulting model and provide convergence analysis of the developed scheme. Numerical experiments are presented to demonstrate that the developed method outperforms the TV regularized reconstruction method in terms of suppressing noise for CT reconstruction.

Fixed point characterizations of continuous univariate probability distributions and their applications

By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein’s method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavors, we focus on explicit representations given through a formula for the density- or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known. To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.

A Krasnoselskii-Mann Algorithm With an Improved EM Preconditioner for PET Image Reconstruction.

This paper presents a preconditioned Krasnoselskii-Mann (KM) algorithm with an improved EM preconditioner (IEM-PKMA) for higher-order total variation (HOTV) regularized positron emission tomography (PET) image reconstruction. The PET reconstruction problem can be formulated as a three-term convex optimization model consisting of the Kullback-Leibler (KL) fidelity term, a nonsmooth penalty term, and a nonnegative constraint term which is also nonsmooth. We develop an efficient KM algorithm for solving this optimization problem based on a fixed-point characterization of its solution, with a preconditioner and a momentum technique for accelerating convergence. By combining the EM precondtioner, a thresholding, and a good inexpensive estimate of the solution, we propose an improved EM preconditioner that can not only accelerate convergence but also avoid the reconstructed image being "stuck at zero." Numerical results in this paper show that the proposed IEM-PKMA outperforms existing state-of-the-art algorithms including, the optimization transfer descent algorithm and the preconditioned L-BFGS-B algorithm for the differentiable smoothed anisotropic total variation regularized model, the preconditioned alternating projection algorithm, and the alternating direction method of multipliers for the nondifferentiable HOTV regularized model. Encouraging initial experiments using clinical data are presented.

Bisimulations for fuzzy transition systems revisited

Bisimulation is a well-known behavioral equivalence for discrete event systems, and has recently been adopted and developed in fuzzy systems. In this paper, we propose a new bisimulation, i.e., the group-by-group fuzzy bisimulation, for fuzzy transition systems. It relaxes the fully matching requirement of the bisimulation definition proposed by Cao et al. (2010) [2], and can equate more pairs of states which are deemed to be equivalent intuitively, but which cannot be equated in previous definitions. We carry out a systematic investigation on this new notion of bisimulation. In particular, a fixed point characterization of the group-by-group fuzzy bisimilarity is given, based on which, we provide a polynomial-time algorithm to check whether two states in a fuzzy transition system are group-by-group fuzzy bisimilar. Moreover, a modal logic, which is an extension of the Hennessy–Milner logic, is presented to completely characterize the group-by-group fuzzy bisimilarity.

Thresholded semantic framework for a fully integrated fuzzy logic language

This work proposes a declarative semantics based on a fuzzy variant of the classical notion of least Herbrand model for the so-called FASILL programming language (acronym of “Fuzzy Aggregators and Similarity Into a Logic Language”) which has been recently designed and implemented for coping with implicit/explicit truth degree annotations, a great variety of connectives and unification by similarity. Also, we define an immediate consequence operator that allows us to give a fixpoint characterization of the least Herbrand model for a FASILL program. A remarkable aspect of our declarative and fixpoint semantics is the fact that both have been enriched with the use of thresholds thanks to the natural ability of the underlying fuzzy language for managing such constructs. Moreover, we have connected both semantics with an operational one, which also manages thresholds in an efficient way, by proving the correctness of the whole semantic framework.