Fixed Point Operator Research Articles

Markovian equilibrium in infinite horizon economies with incomplete markets and public policy

We develop an isotone recursive approach to the problem of existence, computation, and characterization of nonsymmetric locally Lipschitz continuous (and, therefore, Clarke-differentiable) Markovian equilibrium for a class of infinite horizon multiagent competitive equilibrium models with capital, aggregate risk, public policy, externalities, one sector production, and incomplete markets. The class of models we consider is large, and examples have been studied extensively in the applied literature in public economics, macroeconomics, and financial economics. We provide sufficient conditions that distinguish between economies with isotone Lipschitz Markov equilibrium decision processes (MEDPs) and those that have only locally Lipschitz (but not necessarily isotone) MEDPs. As our fixed point operators are based upon order continuous and compact nonlinear operators, we are able to provide sufficient conditions under which isotone iterative fixed point constructions converge to extremal MEDPs via successive approximation. We develop a first application of a new method for computing MEDPs in a system of Euler inequalities using isotone fixed point theory even when MEDPs are not necessarily isotone. The method is a special case of a more general mixed monotone recursive approach. We show MEDPs are unique only under very restrictive conditions. Finally, we prove monotone comparison theorems in Veinott’s strong set order on the space of public policy parameters and distorted production functions.

From linear time to branching time

Model checking is a method for the verification of systems with respect to their specifications. Symbolic model-checking, which enables the verification of large systems, proceeds by calculating fixed-point expressions over the system's set of states. The μ-calculus is a branching-time temporal logic with fixed-point operators. As such, it is a convenient logic for symbolic model-checking tools. In particular, the alternation-free fragment of μ-calculus has a restricted syntax, making the symbolic evaluation of its formulas computationally easy. Formally, it takes time that is linear in the size of the system. On the other hand, specifiers find the μ-calculus inconvenient. In addition, specifiers often prefer to use linear-time formalisms. Such formalisms, however, cannot in general be translated to the alternation-free μ-calculus, and their symbolic evaluation involves nesting of fixed-points, resulting in time complexity that is quadratic in the size of the system. In this article, we characterize linear-time properties that can be specified in the alternation-free μ-calculus. We show that a linear-time property can be specified in the alternation-free μ-calculus iff it can be recognized by a deterministic Büchi automaton. We study the problem of deciding whether a linear-time property, specified by either an automaton or an LTL formula, can be translated to an alternation-free μ-calculus formula, and describe the translation, when possible.

UppDMC: A Distributed Model Checker for Fragments of the μ-Calculus

We present UppDMC, a distributed model-checking tool. It is tailored for checking finite-state systems and μ-calculus specifications with at most one alternation of minimal and maximal fixed-point operators. This fragment is also known as Lμ2. Recently, efficient game-based algorithms for this logic have been outlined.We describe the implementation of these algorithms within UppDMC and study their performance on practical examples. Running UppDMC on a simple workstation cluster, we were able to check liveness properties of the largest examples given in the VLTS Benchmark Suite, for which no answers were previously known.

Characterization of co-observable languages and formulas for their super/sublanguages

We present fixed-point based characterization of several classes of co-observable languages that are of interest in the context of decentralized supervisory control of discrete-event systems, including C&P /spl or/ D&A co-observable languages, C&P co-observable languages, and D&A co-observable languages. We also provide formulas for computing super/sublanguages for each of these classes. In cases where the class of co-observable languages is not closed under intersection/union, we provide upper/lower bound of the super/sublanguage formula we present. The computation of super/sublanguages and also computation of their upper/lower bounds has lead to the introduction of other classes of co-observable languages, namely, strongly C&P co-observable languages, strongly D&A co-observable languages, locally observable languages, and strongly locally observable languages. Fixed-point based characterization of all the above language classes is also given, and their closure under intersection/union is investigated. We also study whether the fixed-point operator preserves prefix closure, relative closure (also called L/sub m/(G)-closure), and controllability.

A Modal Logic for π-Calculus and Model Checking Algorithm

The π-calculus is one of the most important mobile process calculi and has been well studied in the literatures. Temporal logic is thought as a good compromise between description convenience and abstraction and can support useful computational applications, such as model-checking. In this paper, we use symbolic transition graph inherited from π-calculus to model concurrent systems. A wide class of processes, that is, the finite-control processes can be represented as finite symbolic transition graph. A new version π-μ-Logic is introduced as an appropriate temporal logic for the π-calculus. Since we make a distinction between proposition and predicate, the possible interactions between recursion and first-order quantification can be solved. A concise semantics interpretation for our modal logic is given. Based on the above work, we provide a model checking algorithm for the logic, which follows the well-known Winskel's tag set method to deal with fixpoint operator. As for the problem of name instantiating, our algorithm follows the 'on-the-fly' style, and systematically employs schematic names. The correctness of the algorithm is shown.

On the Recursive Enumerability of Fixed-Point Combinators

We show that the set of fixed-point combinators forms a recursively-enumerable subset of a larger set of terms we call non-standard fixed-point combinators. These terms are observationally equivalent to fixed-point combinators in any computable context, but the set of on-standard fixed-point combinators is not recursively enumerable.

Markovian Equilibrium in Infinite Horizon Economies with Incomplete Markets and Public Policy

We develop an isotone recursive approach to the problem of existence, computation, and characterization of nonsymmetric locally Lipschitz continuous (and, therefore, Clarke-differentiable) Markovian equilibrium for a class of infinite horizon multiagent competitive equilibrium models with capital, aggregate risk, public policy, externalities, one sector production, and incomplete markets. The class of models we consider is large, and examples have been studied extensively in the applied literature in public economics, macroeconomics, and financial economics. We provide sufficient conditions that distinguish between economies with isotone Lipschitizian Markov equilibrium decision processes (MEDPs) and those that have only locally Lipschitzian (but not necessarily isotone) MEDPs. As our fixed point operators are based upon order continuous and compact non-linear operators, we are able to provide sufficient conditions under which isotone iterative fixed point constructions converge to extremal MEDPs via successive approximation. We develop a first application of a new method for computing MEDPs in a system of Euler inequalities using isotone fixed point theory even when MEDPs are not necessarily isotone. The method is a special case of a more general mixed monotone recursive approach. We show MEDPs are unique only under very restrictive conditions. Finally, we prove monotone comparison theorems in Veinott's strong set order on the space of public policy parameters and distorted production functions.

A Theory of Local Set Queries

A relational database can be considered as a finite structure for a finite relational signature in first-order logic. This allows finite model theory to be used for investigating database theory. However, database research in the last decade has shifted its emphasis from the relational datamodel to higher-order datamodels including object oriented databases, object-relational databases, and most recently the model of semi-structured data and XML. A common characteristic of these new datamodels of interest is their richness in structure. In order to find a suitable generalisation of database theory that can cope with this structural richness a suitable logical ground is needed. In this paper local set theory, which can be considered as a version of higher-order intuitionistic logic, is investigated for this purpose. The central notions of database schema and instance for a slighly restricted class of higher-order datamodels are re-defined in local languages using a finite set interpretation. Terms of “set type” define simple queries, which provide the analogue of relational calculus queries in the relational model. Then the work on computable queries can be generalised as well. An analogue of the reflective relational machine model called LQ machine is defined and proven to be complete, i.e., able to express exactly the computable queries over local languages. Finally, fixed-point operators are added to simple queries. However, adding fixed-points will not increase expressiveness.

The Knower Paradox in the light of provability interpretations of modal logic

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368--387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338-- 355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287--304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy.

Algebraically complete semirings and Greibach normal form

We give inequational and equational axioms for semirings with a fixed-point operator and formally develop a fragment of the theory of context-free languages. In particular, we show that Greibach’s normal form theorem depends only on a few equational properties of least pre-fixed points in semirings, and eliminations of chain and deletion rules depend on their inequational properties (and the idempotence of addition). It follows that these normal form theorems also hold in non-continuous semirings having enough fixed points.

On the Recursive Enumerability of Fixed-Point Combinators

We show that the set of fixed-point combinators forms a recursively-enumerable subset of a larger set of terms that is (A) not recursively enumerable, and (B) the terms of which are observationally equivalent to fixed-point combinators in any computable context.

The Uniformity Principle on Traced Monoidal Categories

The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin’s principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits.

Model checking linear logic specifications

The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for first order linear logic specifications. The fragment of linear logic we consider in this paper is based on the linear logic programming language called LO (Andreoli and Pareschi, 1990) enriched with universally quantified goal formulas. Although LO was originally introduced as a theoretical foundation for extensions of logic programming languages, it can also be viewed as a very general language to specify a wide range of infinite-state concurrent systems (Andreoli, 1992; Cervesato, 1995). Our approach is based on the relation between backward reachability and provability highlighted in our previous work on propositional LO programs (Bozzano et al., 2002). Following this line of research, we define here a general framework for the bottom-up. evaluation of first order linear logic specifications. The evaluation procedure is based on an effective fixpoint operator working on a symbolic representation of infinite collections of first order linear logic formulas. The theory of well quasi-orderings Abdulla et al., 1996; Finkel and Schnoebelen, 2001) can be used to provide sufficient conditions for the termination of the evaluation of non trivial fragments of first order linear logic.

On the λY calculus

The λY calculus is the simply typed λ calculus augmented with the fixed point operators. We show three results about λY: (a) the word problem is undecidable, (b) weak normalisability is decidable, and (c) higher type fixed point operators are not definable from fixed point operators at smaller types.

Inflationary fixed points in modal logic

We consider an extension of modal logic with an operator for constructing inflationary fixed points, just as the modal μ-calculus extends basic modal logic with an operator for least fixed points. Least and inflationary fixed-point operators have been studied and compared in other contexts, particularly in finite model theory, where it is known that the logics IFP and LFP that result from adding such fixed-point operators to first-order logic have equal expressive power. As we show, the situation in modal logic is quite different, as the modal iteration calculus (MIC), we introduce has much greater expressive power than the μ-calculus. Greater expressive power comes at a cost: the calculus is algorithmically much less manageable.

Logic of global synchrony

An intermediate-level specification formalism (i.e., specification language supported by laws and a semantic model), Logs, is presented for PRAM and BSP styles of parallel programming. It extends pre-post sequential semantics to reveal states at points of global synchronization. The result is an integration of the pre-post and reactive-process styles of specification. The language consists of only six commands from which other useful commands can be derived. Parallel composition is simply logical conjunction and hence compositional. A simple predicative semantics and a complete set of algebraic laws are presented. Novel ingredients include the separation, in our reactive context, of the processes for nontermination and for abortion which coincide in standard programming models; the use of partitions, combining the terminating behavior of one program with the nonterminating behavior of another; and a fixpoint operator, the partitioned fixpoint. Our semantics benefits from the recent "healthiness function" approach for predicative semantics. Use of Logs, along with the laws for reasoning about it, is demonstrated on two problems: matrix multiplication (a terminating numerical computation) and the dining philosophers (a reactive computation). The style of reasoning is so close to programming practice that direct transformation from Logs specifications to real PRAM and BSP programs becomes possible.

Intersection types and domain operators

We use intersection types as a tool for obtaining λ-models. Relying on the notion of easy intersection type theory, we successfully build a λ-model in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This allows us to prove two results. The first gives a proof of consistency of the λ-theory where the λ-term ( λx. xx)( λx. xx) is forced to behave as the join operator. This result has interesting consequences on the algebraic structure of the lattice of λ-theories. The second result is that for any simple easy term, there is a λ-model, where the interpretation of the term is the minimal fixed point operator.

Parallel Model Checking for LTL, CTL∗, and L2μ

We describe a parallel model-checking algorithm for the fragment of the μ-calculus that allows one alternation of minimal and maximal fixed-point operators. This fragment is also known as L2μ. Since LTL and CTL∗ can be encoded in this fragment, we obtain parallel model checking algorithms for practically important temporal logics.Our solution is based on a characterization of this problem in terms of two player games. We exhibit the structure of their game graphs and show that we can iteratively work with game graphs that have the same special structure as the ones obtained for L1μ-formulae. Since good parallel algorithms for colouring game-graphs for L1μ-formulae exist, it is straightforward to implement this algorithm in parallel and good run-time results can be expected.

Quenched QCD with fixed-point and chirally improved fermions

In this contribution we present results from quenched QCD simulations with the parameterized fixed-point (FP) and the chirally improved (CI) Dirac operator. Both these operators are approximate solutions of the Ginsparg-Wilson equation and have good chiral properties. We focus our discussion on observables sensitive to chirality. In particular we explore pion masses down to 210 MeV in light hadron spectroscopy, quenched chiral logs, the pion decay constant and the pion scattering length. We discuss finite volume effects, scaling properties of the FP and CI operators and performance issues in their numerical implementation.

The Uniformity Principle on Traced Monoidal Categories

The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin's principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits.