Tarski's Fixpoint Research Articles

A generic approach to measuring the strength of completeness/compactness of various types of spaces and ordered structures

With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions of functions being contractive in these spaces and structures. As a sample of possible applications we discuss metric spaces, ultrametric spaces, ordered groups and fields, topological spaces, partially ordered sets, and lattices. We describe several notions of completeness in these spaces and structures and determine their respective strengths. In order to illustrate some consequences of the levels of strength, we give examples of generic fixed point theorems which then can be specialized to theorems in various applications which work with contracting functions and some completeness property of the underlying space. Ball spaces are nonempty sets of nonempty subsets of a given set. They are called spherically complete if every chain of balls has a nonempty intersection. This is all that is needed for the encoding of completeness notions. We discuss operations on the sets of balls to determine when they lead to larger sets of balls; if so, then the properties of the so obtained new ball spaces are determined. The operations can lead to increased level of strength, or to ball spaces of newly constructed structures, such as products. Further, the general framework makes it possible to transfer concepts and approaches from one application to the other; as examples we discuss theorems analogous to the Knaster–Tarski Fixed Point Theorem for lattices and theorems analogous to the Tychonoff Theorem for topological spaces.

A note on the Knaster–Tarski Fixpoint Theorem

This note shows that several statements about fixpoints in order theory are equivalent to Knaster–Tarski Fixpoint Theorem for complete lattices. All proofs have been done in Zermelo–Fraenkel set theory without the Axiom of Choice.

Towards a unified view of finite automata and semi-Markov flowgraph models

Two disparate areas of knowledge are interrelated in this article: (a) Theory of Finite Automata and (b) semi-Markov Flowgraphs. These two areas, which at first sight do not have anything in common, are shown to have similar structures and to be governed by similar rules. It is shown that the renewal argument applies in both areas, resulting in equations marking at certain points in ‘time’ system regeneration. The corresponding sets of equations look formally the same but hold in entirely different domains, (a) language equations and (b) equations for Laplace transforms of waiting time distributions. The equations thus obtained can be solved iteratively, as their solutions are Fixed Points. There are established theories addressing this type of problem in each field, (a) Knaster–Tarski Fixed Point Theorem and (b) Contraction Mapping Principle. Even though the mathematical structures in both areas are different ((a): partially ordered set; (b): complete metric space), the formal steps of arriving at the equations, their structure and solution techniques used are remarkably similar. The paper has two goals. The first is its cross-disciplinary contribution, identifying structural similarities in two distant areas. This structural result is hoped to be the first step towards finding a single theoretical framework encompassing both fields. Furthermore, the analogy observed here may in the future allow known techniques from one field to be ‘carried over’ to the other, thus generating new results. The paper's second contribution is educational. It is instructive to draw learners' attention to analogous structures in another field as it reinforces the learning experience.

An Efficient Distributed Power Control for Infeasible Downlink Scenarios--Global-Local Fixed-Point-Approximation Technique

In this paper, we present an efficient downlink power control scheme, for wireless networks, based on two key ideas: (i) global-local fixed-point-approximation technique (GLOFPAT) and (ii) bottleneck removal criterion (BRC). The proposed scheme copes with all scenarios including infeasible case where no power allocation can provide all multiple accessing users with target quality of service (QoS). For feasible case, the GLOFPAT efficiently computes a desired power allocation which corresponds to the allocation achieved by conventional algorithms. For infeasible case, the GLOFPAT offers valuable information to detect bottleneck users, to be removed based on the BRC, which deteriorate overall QoS. The GLOFPAT is a mathematically-sound distributed algorithm approximating desired power allocation as a unique fixed-point of an isotone mapping. The unique fixed-point of the global mapping is iteratively computed by fixed-point-approximations of multiple distributed local mappings, which can be computed in parallel by base stations respectively. For proper detection of bottleneck users, complete analysis of the GLOFPAT is presented with aid of the Tarski's fixed-point theorem. Extensive simulations demonstrate that the proposed scheme converges faster than the conventional algorithm and successfully increases the number of happy users receiving target QoS.

Tarski's Fixed-Point Theorem And Lambda Calculi With Monotone Inductive Types

The new concept of lambda calculi with monotone inductive types is introduced byhelp of motivations drawn from Tarski's fixed-point theorem (in preorder theory) andinitial algebras and initial recursive algebras from category theory. They are intendedto serve as formalisms for studying iteration and primitive recursion ongeneral inductively given structures. Special accent is put on the behaviour ofthe rewrite rules motivated by the categorical approach, most notably on thequestion of strong normalization (i.e., the impossibility of an infinitesequence of successive rewrite steps). It is shown that this key propertyhinges on the concrete formulation. The canonical system of monotone inductivetypes, where monotonicity is expressed by a monotonicity witness beinga term expressing monotonicity through its type, enjoys strong normalizationshown by an embedding into the traditional system of non-interleavingpositive inductive types which, however, has to be enriched by the parametricpolymorphism of system F. Restrictions to iteration on monotone inductive typesalready embed into system F alone, hence clearly displaying the differencebetween iteration and primitive recursion with respect to algorithms despitethe fact that, classically, recursion is only a concept derived from iteration.

Constructive design of a hierarchy of semantics of a transition system by abstract interpretation

We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the big-step semantics, termination and nontermination semantics, Plotkin's natural, Smyth's demoniac and Hoare's angelic relational semantics and equivalent nondeterministic denotational semantics (with alternative powerdomains to the Egli–Milner and Smyth constructions), D. Scott's deterministic denotational semantics, the generalized and Dijkstra's conservative/liberal predicate transformer semantics, the generalized/total and Hoare's partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these semantics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarski fixpoint approximation transfer theorems.

From Set-theoretic Coinduction to Coalgebraic Coinduction: some results, some problems

We investigate the relation between the set-theoretical description of coinduction based on Tarski Fixpoint Theorem, and the categorical description of coinduction based on coalgebras. In particular, we introduce set-theoretic generalizations of the coinduction proof principle, in the spirit of Milner's bisimulation “up-to”, and we discuss categorical counterparts for these. Moreover, we investigate the connection between these and the equivalences induced by T-coiterative functions. These are morphisms into final coalgebras, satisfying the T-coiteration scheme, which is a generalization of the corecursion scheme. We show how to describe coalgebraic F-bisimulations as set-theoretical ones. A list of examples of set-theoretic coinductions which appear not to be easily amenable to coalgebraic terms are discussed.

An Existence Theorem for a Parabolic Differential Equation in l∞{A) Based on the Tarski Fixed Point Theorem

An Existence Theorem for a Parabolic Differential Equation in l∞{A) Based on the Tarski Fixed Point Theorem

Tarski's Fixpoint Lemma and combinatorial games

Using Tarski's Fixpoint Lemma for order preserving maps of a complete lattice into itself, a new, lattice theoretic proof is given for the existence of persistent strategies for combinatorial games as well as for games with a topological tolerance and games on lattices. Further, the existence of winning strategies is obtained for games on superalgebraic lattices, which includes the case of ordinary combinatorial games. Finally, a basic representation theorem is presented for those lattices.

A lattice-theoretic approach to a class of dynamic games

We consider a general class of dynamic games arising in strategic capital accumulation/resource extraction, and show that they have Nash equilibria in pure stationary strategies. The approach is new and relies on Topkis' results in lattice programming and Tarski's fixed-point theorem.

Remarks on a nonlinear complementarity problem

A class of nonlinear complementarity problems defined by monotone operators is considered in the spaceLp[a,b], and an existence theorem is established using Tarski's fixed-point theorem.

THOUGHTS ON THE CANTOR-BERNSTEIN THEOREM

The usual proofs of the well-known set-theoretical theorem “Given one-one maps f: A → B and g:B → A, there exists a one-one onto map h:A → B” actually produce a map h:A → B contained in the relation f U g−1. Considering Tarski's Fixpoint Theorem as the implicit basic ingredient of such proofs. We examine several classical proofs/starting with Dedekind (1887), and illuminate their common feature by means of the categorical notion of a natural fixpoint. We consider a categorical form (CBT) of the theorem (with h ⊆ f Ug−1) in a variety of contexts, obtaining some examples of categories where CBT holds and others where it fails. Among other results we prove for a topos E, (1) CBT holds if E is Boolean, and conversely if E has a natural number object; (2) The Axiom of Choice in E implies a dual version of CBTI and conversely if E has splitting supports and a natural number object.