Infinite Reduction Research Articles

Constructing abelian varieties from rank 2 Galois representations

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let ${\mathbb {L}}$ be a rank $2$, geometrically irreducible lisse $\overline {{\mathbb {Q}}}_\ell$-sheaf on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \overline {\mathbb {Q}}_{\ell }$, and has bad, infinite reduction at some closed point $x$ of $X\setminus U$. We show that ${\mathbb {L}}$ occurs as a summand of the cohomology of a family of abelian varieties over $U$. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when $E=\mathbb {Q}$, then ${\mathbb {L}}$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.

A flexure-based displacement reducer capable of achieving very large reduction ratio

To achieve positioning of nanometer or even sub-nanometer resolution from traditional piezoelectric actuators and giant magnetostrictive actuators, a bridge-type differential displacement reducer, which shows the capability of obtaining very large reduction ratio, is proposed. The reducer contains two bridge-type flexure mechanisms connected in a way that the output of the reducer equals the difference of the outputs of the two mechanisms. If the two bridge-type flexure mechanisms are identical, no matter how large the input is, the output will always be 0, indicating an infinite reduction ratio theoretically. Therefore, introducing a slight difference between the two mechanisms can yield a very large reduction ratio. A kinetostatic model for the reducer is developed, base on which a reducer exhibiting a reduction ratio of 100 is designed and prototyped. The results of the kinetostatic model, the finite element model and the experiment agree well with each other, validating the effectiveness of the proposed reducer and the correctness of the kinetostatic model.

Comments on the quantum field theory of the Coulomb gas formalism

The holomorphic Coulomb gas formalism, as developed by Feigin-Fuchs, Dotsenko-Fateev and Felder, is a set of rules for computing minimal model observables using free field techniques. We attempt to derive and clarify these rules using standard techniques of quantum field theory. We begin with a careful examination of the timelike linear dilaton. Although the background charge of the model breaks the scalar field’s continuous shift symmetry, the exponential of the action remains invariant under a discrete shift because the background charge is imaginary. Gauging this symmetry makes the dilaton compact and introduces winding modes into the spectrum. One of these winding operators corresponds to the anti-holomorphic completion of the BRST current first introduced by Felder, and the full left/right cohomology of this BRST charge isolates the irreducible representations of the Virasoro algebra within the degenerate Fock space of the linear dilaton. The “supertrace” in the BRST complex reproduces the minimal model partition function and exhibits delicate cancellations between states with both momentum and winding. The model at the radius R=sqrt{pp^{prime }} has two marginal operators corresponding to the Dotsenko-Fateev “screening charges”. Deforming by them, we obtain a model that might be called a “BRST quotiented compact timelike Liouville theory”. The Hamiltonian of the zero-mode quantum mechanics of this model is not Hermitian, but it is PT-symmetric and exactly solvable. Its eigenfunctions have support on an infinite number of plane waves, suggesting an infinite reduction in the number of independent states in the full quantum field theory. Applying conformal perturbation theory to the exponential interactions reproduces the Coulomb gas calculations of minimal model correlation functions. In contrast to spacelike Liouville, these “resonance correlators” are finite because the zero mode is compact. We comment on subtleties regarding the reflection operator identification, as well as naive violations of truncation in correlators with multiple reflection operators inserted. This work is part of an attempt to understand the relationship between the JT model of two dimen- sional gravity and the worldsheet description of the (2, p) minimal string as suggested by Seiberg and Stanford.

Computing with Infinite Terms and Infinite Reductions

Computing with Infinite Terms and Infinite Reductions

Projections for infinitary rewriting (extended version)

Proof terms in term rewriting are a representation means for reduction sequences, and more in general for contraction activity, allowing to distinguish e.g. simultaneous from sequential reduction. Proof terms for finitary, first-order, left-linear term rewriting are described in the literature. In a previous work, we defined an extension of the finitary proof-term formalism, that allows to describe contractions in infinitary first-order term rewriting, and gave a characterisation of permutation equivalence.In this work, we discuss how projections of possibly infinite rewrite sequences can be modelled using proof terms. Again, the foundation is a characterisation of projections for finitary rewriting. We extend this characterisation to infinitary rewriting and also refine it, by describing precisely the role that structural equivalence plays in the development of the notion of projection. The characterisation we propose yields a definite expression, i.e. a proof term, that describes the projection of an infinitary reduction over another.To illustrate the working of projections, we show how a common reduct of a (possibly infinite) reduction and a single step that makes part of it can be obtained via their respective projections. We show, by means of several examples, that the proposed definition yields the expected behaviour also in cases beyond those covered by this result. Finally, we discuss how the notion of limit is used in our definition of projection for infinite reduction.

Convergence in infinitary term graph rewriting systems is simple

Term graph rewriting provides a formalism for implementing term rewriting in an efficient manner by emulating duplication via sharing. Infinitary term rewriting has been introduced to study infinite term reduction sequences. Such infinite reductions can be used to modelnon-strict evaluation. In this paper, we unify term graph rewriting and infinitary term rewriting thereby addressing both components oflazy evaluation: non-strictness and sharing. In contrast to previous attempts to formalise infinitary term graph rewriting, our approach is based on a simple and natural generalisation of the modes of convergence of infinitary term rewriting. We show that this new approach is better suited for infinitary term graph rewriting as it is simpler and more general. The latter is demonstrated by the fact that our notions of convergence give rise to two independent canonical and exhaustive constructions of infinite term graphs from finite term graphs via metric and ideal completion. In addition, we show that our notions of convergence on term graphs are sound w.r.t. the ones employed in infinitary term rewriting in the sense that convergence is preserved by unravelling term graphs to terms. Moreover, the resulting infinitary term graph calculi provide a unified framework for both infinitary term rewriting and term graph rewriting, which makes it possible to study the correspondences between these two worlds more closely.

Projections for Infinitary Rewriting

Proof terms in term rewriting are a representation means for reduction sequences, and more in general for contraction activity, allowing to distinguish e.g. simultaneous from sequential reduction. Proof terms for finitary, first-order, left-linear term rewriting are described in [Terese. Term Rewriting Systems, volume 55 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, UK, 2003], ch. 8. In a previous work [C. Lombardi, A. Ríos, and R. de Vrijer. Proof terms for infinitary rewriting. In G. Dowek, editor, RTA-TLCA'14, volume 8560 of Lecture Notes in Computer Science, pages 303–318. Springer, 2014] we defined an extension of the finitary proof-term formalism, that allows to describe contractions in infinitary first-order term rewriting, and gave a characterisation of permutation equivalence. In this work, we discuss how projections of possibly infinite rewrite sequences can be modeled using proof terms. Again, the foundation is a characterisation of projections for finitary rewriting described in [Terese. Term Rewriting Systems, volume 55 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, UK, 2003], Sec. 8.7. We extend this characterisation to infinitary rewriting and also refine it, by describing precisely the role that structural equivalence plays in the development of the notion of projection. The characterisation we propose yields a definite expression, i.e. a proof term, that describes the projection of an infinitary reduction over another. To illustrate the working of projections, we show how a common reduct of a (possibly infinite) reduction and a single step that makes part of it can be obtained via their respective projections. We show, by means of several examples, that the proposed definition yields the expected behavior also in cases beyond those covered by this result. Finally, we discuss how the notion of limit is used in our definition of projection for infinite reduction.

Development of a New Backdrivable Actuator for Haptic Interfaces and Collaborative Robots

Industrial robots are most often position controlled and insensitive to external forces. In many robotic applications, however, such as teleoperation, haptics for virtual reality, and collaborative robotics, a close cooperation between humans and robots is required. For such applications, force sensing and control capabilities are required for stable interactions with the operator and environment. The robots must also be backdrivable, i.e., the robot must be able to follow user’s induced movements with the least possible resistance. High force efficiency is also desirable. These requirements are different from the design drivers of traditional industrial robots and call for specific actuators and reducers. Many such devices were proposed in the literature. However, they suffer from several drawbacks, offering either a limited reduction ratio or being complex and bulky. This paper introduces a novel solution to this problem. A new differential cable drive reducer is presented. It is backdrivable, has a high efficiency, and a potentially infinite reduction ratio. A prototype actuator using such a reducer has been developed and integrated on a test bench. The experimental characterization of its performance confirms its theoretical advantages.

Relative Termination via Dependency Pairs

A term rewrite system is terminating when no infinite reduction sequences are possible. Relative termination generalizes termination by permitting infinite reductions as long as some distinguished rules are not applied infinitely many times. Relative termination is thus a fundamental notion that has been used in a number of different contexts, like analyzing the confluence of rewrite systems or the termination of narrowing. In this work, we introduce a novel technique to prove relative termination by reducing it to dependency pair problems. To the best of our knowledge, this is the first significant contribution to Problem #106 of the RTA List of Open Problems. We first present a general approach that is then instantiated to provide a concrete technique for proving relative termination. The practical significance of our method is illustrated by means of an experimental evaluation.

The conservation theorem for differential nets

We prove the conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL). The conservation theorem states that the property of having infinite reductions (here infinite chains of cut elimination steps) is preserved by non-erasing steps. This turns the quest for strong normalisation (SN) into one for non-erasing weak normalisation (WN), and indeed we use this result to prove SN of simply typed DiLL (with promotion). Along the way to the theorem we achieve a number of additional results having their own interest, such as a standardisation theorem and a slightly modified system of nets, DiLL∂ϱ.

Infinite reduction of divisors on metric graphs

We demonstrate that the greedy algorithm for reduction of divisors on metric graphs need not terminate by modeling the Euclidean algorithm in this context. We observe that any infinite reduction has a well defined limit, allowing us to treat the greedy reduction algorithm as a transfinite algorithm and to analyze its running time via ordinal numbers. Matching lower and upper bounds on worst case running time of O(ωn) are provided.

Infinitary rewriting: closure operators, equivalences and models

Infinitary Term Rewriting allows to express infinite terms and transfinite reductions that converge to those terms. Underpinning the machinery of infinitary rewriting are closure operators on relations that facilitate the formation of transfinite reductions and transfinite equivalence proofs. The literature on infinitary rewriting has largely neglected to single out such closure operators, leaving them implicit in definitions of (transfinite) rewrite reductions, or equivalence relations. This paper unpicks some of those definitions, extracting the underlying closure principles used, as well as constructing alternative operators that lead to alternative notions of reduction and equivalence. A consequence of this unpicking is an insight into the abstraction level at which these operators can be defined. Some of the material in this paper already appeared in Kahrs (2010). The paper also generalises the notion of equational model for infinitary rewriting. This leads to semantics-based notions of equivalence that tie in with the equivalences constructed from the closure operators.

Strong normalization of [formula omitted] via a calculus of coercions

ML F is a type system extending ML with first-class polymorphism as in system F . The main goal of the present paper is to show that ML F enjoys strong normalization, i.e., it has no infinite reduction paths. The proof of this result is achieved in several steps. We first focus on xML F , the Church-style version of ML F , and show that it can be translated into a calculus of coercions: terms are mapped into terms and instantiations into coercions. This coercion calculus can be seen as a decorated version of system F , so that the simulation result entails strong normalization of xML F through the same property of system F . We then transfer the result to all other versions of ML F using the fact that they can be compiled into xML F and showing there is a bisimulation between the two. We conclude by discussing what results and issues are encountered when using the candidates of reducibility approach to the same problem.

Strong non-Noetherity of polynomial reduction

It is a well-known result by A. Reeves and B. Sturmfels that the reduction modulo a marked set of polynomials is Noetherian if and only if the marking is induced from an admissible term ordering. For finite sets of polynomials with a nonadmissible ordering, there is a constructive proof of the existence of an infinite reduction sequence, although a finite one might still be possible. On the base of our specialized software for combinatorics of monomial orderings, we have found some examples for which there is no finite reduction sequence. This is what we call “strong” non-Noetherity. Bibliography: 3 titles.

Classical proof forestry

Classical proof forests are a proof formalism for first-order classical logic based on Herbrand’s Theorem and backtracking games in the style of Coquand. First described by Miller in a cut-free setting as an economical representation of first-order and higher-order classical proof, defining features of the forests are a strict focus on witnessing terms for quantifiers and the absence of inessential structure, or ‘bureaucracy’. This paper presents classical proof forests as a graphical proof formalism and investigates the possibility of composing forests by cut-elimination. Cut-reduction steps take the form of a local rewrite relation that arises from the structure of the forests in a natural way. Yet reductions, which are significantly different from those of the sequent calculus, are combinatorially intricate and do not exclude the possibility of infinite reduction traces, of which an example is given. Cut-elimination, in the form of a weak normalisation theorem, is obtained using a modified version of the rewrite relation inspired by the game-theoretic interpretation of the forests. It is conjectured that the modified reduction relation is, in fact, strongly normalising.

An insertion operator preserving infinite reduction sequences

A common way to show the termination of the union of two abstract reduction systems, provided both systems terminate, is to prove that they enjoy a specific property (some sort of ‘commutation’ for instance). This specific property is actually used to show that, for the union not to terminate, one of the systems must itself be non-terminating, which leads to a contradiction. Unfortunately, the property may be impossible to prove because some of the objects that are reduced do not enjoy an adequate form.Hence the purpose of this paper is threefold:–First, it introduces an operator enabling us to insert a reduction step on such an object, and therefore to change its shape, while still preserving the ability to use the property. Of course, some new properties will need to be verified.–Second, as an instance of our technique, the operator is applied to relax a well-known lemma stating the termination of the union of two termination abstract reduction systems.–Finally, this lemma is applied in a peculiar and then in a more general way to show the termination of some lambda calculi with inductive types augmented with specific reductions dealing with:(i)copies of inductive types;(ii)the representation of symmetric groups.

Towards a Systematic Method for Proving Termination of Graph Transformation Systems

We describe a method for proving the termination of graph transformation systems. The method is based on the fact that infinite reductions must include infinite ‘creation chains’, that is chains of edges in different graphs of the reduction sequence, such that each edge is involved in creating the next edge. In our approach, the length of such creation chains is recorded by associating with each edge label a creation depth, which denotes the minimal length of a creation chain from an edge in the initial graph to that edge. We develop an algorithm which can prove the absence of such infinite chains (and therefore termination), analyse problems of the approach and propose possible solutions.

Schreier sets in Ramsey theory

We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k k -tuples of words (for every natural number k k ) over a finite alphabet, can be extended to one for partitions on Schreier-type sets of words (of every countable ordinal). Indeed, we establish an extension of the partition theorem of Carlson about words and of the (more general) partition theorem of Furstenberg-Katznelson about combinatorial subspaces of the set of words (generated from k k -tuples of words for any fixed natural number k k ) into a partition theorem about combinatorial subspaces (generated from Schreier-type sets of words of order any fixed countable ordinal). Furthermore, as a result we obtain a strengthening of Carlson’s infinitary Nash-Williams type (and Ellentuck type) partition theorem about infinite sequences of variable words into a theorem, in which an infinite sequence of variable words and a binary partition of all the finite sequences of words, one of whose components is, in addition, a tree, are assumed, concluding that all the Schreier-type finite reductions of an infinite reduction of the given sequence have a behavior determined by the Cantor-Bendixson ordinal index of the tree-component of the partition, falling in the tree-component above that index and in its complement below it.

Dimensionally continued infinite reduction of couplings

The infinite reduction of couplings is a tool to consistently renormalize a wide class of non-renormalizable theories with a reduced, eventually finite, set of independent couplings, and classify the non-renormalizable interactions. Several properties of the reduction of couplings, both in renormalizable and non-renormalizable theories, can be better appreciated working at the regularized level, using the dimensional-regularization technique. We show that, when suitable invertibility conditions are fulfilled, the reduction follows uniquely from the requirement that both the bare and renormalized reduction relations be analytic in epsilon=D-d, where D and d are the physical and continued spacetime dimensions, respectively. In practice, physically independent interactions are distinguished by relatively non-integer powers of epsilon. We discuss the main physical and mathematical properties of this criterion for the reduction and compare it with other equivalent criteria. The leading-log approximation is solved explicitly and contains sufficient information for the existence and uniqueness of the reduction to all orders.

Optimal virtual sensing for active noise control in a rigid-walled acoustic duct

The performance of local active noise control systems is generally limited by the small sizes of the zones of quiet created at the error sensors. This is often exacerbated by the fact that the error sensors cannot always be located close to an observer’s ears. Virtual sensing is a method that can move the zone of quiet away from the physical location of the transducers to a desired location, such as an observer’s ear. In this article, analytical expressions are derived for optimal virtual sensing in a rigid-walled acoustic duct with arbitrary termination conditions. The expressions are derived for tonal excitations, and are obtained by employing a traveling wave model of a rigid-walled acoustic duct. It is shown that the optimal solution for the virtual sensing microphone weights is independent of the source location and microphone locations. It is also shown that, theoretically, it is possible to obtain infinite reductions at the virtual location. The analytical expressions are compared with forward difference prediction techniques. The results demonstrate that the maximum attenuation, that theoretically can be obtained at the virtual location using forward difference prediction techniques, is expected to decrease for higher excitation frequencies and larger virtual distances.