Modulo Equivalence Research Articles

On when the union of two algebraic sets is algebraic

AbstractIn universal algebraic geometry, an algebra is called an equational domain if the union of two algebraic sets is algebraic. We characterize equational domains, with respect to polynomial equations, inside congruence permutable varieties, and with respect to term equations, among all algebras of size two and all algebras of size three with a cyclic automorphism. Furthermore, for each size at least three, we prove that, modulo term equivalence, there is a continuum of equational domains of that size.

On the M�bius invariant principal functions of Pincus

In this semi-expository short note, we prove that the only homogeneous pure hyponormal operator \(T\) with \(\operatorname{rank} (T^*T-TT^*) =1\), modulo unitary equivalence, is the unilateral shift.

With or Without You: Programming with Effect Exclusion

Type and effect systems have been successfully used to statically reason about effects in many different domains, including region-based memory management, exceptions, and algebraic effects and handlers. Such systems’ soundness is often stated in terms of the absence of effects. Yet, existing systems only admit indirect reasoning about the absence of effects. This is further complicated by effect polymorphism which allows function signatures to abstract over arbitrary, unknown sets of effects. We present a new type and effect system with effect polymorphism as well as union, intersection, and complement effects. The effect system allows us to express effect exclusion as a new class of effect polymorphic functions: those that permit any effects except those in a specific set. This way, we equip programmers with the means to directly reason about the absence of effects. Our type and effect system builds on the Hindley-Milner type system, supports effect polymorphism, and preserves principal types modulo Boolean equivalence. In addition, a suitable extension of Algorithm W with Boolean unification on the algebra of sets enables complete type and effect inference. We formalize these notions in the λ ∁ calculus. We prove the standard progress and preservation theorems as well as a non-standard effect safety theorem: no excluded effect is ever performed. We implement the type and effect system as an extension of the Flix programming language. We conduct a case study of open source projects identifying 59 program fragments that require effect exclusion for correctness. To demonstrate the usefulness of the proposed type and effect system, we recast these program fragments into our extension of Flix.

Counting inversion sequences by parity successions and runs

Let In denote the set of inversion sequences of length n. In this paper, we enumerate members of In represented using positive integers according to parameters which track the number of runs and successions of letters of a given parity. To this end, we consider a four-parameter joint distribution on In and derive several general results for this distribution, providing further treatment of some particular cases. In order to prove our results, we make use of a non-standard generating function which allows for the system of recurrences satisfied by various refinements of the aforementioned joint distribution to be translated into a system of linear differential equations that can be solved explicitly. Finally, the more general problem of enumerating the members of In according to adjacencies involving two elements that are equivalent modulo m is considered.

On the “traveling pulses” of the limit of the FitzHugh–Nagumo equation when [formula omitted

A solution (u(s),v(s)) of the differential system u′=v,v′=−cv−u(u−a)(1−u)+w,w′=−(ɛ/c)(u−γw).with a,c,ɛ∈R such that (u(s),v(s))→(0,0) when s→±∞ is a traveling pulse of the FitzHugh–Nagumo equation. The limit of this differential system when ɛ→0 gives rise to the polynomial differential system u′=v,v′=−cv−u(u−a)(1−u)+w,where now a,c,w∈R. We give the complete description of its phase portraits in the Poincaré disc (i.e. in the compactification of R2 adding the circle S1 of the infinity) modulo topological equivalence.

Forward-modulated damping estimates and nonlocalized stability of periodic Lugiato–Lefever waves

In an interesting recent analysis, Haragus–Johnson–Perkins–de Rijk have shown modulational stability under localized perturbations of steady periodic solutions of the Lugiato–Lefever equation (LLE), in the process pointing out a difficulty in obtaining standard “nonlinear damping estimates” on modulated perturbation variables to control regularity of solutions. Here, we point out that in place of standard “inverse-modulated” damping estimates, one can alternatively carry out a damping estimate on the “forward-modulated” perturbation, noting that norms of forward- and inverse-modulated variables are equivalent modulo absorbable errors, thus recovering the classical argument structure of Johnson–Noble–Rodrigues–Zumbrun for parabolic systems. This observation seems of general use in situations of delicate regularity. Applied in the context of (LLE), it gives the stronger result of stability and asymptotic behavior with respect to nonlocalized perturbations.

Coupling of Zinc Dose to Frequency in a Regularly Recurrent Pattern Shows a Limited Capacity of Excessive Dietary Zinc to Compensate for a Previously Deficient Intake

The impact of feeding a proportional dose of zinc in a regularly recurrent pattern (idiorrhythmic feeding) on body growth of weanling rats was studied to assess the extent to which an intake exceeding the requirement could compensate for a previously deficient intake. The idiorrhythmic regimen (I) was designed so that the overall dose-time equivalent (modulo; Mx) was kept constant over a selected period of time (epoch; E), whereas the actual dose and frequency varied regularly according to a predetermined pattern; that is, I = dnth(Mx)/dnth, where Mx and dnth are the selected dose-time equivalent and the sequential number of zinc-dosing days, respectively. For example, 3 mg Zn·kg-1 diet fed every day (I = 3/1) and 24 mg Zn·kg-1 diet fed 1 d in an 8-d period (7 d of feeding a zinc-deficient diet) (I = 24/8) are dose-time equivalents because they both provide 72 mg Zn·kg-1 over a 24-d E. Modulos of M3, M6, M12 and M24 were used, which provided 3, 6, 12 and 24 mg Zn·kg-1·d-1 dose-time equivalents during a 24-d E. Each modulo had seven analogous idiorrhythms: I = Mx/1, 2Mx/2, 3Mx/3, 4Mx/4, 5Mx/5, 6Mx/6 and 8Mx/8; for example, with 8 Mx/8, 24, 48, 96 or 192 mg Zn·kg-1 diet was fed every 8th d; on the other 7 d a diet without a zinc supplement was fed for M3, M6, M12 and M24, respectively. Zinc dose-rate idiorrhythm generated a complex, gestalt-like, biphasic growth response pattern where an intake of dietary zinc exceeding requirements had either no effect (M3 or average of 3 mg kg-1 diet) or fully compensated for only 1 (M6 and M12 or average of 6 and 12 mg Zn·kg-1 diet) or 2 (M24 or average of 24 mg Zn·kg-1 diet) d.

Babble: Learning Better Abstractions with E-Graphs and Anti-unification

Library learning compresses a given corpus of programs by extracting common structure from the corpus into reusable library functions. Prior work on library learning suffers from two limitations that prevent it from scaling to larger, more complex inputs. First, it explores too many candidate library functions that are not useful for compression. Second, it is not robust to syntactic variation in the input. We propose library learning modulo theory (LLMT), a new library learning algorithm that additionally takes as input an equational theory for a given problem domain. LLMT uses e-graphs and equality saturation to compactly represent the space of programs equivalent modulo the theory, and uses a novel e-graph anti-unification technique to find common patterns in the corpus more directly and efficiently. We implemented LLMT in a tool named babble. Our evaluation shows that babble achieves better compression orders of magnitude faster than the state of the art. We also provide a qualitative evaluation showing that babble learns reusable functions on inputs previously out of reach for library learning.

Approximate equivalence in von Neumann algebras

Suppose $\mathcal{A}$ is a separable unital ASH C*-algebra, $\mathcal{R}$ is a sigma-finite II$_{\infty}$ factor von Neumann algebra, and $\pi,\rho :\mathcal{A}\rightarrow\mathcal{R}$ are unital $\ast$-homomorphisms such that, for every $a\in\mathcal{A}$, the range projections of $\pi\left( a\right) $ and $\rho\left( a\right) $ are Murray von Neuman equivalent in $\mathcal{R}% $. We prove that $\pi$ and $\rho$ are approximately unitarily equivalent modulo $\mathcal{K}_{\mathcal{R}}$, where $\mathcal{K}_{\mathcal{R}}$ is the norm closed ideal generated by the finite projections in $\mathcal{R}$. We also prove a very general result concerning approximate equivalence in arbitrary finite von Neumann algebras.

Lonely Points in Simplices

Given a lattice Lsubseteq mathbb Z^m and a subset Asubseteq mathbb R^m, we say that a point in A is lonely if it is not equivalent modulo L to another point of A. We are interested in identifying lonely points for specific choices of L when A is a dilated standard simplex, and in conditions on L which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.

Covariant holographic negativity from the entanglement wedge in AdS3/CFT2

We propose a covariant holographic construction based on the extremal entanglement wedge cross section, for the entanglement negativity of bipartite states in CFT1+1s dual to non static bulk AdS3 geometries. Utilizing our proposal we obtain the holographic entanglement negativity for bipartite mixed states in CFT1+1s dual to bulk extremal and non extremal rotating BTZ black holes and the time dependent Vaidya-AdS3 geometries. Our results correctly reproduce the replica technique results modulo certain constants for the corresponding dual CFT1+1s in the large central charge limit for the bulk black hole geometries. Furthermore for all the cases our results also match upto such constants with the corresponding results from an earlier alternate covariant holographic construction in the literature. This demonstrates the equivalence modulo constants of the two constructions for the AdS3/CFT2 scenario.

Friedman-reflexivity

In the present paper, we explore an idea of Harvey Friedman to obtain a coordinate-free presentation of consistency. For some range of theories, Friedman's idea delivers actual consistency statements (modulo provable equivalence). For a wider range, it delivers consistency-like statements.We say that a sentence C is an interpreter of a finitely axiomatised A over U iff it is the weakest statement C over U, with respect to U-provability, such that U+C interprets A. A theory U is Friedman-reflexive iff every finitely axiomatised A has an interpreter over U. Friedman shows that Peano Arithmetic, PA, is Friedman-reflexive.We study the question which theories are Friedman-reflexive. We show that a very weak theory, Peano Corto, is Friedman-reflexive. We do not get the usual consistency statements here, but bounded, cut-free, or Herbrand consistency statements. We illustrate that Peano Corto as a base theory has additional desirable properties.We prove a characterisation theorem for the Friedman-reflexivity of sequential theories. We provide an example of a Friedman-reflexive sequential theory that substantially differs from the paradigm cases of Peano Arithmetic and Peano Corto.Interpreters over a Friedman-reflexive U can be used to define a provability-like notion for any finitely axiomatised A that interprets U. We explore what modal logics this idea gives rise to. We call such logics interpreter logics. We show that, generally, these logics satisfy the Löb Conditions, aka K4. We provide conditions for when interpreter logics extend S4, K45, and Löb's Logic. We show that, if either U or A is sequential, then the condition for extending Löb's Logic is fulfilled. Moreover, if our base theory U is sequential and if, in addition, its interpreters can be effectively found, we prove Solovay's Theorem. This holds even if the provability-like operator is not necessarily representable by a predicate of Gödel numbers.At the end of the paper, we briefly discuss how successful the coordinate-free approach is.

Extreme Points of Gram Spectrahedra of Binary Forms

The Gram spectrahedron mathrm {Gram}(f) of a form f with real coefficients is a compact affine-linear section of the cone of psd symmetric matrices. It parametrizes the sum of squares decompositions of f, modulo orthogonal equivalence. For f a sufficiently general positive binary form of arbitrary degree, we show that mathrm {Gram}(f) has extreme points of all ranks in the Pataki range. We also calculate the dimension of the set of rank r extreme points, for any r. Moreover, we determine the pairs of rank two extreme points for which the connecting line segment is an edge of mathrm {Gram}(f). The proof of the main result relies on a purely algebraic fact of independent interest: Whenever d,rge 1 are integers with left( {begin{array}{c}r+1 2end{array}}right) le 2d+1, there exists a length r sequence f_1,dots ,f_r of binary forms of degree d for which the left( {begin{array}{c}r+1 2end{array}}right) pairwise products f_if_j, ile j, are linearly independent.

Factorial relative commutants and the generalized Jung property for II1 factors

The findings reported in this paper aim to garner the interest of both model theorists and operator algebraists alike. Using a novel blend of model theoretic and operator algebraic methods, we show that the family of II1 factors elementarily equivalent to the hyperfinite II1 factor R all admit embeddings into RU with factorial relative commutant. This answers a long standing question of Popa for an uncountable family of II1 factors. We introduce the notion of a generalized Jung factor: a II1 factor M for which any two embeddings of M into its ultrapower MU are equivalent by an automorphism of MU. As an application of the result above, we show that R is the unique RU-embeddable generalized Jung factor. Using the concept of building von Neumann algebras by games and the recent refutation of the Connes embedding problem, we also show that there exists a generalized Jung factor which does not embed into RU. Moreover, we find that there are uncountably many non RU-embeddable generalized Jung type II1 von Neumann algebras. We study the space of embeddings modulo automorphic equivalence of a II1 factor N into an ultrapower II1 factor MU and equip it with a natural topometric structure, yielding cardinality results for this space in certain cases. These investigations are naturally connected to the super McDuff property for II1 factors: the property that the central sequence algebra is a II1 factor. We provide new examples, classification results, and assemble the present landscape of such factors. Finally, we prove a transfer theorem for inducing factorial commutants on embeddings with several applications.

Relational nullable types with Boolean unification

We present a simple, practical, and expressive relational nullable type system. A relational nullable type system captures whether an expression may evaluate to null based on its type, but also based on the type of other related expressions. The type system extends the Hindley-Milner type system with Boolean constraints, supports parametric polymorphism, and preserves principal types modulo Boolean equivalence. We show how to support full Hindley-Milner style type inference with an extension of Algorithm W. We conduct a preliminary study of open source projects showing that there is a need for relational nullable type systems across a wide range of programming languages. The most important findings from the study are: (i) programmers use programming patterns where the nullability of one expression depends on the nullability of other related expressions, (ii) such invariants are commonly enforced with run-time exceptions, and (iii) reasoning about these programming patterns requires not only knowledge of when an expression may evaluate to null, but also when it may evaluate to a non-null value. We incorporate these observations in the design of the proposed relational nullable type system.

Bifurcations of the Riccati Quadratic Polynomial Differential Systems

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

Not by equations alone: Reasoning with extensible effects

Abstract The challenge of reasoning about programs with (multiple) effects such as mutation, jumps, or IO dates back to the inception of program semantics in the works of Strachey and Landin. Using monads to represent individual effects and the associated equational laws to reason about them proved exceptionally effective. Even then it is not always clear what laws are to be associated with a monad—for a good reason, as we show for non-determinism. Combining expressions using different effects brings challenges not just for monads, which do not compose, but also for equational reasoning: the interaction of effects may invalidate their individual laws, as well as induce emerging properties that are not apparent in the semantics of individual effects. Overall, the problems are judging the adequacy of a law; determining if or when a law continues to hold upon addition of new effects; and obtaining and easily verifying emergent laws. We present a solution relying on the framework of (algebraic, extensible) effects, which already proved itself for writing programs with multiple effects. Equipped with a fairly conventional denotational semantics, this framework turns useful, as we demonstrate, also for reasoning about and optimizing programs with multiple interacting effects. Unlike the conventional approach, equational laws are not imposed on programs/effect handlers, but induced from them: our starting point hence is a program (model), whose denotational semantics, besides being used directly, suggests and justifies equational laws and clarifies side conditions. The main technical result is the introduction of the notion of equivalence modulo handlers (“modulo observation”) or a particular combination of handlers—and proving it to be a congruence. It is hence usable for reasoning in any context, not just evaluation contexts—provided particular conditions are met. Concretely, we describe several realistic handlers for non-determinism and elucidate their laws (some of which hold in the presence of any other effect). We demonstrate appropriate equational laws of non-determinism in the presence of global state, which have been a challenge to state and prove before.

The Marriage of Univalence and Parametricity

Reasoning modulo equivalences is natural for everyone, including mathematicians. Unfortunately, in proof assistants based on type theory, which are frequently used to mechanize mathematical results and carry out program verification efforts, equality is appallingly syntactic, and as a result, exploiting equivalences is cumbersome at best. Parametricity and univalence are two major concepts that have been explored in the literature to transport programs and proofs across type equivalences, but they fall short of achieving seamless, automatic transport. This work first clarifies the limitations of these two concepts when considered in isolation and then devises a fruitful marriage between both. The resulting concept, called univalent parametricity , is an extension of parametricity strengthened with univalence that fully realizes programming and proving modulo equivalences. Our approach handles both type and term dependency, as well as type-level computation. In addition to the theory of univalent parametricity, we present a lightweight framework implemented in the Coq proof assistant that allows the user to transparently transfer definitions and theorems for a type to an equivalent one, as if they were equal. For instance, this makes it possible to conveniently switch between an easy-to-reason-about representation and a computationally efficient representation as soon as they are proven equivalent. The combination of parametricity and univalence supports transport à la carte : basic univalent transport, which stems from a type equivalence, can be complemented with additional proofs of equivalences between functions over these types, in order to be able to transport more programs and proofs, as well as to yield more efficient terms. We illustrate the use of univalent parametricity on several examples, including a recent integration of native integers in Coq. This work paves the way to easier-to-use proof assistants by supporting seamless programming and proving modulo equivalences.

ISOTROPIC MOTIVES

AbstractIn this article we introduce the local versions of the Voevodsky category of motives with$\mathbb{F} _p$-coefficients over a fieldk, parametrized by finitely generated extensions ofk. We introduce the so-calledflexiblefields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructedlocalmotivic categories are much simpler than theglobalone and more reminiscent of a topological counterpart. This provides handy ‘local’ invariants from which one can read motivic information. We compute the local motivic cohomology of a point for$p=2$and study thelocal Chow motivic category. We introducelocal Chow groupsand conjecture that over flexible fields these should coincide withChow groups modulo numerical equivalence with$\mathbb{F} _p$-coefficients, which implies thatlocal Chow motivescoincide withnumerical Chow motives. We prove this conjecture in various cases.

Polymorphic types and effects with Boolean unification

We present a simple, practical, and expressive type and effect system based on Boolean constraints. The effect system extends the Hindley-Milner type system, supports parametric polymorphism, and preserves principal types modulo Boolean equivalence. We show how to support type inference by extending Algorithm W with Boolean unification based on the successive variable elimination algorithm. We implement the type and effect system in the Flix programming language. We perform an in-depth evaluation on the impact of Boolean unification on type inference time and end-to-end compilation time. While the computational complexity of Boolean unification is NP-hard, the experimental results demonstrate that it works well in practice. We find that the impact on type inference time is on average a 1.4x slowdown and the overall impact on end-to-end compilation time is a 1.1x slowdown.