Fixed Point Operator Research Articles

Complexity results for modal logic with recursion via translations and tableaux

This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the $\mu$-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce terminating and non-terminating tableau systems for the logics we study, based on Kozen's tableau for the $\mu$-calculus and the one of Fitting and Massacci for modal logic. Finally, we describe these tableaux with $\mu$-calculus formulas, thus reducing the satisfiability of each of these logics to the satisfiability of the $\mu$-calculus, resulting in a general 2EXP upper bound for satisfiability testing.

Non-stationary Anderson acceleration with optimized damping

Anderson acceleration (AA) has a long history of use and a strong recent interest due to its potential ability to dramatically improve the linear convergence of the fixed-point iteration. Most authors are simply using and analyzing the stationary version of Anderson acceleration (sAA) with a constant damping factor or without damping. Little attention has been paid to nonstationary algorithms. However, damping can be useful and is sometimes crucial for simulations in which the underlying fixed-point operator is not globally contractive. The role of this damping factor has not been fully understood. In the present work, we consider the non-stationary Anderson acceleration algorithm with optimized damping (AAoptD) in each iteration to further speed up linear and nonlinear iterations by applying one extra inexpensive optimization. We analyze the convergence rate this procedure and develop an efficient and inexpensive implementation scheme. We show by extensive numerical experiments that the proposed non-stationary Anderson acceleration with optimized damping procedure often converges much faster than stationary AA with constant damping, adaptive damping or without damping, especially in the cases larger window sizes are needed. We also observe that simple strategies like using constant damping factors and adaptive damping factors, sometimes, work very well for some problems while sometimes they are even worse than AA without damping. Our proposed method is usually more robust than AA with constant damping and adaptive damping. Moreover, we also observed from our numerical results that damping can be good, but choosing the wrong damping factors may slow down the convergence rate. Theoretical analysis of the effects of damping factors are needed and important.

On undecidability of unary non-nested PFP-operators for one successor function theory

We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.

Dynamic Tangled Derivative Logic of Metric Spaces

Dynamical systems are abstract models of interaction between space and time. They are often used in fields such as physics and engineering to understand complex processes, but due to their general nature, they have found applications for studying computational processes, interaction in multi-agent systems, machine learning algorithms and other computer science related phenomena. In the vast majority of applications, a dynamical system consists of the action of a continuous `transition function' on a metric space. In this work, we consider decidable formal systems for reasoning about such structures. Spatial logics can be traced back to the 1940's, but our work follows a more dynamic turn that these logics have taken due to two recent developments: the study of the topological mu-calculus, and the the integration of linear temporal logic with logics based on the Cantor derivative. In this paper, we combine dynamic topological logics based on the Cantor derivative and the `next point in time' operators with an expressively complete fixed point operator to produce a combination of the topological mu-calculus with linear temporal logic. We show that the resulting logics are decidable and have a natural axiomatisation. Moreover, we prove that these logics are complete for interpretations on the Cantor space, the rational numbers, and subspaces thereof.

Frugal and decentralised resolvent splittings defined by nonexpansive operators

Frugal resolvent splittings are a class of fixed point algorithms for finding a zero in the sum of the sum of finitely many set-valued monotone operators, where the fixed point operator uses only vector addition, scalar multiplication and the resolvent of each monotone operator once per iteration. In the literature, the convergence analyses of these schemes are performed in an inefficient, algorithm-by-algorithm basis. In this work, we address this by developing a general framework for frugal resolvent splitting which simultaneously covers and extends several important schemes in the literature. The framework also yields a new resolvent splitting algorithm which is suitable for decentralised implementation on regular networks.

Higher-Order Property-Directed Reachability

The property-directed reachability (PDR) has been used as a successful method for automated verification of first-order transition systems. We propose a higher-order extension of PDR, called HoPDR, where higher-order recursive functions may be used to describe transition systems. We formalize HoPDR for the validity checking problem for conjunctive nu-HFL(Z), a higher-order fixpoint logic with integers and greatest fixpoint operators. The validity checking problem can also be viewed as a higher-order extension of the satisfiability problem for Constrained Horn Clauses (CHC), and safety property verification of higher-order programs can naturally be reduced to the validity checking problem. We have implemented a prototype verification tool based on HoPDR and confirmed its effectiveness. We also compare our HoPDR procedure with the PDR procedure for first-order systems and previous methods for fully automated higher-order program verification.

New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems

In this paper we propose and analyze Banach spaces-based fully-mixed approaches yielding new finite element methods for numerically solving the coupled partial differential equations describing the pseudostress-assisted diffusion of a solute into an elastic material. Two mixed formulations employing the diffusive flux as an additional variable are introduced for the diffusion equation, and the concentration gradient is considered as an auxiliary unknown of the second one of them. The resulting coupled systems are rewritten as equivalent fixed point operator equations, so that the respective unique solvabilities are proved by applying the classical Banach theorem along with the Babuška-Brezzi theory. The nonlinear dependency on the elastic variables of the diffusion coefficient and its source term, as well as the nonlinear dependency on the concentration of the elastic source term, suggest, for appropriate continuous and discrete analyses, that the unknowns be sought in suitable Lebesgue spaces. The associated Galerkin schemes are addressed similarly, and the Brouwer theorem yields the existence of discrete solutions. A priori error estimates are derived for both approaches, and rates of convergence for specific finite element subspaces satisfying the required discrete inf-sup conditions, are established in 2D. Finally, several numerical examples illustrating the performance of the two methods and confirming the theoretical findings, are reported.

Recursion-free modular arithmetic in the lambda-calculus

We investigate encodings for modular arithmetic in the lambda-calculus. There are two approaches: adapting well-known numeral systems, and building a new one. This paper focuses on providing original techniques to encode modular arithmetic directly. We present a modular arithmetic numeral system complete with multiplication and an implementation of the Chinese remainder theorem, all without recursion i.e., without using fixed-point operators.

Anderson acceleration for a regularized Bingham model

AbstractThis article studies a finite element discretization of the regularized Bingham equations that describe viscoplastic flow. An efficient nonlinear solver for the discrete model is then proposed and analyzed. The solver is based on Anderson acceleration (AA) applied to a Picard iteration, and we show accelerated convergence of the method by applying AA theory (recently developed by the authors) to the iteration, after showing sufficient smoothness properties of the associated fixed point operator. Numerical tests of spatial convergence are provided, as are results of the model for 2D and 3D driven cavity simulations. For each numerical test, the proposed nonlinear solver is also tested and shown to be very effective and robust with respect to the regularization parameter as it goes to zero.

HFL(Z) Validity Checking for Automated Program Verification

We propose an automated method for checking the validity of a formula of HFL(Z), a higher-order logic with fixpoint operators and integers. Combined with Kobayashi et al.'s reduction from higher-order program verification to HFL(Z) validity checking, our method yields a fully automated, uniform verification method for arbitrary temporal properties of higher-order functional programs expressible in the modal mu-calculus, including termination, non-termination, fair termination, fair non-termination, and also branching-time properties. We have implemented our method and obtained promising experimental results.

Solvability and stability analysis of a coupled system involving generalized fractional derivatives

<abstract><p>In this article, we investigate the existence of unique maximal and minimal solutions for a coupled differential system in terms of generalized fractional derivative with arbitrary order. The iterative technique of a fixed point operator together with the properties of green's function are used basically. Moreover, we investigate the generalized Ulam-Hyers stability of the solution for the given coupled system. Finally, some examples are given to illustrate the theoretic results.</p></abstract>

Resolvent splitting for sums of monotone operators with minimal lifting

In this work, we study fixed point algorithms for finding a zero in the sum of \(n\ge 2\) maximally monotone operators by using their resolvents. More precisely, we consider the class of such algorithms where each resolvent is evaluated only once per iteration. For any algorithm from this class, we show that the underlying fixed point operator is necessarily defined on a d-fold Cartesian product space with \(d\ge n-1\). Further, we show that this bound is unimprovable by providing a family of examples for which \(d=n-1\) is attained. This family includes the Douglas–Rachford algorithm as the special case when \(n=2\). Applications of the new family of algorithms in distributed decentralised optimisation and multi-block extensions of the alternation direction method of multipliers (ADMM) are discussed.

Reliable numerical analysis for stochastic reaction-diffusion system

In this article, stochastic behavior of reaction diffusion brusselator model is under consideration. There are many physical phenomena which are related to chemical concentrations. One chemical concentration coincide with the other chemical concentration and their inter-diffusion is a major question to be addressed and to be understood. So, that is why Brusselator model is very proto-type and standard model that lays the foundation of any kind of that matter chemical concentrations of different substances. It also has the application in physical species as well. That is why we are considering such model. The existence of solution is guaranteed with fix-point operator, self mapping and pre-compact conditions. Nonstandard finite difference scheme and Crank-Nicolson schemes are used to show the graphical behavior of the model. The consistency and stability of the schemes are discussed and both schemes are unconditionally stable. The 3D and 2D graphs represents the concentration of the models.

A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.

Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation

We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1, 2, 3) that it is well-defined in a tensor product of appropriate local function spaces, each with L^2 impedance boundary data. We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance maps that ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. At least for a model case with two subdomains, these results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedance-to-impedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graph-partitioning software.

Complexity through Translations for Modal Logic with Recursion

This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the mu-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce a terminating tableau system for the logics we study, based on Kozen's tableau for the mu-calculus, and the one of Fitting and Massacci for modal logic.

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.

A closed-form bound on the asymptotic linear convergence of iterative methods via fixed point analysis

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the asymptotic linear rate of convergence, it does not explain the non-linear behavior of these algorithms in the non-asymptotic regime. In this letter, we take into account the effect of the first-order approximation error and present a closed-form bound on the convergence in terms of the number of iterations required for the distance between the iterate and the limit point to reach an arbitrarily small fraction of the initial distance. Our bound includes two terms: one corresponds to the number of iterations required for the linearized version of the fixed-point operator and the other corresponds to the overhead associated with the approximation error. With a focus on the convergence in the scalar case, the tightness of the proposed bound is proven for positively quadratic first-order difference equations.

An algebraic representation of the fixed-point closure of *-continuous Kleene algebras – A categorical Chomsky–Schützenberger theorem

AbstractThe family ${\mathcal{R}} X^*$ of regular subsets of the free monoid $X^*$ generated by a finite set X is the standard example of a ${}^*$ -continuous Kleene algebra. Likewise, the family ${\mathcal{C}} X^*$ of context-free subsets of $X^*$ is the standard example of a $\mu$ -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator $\mu$ . For arbitrary monoids M, ${\mathcal{C}} M$ is the closure of ${\mathcal{R}}M$ as a $\mu$ -continuous Chomsky algebra, more briefly, the fixed-point closure of ${\mathcal{R}} M$ . We provide an algebraic representation of ${\mathcal{C}} M$ in a suitable product of ${\mathcal{R}} M$ with $C_2'$ , a quotient of the regular sets over an alphabet $\Delta_2$ of two pairs of bracket symbols. Namely, ${\mathcal{C}}M$ is isomorphic to the centralizer of $C_2'$ in the product of ${\mathcal{R}} M$ with $C_2'$ , i.e. the set of those elements that commute with all elements of $C_2'$ . This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets $X\subseteq M$ by regular expressions over $X\cup\Delta_2$ interpreted in the product of ${\mathcal{R}} M$ and $C_2'$ . More generally, for any ${}^*$ -continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of $C_2'$ in the product of K with $C_2'$ .

An axiom system of probabilistic Mu-calculus

Mu-calculus (a.k.a. μTL) is built up from modal/dynamic logic via adding the least fixpoint operator μ. This type of logic has attracted increasing attention since Kozen's seminal work. PμTL is a succinct probabilistic extension of the standard μTL obtained by making the modal operators probabilistic. Properties of this logic, such as expressiveness and satisfiability decision, have been studied elsewhere. We consider another important problem: the axiomatization of that logic. By extending the approaches of Kozen and Walukiewicz, we present an axiom system for PμTL. In addition, we show that the axiom system is complete for aconjunctive formulas.