Non-isomorphic Representations Research Articles

Internalizing representation independence with univalence

In their usual form, representation independence metatheorems provide an external guarantee that two implementations of an abstract interface are interchangeable when they are related by an operation-preserving correspondence. If our programming language is dependently-typed, however, we would like to appeal to such invariance results within the language itself, in order to obtain correctness theorems for complex implementations by transferring them from simpler, related implementations. Recent work in proof assistants has shown that Voevodsky's univalence principle allows transferring theorems between isomorphic types, but many instances of representation independence in programming involve non-isomorphic representations. In this paper, we develop techniques for establishing internal relational representation independence results in dependent type theory, by using higher inductive types to simultaneously quotient two related implementation types by a heterogeneous correspondence between them. The correspondence becomes an isomorphism between the quotiented types, thereby allowing us to obtain an equality of implementations by univalence. We illustrate our techniques by considering applications to matrices, queues, and finite multisets. Our results are all formalized in Cubical Agda, a recent extension of Agda which supports univalence and higher inductive types in a computationally well-behaved way.

Non-isomorphism in efficient coding of complex sound properties

To the extent that sensorineural systems are efficient, stimulus redundancy should be captured in ways that optimize information transmission. Consistent with this principle, neural representations of sounds have been proposed to become "non-isomorphic," increasingly abstract and decreasingly resembling the original (redundant) input. Here, non-isomorphism is tested in perceptual learning using AXB discrimination of novel sounds with two highly correlated complex acoustic properties and a randomly varying third dimension. Discrimination of sounds obeying the correlation became superior to that of sounds violating it despite widely varying physical acoustic properties, suggesting non-isomorphic representation of stimulus redundancy.

Krull Dimension and Deviation in Certain Parafree Groups

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) − n, where μ(G) is the minimum possible number of generators of G. Let G be fg; then Hom(G, SL 2ℂ) inherits the structure of an algebraic variety, denoted by R(G). If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer.

Chimpanzees’ context-dependent tool use provides evidence for separable representations of hand and tool even during active use within peripersonal space

Considerable attention has been devoted to behaviors in which tools are used to perform actions in extrapersonal space by extending the reach. Evidence suggests that these behaviors result in an expansion of the body schema and peripersonal space. However, humans often use tools to perform tasks within peripersonal space that cannot be accomplished with the hands. In some of these instances (e.g., cooking), a tool is used as a substitute for the hand in order to pursue actions that would otherwise be hazardous. These behaviors suggest that even during the active use of tools, we maintain non-isomorphic representations that distinguish between our hands and handheld tools. Understanding whether such representations are a human specialization is of potentially great relevance to understand the evolutionary history of technological behaviors including the controlled use of fire. We tested six captive adult chimpanzees to determine whether they would elect to use a tool, rather than their hands, when acting in potentially hazardous vs. nonhazardous circumstances located within reach. Their behavior suggests that, like humans, chimpanzees represent the distinction between the hand vs. tool even during active use. We discuss the implications of this evidence for our understanding of tool use and its evolution.

Non-homeomorphic Nilmanifolds with Identical Unitary Spectrum

We show in every three-step nilpotent Lie groupGwith all coadjoint orbits flat the existence of a pair of discrete cocompact subgroupsΓ1andΓ2such thatΓ1\\GandΓ2\\Ghave the same unitary spectrum butΓ1is not isomorphic toΓ2. This result generalizes an example of Gornet. We mention without giving a proof a result that would enunciate that for a large category of subgroups of the four dimensional three-step chain group with non-flat coadjoint orbits, this phenomenon of non-isomorphic representation equivalence cannot occur. We also prove some short structural results for three-step nilpotent Lie groups with one-dimensional center.

Frog skin, time series, and other kinds of data: Nonisomorphic representation and analysis

Representational art is spatially congruent, or isomorphic, with objects in the world. In other words, they are the same in some important way. But not all representation is isomorphic. Data graphs are a type of nonisomorphic representation. A major goal of data graphs is to facilitate the rapid comprehension of differences or relations. This is done by providing a kind of analogy between something that can be “seen” (the relative heights of points) and some relevant information. This article discusses some rules about data graphing, then some new visualization tools are briefly critiqued.

Smith equivalence of representations for odd order cyclic groups

THEOREM A. There exist odd order cyclic groups which have Smith equivalent but nonisomorphic representations. After a short background discussion, we shall give a list of conditions (1.1) which are important in the development of this paper and state more precise versions of our results. Theorem B provides nonisomorphic Smith equivalent representations if one has nonisomorphic representations of G which satisfy (1.1). Corollary C shows the existence of infinitely many groups which have nonisomorphic representations which satisfy (1.1). The orders of some cyclic groups of this type are described by Lemma 2.5 and Corollary 2.6. Taken together, Theorem B and Corollary C prove Theorem A. The concept of Smith equivalence is based upon a question posed by P. A. Smith in a 1960 survey article [32, p. 4061. In our terminology the question is whether Smith equivalent representations are linearly isomorphic. Atiyah-Bott [3] and Milnor [17] established an affirmative answer for cyclic groups of odd prime power order and for representations with semifree action. By definition a group acts semifreely if each point is left fixed by only the trivial group element or by each group element. An extension of this work by Sanchez [28] implied the answer also to be affirmative if G is cyclic of order pq where p and q are odd primes. Bredon [6] showed for 2-groups that Smith equivalent representations are isomorphic if their dimension is large in comparison to the order of the group. The first negative answers to Smith’s question were established by the second author for odd order abelian groups with at least four noncyclic Sylow subgroups. See [23], [24]. Subsequently a number of papers have established a negative answer for even order groups, but the case of odd order cyclic groups has remained open. For cyclic groups of even order see the papers [7], [25], [ 111, [3 13, For noncyclic abelian groups and nonabelian 2-groups, see [34] and [S]. There are many interesting results and ideas in these papers. Results and techniques used in answering Smith’s question varj markedly according to the parity of the order of G. In Remark 1.2(ii) we explain the difference of the geometric and number theoretic treatment of even and odd order groups as it arises from the Atiyah-Bott extension of the Lefschetz Fixed Point theorem. One interesting question, untouched here, is to relate the differentiable structure of a homotopy sphere Z and the isotropy representations of an action of G on Z with exactly two fixed points. Some general remarks on this appear in the problem section of the forthcoming proceedings of the 1983 conference on group actions at Boulder [393. It can be shown that the homotopy spheres with group actions that are constructed by our method are all standard spheres. The methods of Schultz (e.g. in [29] and other papers) are relevant to realizing these actions on exotic spheres.