Topological Interpretation Research Articles

Experiments in Theorem Proving for Topological Hybrid Logic

This paper discusses two experiments in theorem proving for hybrid logic under the topological interpretation. We begin by discussing the topological interpretation of hybrid logic and noting what it adds to the topological interpretation of orthodox modal logic. We then examine two implemented proof methods. The first makes use of HyLoBan, a terminating theorem prover that searches for a winning search strategy in certain topologically motivated games. The second is a translation-based approach that makes use HyLoTab [J. van Eijck. HyLoTab - Tableau-based theorem proving for hybrid logics. Manuscript, CWI, available from http://www.cwi.nl/~jve/hylotab, 2002], a tableaux-based theorem prover for hybrid logic under the standard relational interpretation. We compare the two methods, and note a number of directions for further work.

Dominant K-theory and integrable highest weight representations of Kac–Moody groups

We give a topological interpretation of the highest weight representations of Kac–Moody groups. Given the unitary form G of a Kac–Moody group (over C ), we define a version of equivariant K-theory, K G on the category of proper G -CW complexes. We then study Kac–Moody groups of compact type in detail (see Section 2 for definitions). In particular, we show that the Grothendieck group of integrable highest weight representations of a Kac–Moody group G of compact type, maps isomorphically onto K ˜ G ∗ ( E G ) , where E G is the classifying space of proper G -actions. For the affine case, this agrees very well with recent results of Freed–Hopkins–Teleman. We also explicitly compute K G ∗ ( E G ) for Kac–Moody groups of extended compact type, which includes the Kac–Moody group E 10.

Topological interpretation of Barbero-Immirzi parameter

We set up a canonical Hamiltonian formulation for a theory of gravity based on a Lagrangian density made up of the Hilbert-Palatini term and, instead of the Holst term, the Nieh-Yan topological density. The resulting set of constraints in the time gauge are shown to lead to a theory in terms of a real SU(2) connection which is exactly the same as that of Barbero and Immirzi with the coefficient of the Nieh-Yan term identified as the inverse of the Barbero-Immirzi parameter. This provides a topological interpretation for this parameter. Matter coupling can then be introduced in the usual manner, without changing the universal topological Nieh-Yan term.

Augmented dimension groups and ordered cohomology

AbstractWe show that the ordered MatsumotoK0-group of a substitutive system is derivable from the ordered Čech cohomology of a simple ‘augmentation’ of the tiling space associated with the substitution. Besides supplying a topological interpretation of the MatsumotoK0-group, this leads to finer invariants of the flow equivalence class of a substitutive system.

Scalar parabolic PDEs and braids

The comparison principle for scalar second order parabolic PDEs on functions u ( t , x ) u(t,x) admits a topological interpretation: pairs of solutions, u 1 ( t , ⋅ ) u^1(t,\cdot ) and u 2 ( t , ⋅ ) u^2(t,\cdot ) , evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions { u α ( t , ⋅ ) } α = 1 n \{u^\alpha (t,\cdot )\}_{\alpha =1}^n . By lifting the graphs to Legendrian braids, we give a global version of the comparison principle: the curves u α ( t , ⋅ ) u^\alpha (t,\cdot ) evolve so as to (weakly) decrease the algebraic length of the braid. We define a Morse-type theory on Legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic PDEs. This is done via discretization to a finite-dimensional system and a suitable Conley index for discrete braids. The result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic PDEs. We give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time-periodic solutions.

The Gap in the Law and the Border-Breaching Function of the Exception

This paper evaluates the status and function of the border in the two polemics that bookend the long history of political theology: Paul’s polemic against the Jewish law, and Carl Schmitt’s critique of constitutional liberalism. Paul and Schmitt both challenge spatial notions of law that establish a boundary between an “inside” and an “outside” by topologizing “inside” and “outside” as continuous: through the “fulfillment of the law” in Paul, and through the strategy of sovereign exception in Schmitt. Schmitt even describes sovereignty as a “border concept,” a Grenzbegriff: a concept that pertains to borderline cases, but perhaps at the same time a concept of the border, one that proposes a particular interpretation of the border and its logical function. The essay argues that Schmitt’s contribution to the logical problem of law as border is to claim that the border need not be written, and that the sovereign exception is not of the order of a writing, but a “miracle” that is by nature unwritten and unwritable. The Hebraic tradition of law that is the object of Paul’s polemic offers another way of considering the border, as the function of the written law. It conceives of law not as a normative representation of an existing situation, but as the delimination of a space or gap that may not be transgressed. While the fulfillment of the law and the sovereign exception bypass the limit implied in the law, offering a topological interpretation of the border as what integrates “inside” and “outside” into a new whole, the Hebraic law aligns the border with the logical function of negation. The essay approaches these different accounts of the border by way of Lacan’s distinction between the imaginary and symbolic dimensions of law. The imaginary is the “incarnated” dimension of law, while the symbolic corresponds to the function of speech as a barrier or limit, and involves a spacing and negation, the introduction of a gap or emptiness that precludes anything like a “whole.”

Non-deterministic semantics for dynamic topological logic

Dynamic Topological Logic ( D T L ) is a combination of S 4 , under its topological interpretation, and the temporal logic L T L interpreted over the natural numbers. D T L is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models, which are tuples < X , T , f , V > , where < X , T > is a topological space, f a function on X and V a truth valuation assigning subsets of X to propositional variables. Our main result is that the set of valid formulas of D T L over spaces with continuous functions is recursively enumerable. We show this by defining alternative semantics for D T L . Under standard semantics, D T L is not complete for Kripke frames. However, we introduce the notion of a non-deterministic quasimodel, where the function f is replaced by a binary relation g assigning to each world multiple temporal successors. We place restrictions on the successors so that the logic remains unchanged; under these alternative semantics, D T L becomes Kripke-complete. We then apply model-search techniques to enumerate the set of all valid formulas.

Decontracted double BRST symmetry on the lattice

We present the Curci-Ferrari model on the lattice. In the massless case the topological interpretation of this model with its double Becchi-Rouet-Stora-Tyutin (BRST) symmetry relates to the Neuberger $0/0$ problem which we extend to include the ghost/antighost symmetric formulation of the nonlinear-covariant Curci-Ferrari gauges on the lattice. The introduction of a Curci-Ferrari mass term, however, serves to regulate the $0/0$ indeterminate form of physical observables observed by Neuberger. While such a mass $m$ decontracts the double BRST/anti-BRST algebra, which is well known to result in a loss of unitarity, observables can be meaningfully defined in the limit $m\ensuremath{\rightarrow}0$ via l'Hospital's rule. At finite $m$, the topological nature of the partition function used as the gauge-fixing device seems lost. We discuss the gauge parameter $\ensuremath{\xi}$ and mass $m$ dependence of the model and show how both cancel when $m\ensuremath{\equiv}m(\ensuremath{\xi})$ is appropriately adjusted with $\ensuremath{\xi}$.

On Dynamic Topological Logic of the Real Line

This article explores the topological interpretations of the modal language with two modalities—□, which is interpreted as the interior operation and ◯ (‘next’) which is interpreted as the pre-image operation for a continuous function. It is known that the □◯ logic S4C is complete with respect to topological interpretations in ℝn for n≥2, yet it is incomplete with respect to topological interpretations in ℝ. We focus on the logic L□◯(ℝ) of all the □◯ formulas that are sound with respect to topological interpretations in ℝ. In this article we present two formulas in L□◯(ℝ)–S4C, and prove that they are sound in ℝ and independent. We also establish that the previously known examples of formulas in L□◯(ℝ)–S4C are instances of a particular consequence of one of the two formulas presented.

ERRATUM

Steve Awodey and Kohei Kishida (2008). Topology and Modality: The Topological Interpretation of First-Order Modal Logic. The Review of Symbolic Logic 1(2): 146-166.On page 148 of this article an error was introduced during the production process. The final equation in the displayed formula 8 lines from the bottom of the page should read,[0, 1) ≠ [0, 1]The publisher regrets this error.

TOPOLOGY AND MODALITY: THE TOPOLOGICAL INTERPRETATION OF FIRST-ORDER MODAL LOGIC

As McKinsey and Tarksi showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the necessity operation is modeled by taking the interior of an arbitrary subset of a topological space. in this paper the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

The topology of justification

Justification Logic is a family of epistemic logical systems obtained from modal logics of knowledge by adding a new type of formula t:F, which is read t is a justification for F. The principal epistemic modal logic S4 includes Tarski’s well-known topological interpretation, according to which the modality 2X is read the Interior of X in a topological space (the topological equivalent of the ‘knowable part of X’). In this paper, we extend Tarski’s topological interpretation from S4 to Justification Logic systems with both modality and justification assertions. The topological semantics interprets t:X as a reachable subset of X (the topological equivalent of ‘test t confirms X’). We establish a number of soundness and completeness results with respect to Kripke topology and the real topology for S4-based systems of Justification Logic.

An equivalence principle for electromagnetics through Clifford's geometric algebras

As already pointed out by several authors, a given electromagnetic medium creates an effective geometry and a given geometry creates an effective medium—what we call the equivalence principle for electromagnetics. This bears some resemblance to the kinematic aspects of the interaction between masses and the spacetime metric of general relativity. In this article, using Clifford's geometric algebras defined over the tangent and cotangent bundles of spacetime, we generalize the topological interpretation of this equivalence principle: introducing a transformation that we call the vacuum form reduction, we are led to a fictitious spacetime corresponding to a special class of bianisotropic media (in the materials interpretation) and which reduces to spacetime vacuum as a particular case (in the topological interpretation). We illustrate our theoretical approach with two examples: transformation and moving media.

Relative pairing in cyclic cohomology and divisor flows

AbstractWe construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to the essential features of the divisor flows, namely homotopy invariance, additivity and integrality, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.

A diagrammatic approach to response problems in composite systems

The bulk macroscopic response of a system of particles or inclusions with field-inducedforces is studied. The susceptibilities and transport coefficients in such a system areexpressed as averages of a multiple-scattering expansion. A special diagrammaticmethod is developed for analyzing the structure of the expansion. The concept ofirreducibility is discussed in detail and shown to be crucial in obtaining macroscopicequations characterizing the system response with coefficients depending solely onlocal properties of the medium. Due to the representation of particles by lines indiagrams, irreducibility is given a particularly simple topological interpretation in thediagrammatic language. The method is illustrated by a discussion of responseproblems in colloidal suspensions in the presence of hydrodynamic interactions.

STATE SUM CONSTRUCTION OF TWO-DIMENSIONAL OPEN-CLOSED TOPOLOGICAL QUANTUM FIELD THEORIES

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

Derivation of Newton’s gravitational fine structure constant from the spectrum of Heterotic superstring theory

The work presents, probably for the first time, a derivation of Newton’s dimensionless gravity fine structure constant from the Green-Schwarz-Witten spectrum of Heterotic superstring theory. Various physical and topological interpretations are given to elucidate the deep meaning of connecting classical gravity to elementary high energy quantum particles.

Knot colouring polynomials

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in \pi_K. Given a finite group G and an element x in G, we consider the set of representations \rho from \pi_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images \rho(l_K). The resulting invariant maps knots to the group ring Z[G]. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses. We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that P can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails in particular that Yang-Baxter invariants can detect non-inversible and non-reversible knots.

Symmetry groups for 3D dynamical systems

We present a systematic way to construct dynamical systems with a specific symmetry group G. Each symmetric strange attractor has a unique image attractor that is locally identical to it but different at the global topological level. Image attractors can be lifted to many inequivalent covering attractors. These are distinguished by an index that has related topological, algebraic and group theoretical interpretations. These methods are used to describe dynamical systems with symmetry groups V4, S4 and S6.

Self-duality and codings for expansive automorphisms

Lind and Schmidt have shown that for certain ergodic Z k -actions on a compact abelian group Γ, the homoclinic group H is isomorphic to the Pontryagin dual of Γ. Einsiedler and Schmidt extended these results and showed that Γ is a quotient of a locally compact ring R modulo H. In this paper, we present a dynamical interpretation of R if k = 1 : it is a product of the stable group and the unstable group of Γ, under a suitable topology. As applications, we give a topological interpretation of the Pisot–Vijayaraghavan theorem and we link the results to tessellation theory.