Homomorphic Representation Research Articles

Ordering motivation and Likert scale ratings: When a numeric scale is not necessarily better.

Measuring psychological attributes, such as motivation, typically involves rating scales, assuming that an attribute can be ordered, and that ratings represent this order. Previously, only the first assumption had been tested, albeit limited. First, we checked the ordinal structure of motivation, looking at whether people can establish transitive relations between motivation levels in pairwise comparisons; and we found different ordering patterns: strict transitive, weak transitive, changing order, and intransitivity. The rate of intransitivity was similar to that found previously and was somewhat higher than we obtained when we asked participants to compare definitely quantitative attributes (such as weight). Second, we checked if specific ordering patterns were related to individual interpretations of the statements that deviated from expected motivation types. Indeed, about a third of participants miscategorized statements, and these deviant interpretations were related to intransitivity as well as weak transitivity. Third, we checked whether Likert ratings represent the order of motives obtained from pairwise comparisons. We found rather homomorphic representation: ratings correlated with the order, but they did not differentiate between different ordering patterns and hierarchies of motives. We conclude that the Likert rating scale provides less information about respondents than pairwise ordering. The findings question the mainstream practice of using rating scales without testing underlying assumptions.

The Role of Non-Positional Arithmetic on Efficient Emerging Cryptographic Algorithms

Number representation systems establish ways in which numbers are mapped to computer architectures, and how operations over the numbers are translated into computer instructions. The efficiency of public-key cryptography is strongly affected by the used number representations, as these systems are constructed from mathematically inspired problems to ensure security, and thus rely on operations over large integers. In this paper, unconventional representations systems, including the Residue Number System (RNS) and stochastic number representations, are systematically reviewed. Homomorphic representations, which allow for parties to operate on data without having access to their plaintext representation, are also considered. The main goal of this survey is to introduce the reader to key aspects of non-traditional number representations that may be exploited for public-key cryptography, without delving too much into the details. Examples of the methods and algorithms herein surveyed include subquadratic modular multiplication for isogeny-based cryptography, the acceleration of Goldreich-Goldwasser-Halevi (GGH) decryption by an order of magnitude, the improvement of the Direct Anonymous Attestation (DAA) protocol both in terms of storage requirements and the time taken to execute it, and efficient algorithm-hiding Fully Homomorphic Encryption (FHE). The implementation of this type of systems in both sequential and parallel platforms is analysed, and their performance is compared with traditional approaches. We hope this work sows the seed of further research on the application of non-positional number arithmetic to other cryptographic use-cases.

When is a bunch of marks on paper a diagram? Diagrams as homomorphic representations

When is a bunch of marks on paper a diagram? Diagrams as homomorphic representations

Coordination of parallel partially ordered processes

Isomorphic representations of finite distributive lattices by lattices of functions and fuzzy sets are constructed. These class es the case of finite lattices are the generalization of the representation of distributive lattices by set lattices whose existence is proved by Birkhoff’s theorem. Homomorphic image and subdirect representations of finite distributive lattices are characterized, and figure of two-dimensional distributive lattices are geometrically characterized. This theoretical basis underlies the general description of systems of linear and partially ordered parallel discrete processes with mutual constraints on step order acting between them. The application of the developed tools is demonstrated on the examples from the graph theory, parallel calculations, artificial intelligence, and power engineering.

A system simulating representation change phenomena while problem solving

This paper deals with a computer programme offering a valid simulation of cognitive human processes related to representation changes while problem solving. An introductory and brief recollection of preceding contributions by psychologists, cognitive scientists and AI researchers provides the necessary background and motivation for our work. A relevant trend of the present research concerns the formal mathematical study of representation phenomena, including precisions about isomorphic and homomorphic representation changes. In order to simplify problem resolution processes, the implemented system performs certain representation changes, by making use of specific procedures which assign and modify a certain relevance value to every attribute involved in the problem representation, on behalf of their respective importance so as to actually solve the problem. The paper includes a full account of mathematical definitions and propositions involved in this system.

Defending the structural concept of representation

The paper defends the structural concept of representation, defined by homomorphisms, against the main objections that have been raised against it: Logical objections, the objection from misrepresentation, the objection from failing necessity, and the copy theory objection. Homomorphic representations are not necessarily 'copies' of their representanda, and thus can convey scientific insight.

APPROXIMATING DEPENDENCY GRAMMARS THROUGH INTERSECTION OF STAR-FREE REGULAR LANGUAGES

The paper formulates the Hays and Gaifman dependency grammar (HGDG) in terms of constraints on a string based encoding of dependency trees and develops an approach to obtain a regular approximation for these grammars. Our encoding of dependency trees uses brackets in a novel fashion: pairs of brackets indicate dependencies between pairs of positions rather than boundaries of phrases. This leads to several advantages: (i) HGDG rules over the balanced bracketing can be expressed using regular languages. (ii) A new homomorphic representation for context-free languages is obtained. (iii) A star-free regular approximation for the original projective dependency grammar is obtained by limiting the number of stacked dependencies. (iv) By relaxing certain constraints, the encoding can be extended to non-projective dependency trees and graphs, (v) strong generative power of HGDGs can now be characterized through sets of bracketed strings.

Parallelizing functional programs by generalization

List homomorphisms are functions that are parallelizable using the divide-and-conquer paradigm. We study the problem of finding homomorphic representations of functions in the Bird–Meertens constructive theory of lists, by means of term rewriting and theorem proving techniques. A previous work proved that to each pair of leftward and rightward sequential representations of a function, based on cons- and snoc-lists, respectively, there is also a representation as a homomorphism. Our contribution is a mechanizable method to extract the homomorphism representation from a pair of sequential representations. The method is decomposed to a generalization problem and an inductive claim, both solvable by term rewriting techniques. To solve the former we present a sound generalization procedure which yields the required representation, and terminates under reasonable assumptions. The inductive claim is provable automatically. We illustrate the method and the procedure by the systematic parallelization of the scan-function (parallel prefix) and of the maximum segment sum problem.

Extracting and implementing list homomorphisms in parallel program development

Homomorphisms are functions that match the divide-and-conquer pattern and are widely used in parallel programming. Two problems are studied for homomorphisms on lists: 1. (1) parallelism extraction: finding a homomorphic representation of a given function; 2. (2) parallelism implementation: deriving an efficient parallel program that computes the function. The proposed approach to parallelism extraction starts by writing two sequential programs for the function, on traditional cons lists and on dual snoc lists; the parallel program is obtained by generalizing sequential programs as terms. For almost-homomorphic functions, e.g., the maximum segment sum problem, our method provides a systematic embedding into a homomorphism. The implementation problem is addressed by introducing the class of distributable homomorphisms and deriving for them a common parallel program schema. The derivation is based on equational reasoning in the Bird-Meertens formalism, which guarantees the correctness of the parallel target program. The approach is illustrated with the function scan (parallel prefix), for which the combination of our two systematic methods yields the optimal hypercube algorithm, usually presented ad hoc in the literature.

Can a quantum process be oblivious to observation?

In this paper time is indexed by a “time group” G that may be either the real line or the integers. A quantum process is a homomorphic representation of G via unitary operators on a Hilbert space L, with the usual continuity restrictions if time is indexed by the reals. Consider a quantum process subject to nonselective observation at all points in time via an observation operator with purely discrete eigenvalues and corresponding simple eigenvectors. Suppose that the process evolves in such a way that the results of observation at a given time are the same, whether or not there have been previous observations. Suppose further that L is spanned by the set of vectors invariant modulo phase under the quantum evolution. If the time group is the integers, then the process is observed to evolve in a classical manner. If the time group is the real line, then the process is never observed to undergo a change of state. The results of this paper are essentially a corollary of von Neumann’s mean ergodic theorem.

Further Remarks on Reduced Languages

Union, homomorphic and inverse homomorphic representations of languages are obtained, starting from reduced languages introduced in earlier formal language approaches to parallel program schemes. The reduced-ness decidability is also investigated (it is decidable for regular and undecidable for linear languages).

On α 0-ν 1-products of automata

The power of α 0-products, ν 1-products, and α 0-ν 1-products of automata is compared. As one of the main results it is shoen that homomorphic representation power of α 0-ν 1-products is the same as that of general products if infinite products are permitted. A detailed analysis of the situation when only finite products are allowable is also provided.

A decidability result for homomorphic representation of automata by 205-1205-1205-1

A decidability result for homomorphic representation of automata by 205-1205-1205-1

KNOWLEDGE REPRESENTATION FROM NEWTON TO MINSKY AND BEYOND

Abstract An attempt is made to see recent developments in knowledge representation, clustered around Minsky 's theory of frames, and the more classical modeling of natural processes, originating in Newton's mechanics, in a unified perspective. Defining knowledge as homomorphic representation, we view classical models as capable of providing behavioral knowledge but lacking in capability to represent important structural knowledge and meta-knowledge. He discuss a framework, inspired by system theoretic concepts, for combining structural and behavioral representation in a form suitable for goal-directed model construction. We conclude by drawing several implications of this approach for system design, artificial intelligence, modeling, and simulation.

A homomorphic representation of simple matrix languages

In this paper we given a homomorphic representation of simple matrix languages similar in form to those of context-free languages, namely: Any simple matrix language L can be written as the homomorphic image of the intersection of the Dyck context-free language and an equal matrix language. We do not obtain a characterization, because the converse is not true.

Functional analytic properties of extremely amenable semigroups

Introduction. Let S be a semigroup m(S) the Banach space of all bounded real functions on S with the norm lf l =sup {ff(s)I; s E S} and m(S)* the conjugate Banach space of m(S). 1 E m(S)* is a mean if > O (i.e. +(f)O> iff O, fe m(S)) and q(1) = 1 (1 stands also for the constant one function on S). The semigroup S is left amenable (LA) if there exists a mean 1 E m(S)* which is left invariant i.e. q(fa) = +(f) for all a E S, f r m(S) (we define fa(S) =f(as), fa(s) =f(sa), laf=fa, raf=fa, La = la*: m(S)* m> m(S)*, Ra = r* for all a, s E S and f cr m(S)). The semigroup S is extremely left amenable (ELA) if there is a left invariant mean 1 on m(S) which is multiplicative (i.e. such that q(fg) = q(f)b(g) for all f, g Er m(S)). In anology one defines (extremely) right amenable semigroups (ERA) RA. ELA semigroups have been studied at first by T. Mitchell in [22] (under different terminology) and afterwards by this author in [15], [16]. They are known to be characterized by the following equivalent properties: 1. S is ELA. 2. S has the common fixed point property on compacta (i.e. for any compact Hausdorff space and any homomorphic representation S' of S as a semigroup of continuous maps from X to X (under functional composition), there is some x0 E X such that s'(x0)=x0 for all s' E S' (Mitchell [22]). 3. For any a, b E S there is some c E S with ac= bc = c [16]. 4. For any function h of the ideal

Relativistic Combination of Internal and Spin Symmetries inS-Matrix Formulation

The relativistic combination of internal and spin symmetries is considered as a restriction on the scattering operator. A physical principle for the statement of symmetries other than space-time is formulated. In the simplest case, the intuitive idea that interactions do not depend on some observables is expressed by the statement that the $S$ operator belongs to a subring of the ring of observables. This principle leads to the usual results for purely internal symmetries, for spin independence of forces, and also for their combinations as in Wigner's supermultiplet theory. The combination of Poincar\'e group, spin independence, and internal symmetry leads to a group which is not a (finite-dimensional) Lie group and not locally compact. Nevertheless, simple homomorphic representations of the little group exist and suffice for physical applications. The nonrelativistic $\mathrm{SU}(6)$ multiplet structure and the selection rules are unchanged in the relativistic case. In addition to the deficiencies of the nonrelativistic model, the relativistic $S$ operator has highly unrealistic properties, although it is not trivial ($S\ensuremath{\ne}1$). A scheme for symmetry breaking is described which may be expected to remedy these deficiencies.