System Of Notations Research Articles

Making Design Tools Like a Weaver: Four Rules

What would happen if we designed CAD systems like a weaver designs cloth? Drawing from our ongoing collaborations with weavers, we suggest four rules to bring these qualities to your own practice: follow the materials, privilege the present and personal, form kinships with the past, and design systems of notations.

A 39,600-year-old leather punch board from Canyars, Gavà, Spain

Puncture alignments are found on Palaeolithic carvings, pendants, and other fully shaped osseous artifacts. These marks were interpreted as abstract decorations, system of notations, and features present on human and animal depictions. Here, we create an experimental framework for the analysis and interpretation of human-made punctures and apply it to a highly intriguing, punctured bone fragment found at Canyars, an Early Upper Palaeolithic coastal site from Catalonia, Spain. Changes of tool and variation in the arrangement and orientation of punctures are consistent with the interpretation of this object as the earliest-known leather work punch board recording six episodes of hide pricking, one of which was to produce a linear seam. Our results indicate that Aurignacian hunters-gatherers used this technology to produce leather works and probably tailored clothes well before the introduction of bone eyed needles in Europe 15,000 years later.

Ukrainian historical thought in the systems of notations and hagiographies of the Renaissance epoch educators

У статті обґрунтована необхідність залучення біографічного матеріалу для уведення в історію українського гуманітарного дискурсу імен представників педагогічного Відродження України (XIV-XV) - просвітителів, які презентують Україну світові. Автор статті основним арсеналом дослідницького пошуку використовує вперше як об'єкт дослідження системи нотацій та агіографій видатних просвітителів українського Відродження. Розглядається діяльність і творча спадщина видатних представників педагогічного компендіуму часів України XIV-XV століть у системах нотацій і агіографій: Павла Русина (бл. 1470-1517), Юрія Котермака (Дрогобича) (бл. 1450-1494) та ін. Автор статті реалізує біографічний метод для вивчення колективних біографій представників видатних просвітителів українського Відродження в історії української наукової думки. Творчість кожного діяча розглянута як унікальне явище й у той самий час розуміється як складова всього процесу розвитку історичної наукової думки України. Узагальнюється в такому випадку не тільки результат виявлення понять доля, життєвий шлях, модель світу, зовнішня і внутрішня детермінація, а також власне середовище, суб'єктивна картина життєвого світу, але й моменти взаємозв'язку дії всіх понять у життєписі персоналій освітян. У статті відстежується походження терміна агіографія через поняття колективна біографія. Охарактеризовано ідею міждисциплінарного підходу в дослідженні життя людини на матеріалі праць, які існують у філософському, культурологічному, психологічному та історичному дискурсі сучасної наукової думки на пострадянському просторі. Запропоновано подальшу перспективу в застосуванні описаного інструментарію в науці.

A Unique Assemblage of Engraved Plaquettes from Ein Qashish South, Jezreel Valley, Israel: Figurative and Non-Figurative Symbols of Late Pleistocene Hunters-Gatherers in the Levant.

Three engraved limestone plaquettes from the recently excavated Epipaleolithic open-air site Ein Qashish South in the Jezreel Valley, Israel comprise unique evidence for symbolic behavior of Late Pleistocene foragers in the Levant. The engravings, uncovered in Kebaran and Geometric Kebaran deposits (ca. 23ka and ca. 16.5ka BP), include the image of a bird—the first figurative representation known so far from a pre-Natufian Epipaleolithic—along with geometric motifs such as chevrons, crosshatchings and ladders. Some of the engravings closely resemble roughly contemporary European finds interpreted as "systems of notations" or "artificial memory systems"–records related to timing of seasonal resources and associated aggregation events of nomadic groups. Moreover, similarly looking signs and patterns are well known from the context of the local Natufian—a final Epipaleolithic culture of sedentary or semi-sedentary foragers who started practicing agriculture. The investigation of the engravings found in Ein Qashish South involves conceptualizations developed in studies of European and local parallels, a selection of ethnographic examples and preliminary microscopic observations of the plaquettes. This shows that the figurative and non-figurative images comprise a coherent assemblage of symbols that might have been applied in order to store, share and transmit information related to social and subsistence realms of mobile bands. It further suggests that the site functioned as a locality of groups' aggregation and indicates social complexity of pre-Natufian foragers in the Levant. While alterations in social and subsistence strategies can explain the varying frequency of image use characterizing different areas of the Late Pleistocene world—the apparent similarity in graphics and the mode of their application support the possibility that symbol-mediated behavior has a common and much earlier origin.

Identification of target system operations. Determination of the time of the actual completion of the target operation

It was found in the paper that the time frames of the studied system operation depend on the research objective. In cases when it comes to problems, related to the physical movement of the input and output products, limits of the study are defined by the time of the beginning of the movement of the first input product of the operation and ends with the time of issue of the last output product of the operation by the system. Time of completion of the physical movement of products is defined in the paper as the time of the physical completion of the operation. Since the purpose of any operation is to increase the value of the output products in relation to the value of input products of the operation, in cases when there is a problem of investigating the target operation, the time of completion of the target operation is determined not by the physics of the process, but its cybernetics. Since for the time of the operation, conversion processes link input products in time, the added value of output products is intended to compensate in time the costs of temporary binding of input products that initially have their value. The time, when the added value of output products of the target operation compensates the value of tight input products for the time of the operation, is defined in the paper as the time of the actual completion of the target operation. A system of notations to describe the system processes of target operations was proposed. The expressions for the numerical and analytical determination of the time of actual completion of the target operation were obtained. A link to the resource, with examples of calculations of the time of the actual completion of the operation with the use of numerical methods and analytic expressions, obtained in the paper was given.

Rice and Rice-Shapiro Theorems for transfinite correction grammars

Hay and, then, Johnson extended the classic Rice and Rice-Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work (with some motivations presented) to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, . Other cases are done for all transfinite notations in a very natural, proper subsystem of , where has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in .

Compartmental modelling for magnetic resonance renography

A basic formalism is presented for generating and interpreting compartmental models for dynamic contrast-enhanced MRI data in the kidney. A graphical convention is introduced to represent and design compartmental models in a transparent and physically intuitive manner. A systematic system of notations and a simple set of rules allows direct translation of the graphical representation into a mathematical solution. The rules are derived from the physical principle of mass conservation, and the interpretation provided by the general tracer-kinetic theory of linear and stationary systems. The power of the formalism is illustrated using examples of models that have been proposed in the literature on perfusion MRI, and by generating a number of advanced models that may be of use in the kidney.

Learning correction grammars

AbstractWe investigate a new paradigm in the context of learning in the limit, namely, learningcorrection grammarsfor classes ofcomputably enumerable (c.e.)languages. Knowing a language may feature a representation of it in terms oftwogrammars. The second grammar is used to make corrections to the first grammar. Such a pair of grammars can be seen as a single description of (or grammar for) the language. We call such grammarscorrection grammars. Correction grammars capture the observable fact that peopledocorrect their linguistic utterances during their usual linguistic activities.We show that learning correction grammars for classes of c.e. languages in theTxtEx-mode(i.e., converging to a single correct correction grammar in the limit) is sometimes more powerful than learning ordinary grammars even in theTxtBc-model (where the learner is allowed to converge to infinitely many syntactically distinct but correct conjectures in the limit). For eachn≥ 0. there is a similar learning advantage, again in learning correction grammars for classes of c.e. languages, but where we compare learning correction grammars that maken+ 1 corrections to those that makencorrections.The concept of a correction grammar can be extended into the constructive transfinite, using the idea of counting-down from notations for transfinite constructive ordinals. This transfinite extension can also be conceptualized as being about learning Ershov-descriptions for c.e. languages. Forua notation in Kleene's general system (O, <o) of ordinal notations for constructive ordinals, we introduce the concept of anu-correction grammar, whereuis used to bound the number of corrections that the grammar is allowed to make. We prove a general hierarchy result: ifuandvare notations for constructive ordinals such thatu<ov. then there are classes of c.e. languages that can beTxtEx-learned by conjecturingv-correction grammars but not by conjecturingu-correction grammars.Surprisingly, we show that—above “ω-many” corrections—it is not possible to strengthen the hierarchy:TxtEx-learningu-correction grammars of classes of c.e. languages, whereuis a notation inOforanyordinal, can be simulated byTxtBc-learningw-correction grammars, wherewis any notation for the smallest infinite ordinalω.

Operational set theory and small large cardinals

A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory is expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations. OST is equivalent in strength to admissible set theory, and a natural extension of OST is equivalent in strength to ZFC. The language of OST provides a framework in which to express “small” large cardinal notions—such as those of being an inaccessible cardinal, a Mahlo cardinal, and a weakly compact cardinal—in terms of operational closure conditions that specialize to the analogue notions on admissible sets. This illustrates a wider program whose aim is to provide a common framework for analogues of large cardinal notions that have appeared in admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics, and systems of recursive ordinal notations that have been used in proof theory.

Große Zahlen in den altakkadischen Königsinschriften

The Old Akkadian system of number notations is still badly understood and – as the author shows – has obviously been so already in antiquity. The system in use differs considerably from the older and even from the contemporary Sumerian one, and previous studies did not result in satisfying explanations of the contradictions and incompatibilities. Among the main problems are the summation of individual numbers, which is often incompatible with the overall sum, as well as misreadings in the texts, which the author is able to improve on several occasions. The discussion of the sources at hand leads the author to the correct interpretation of the system as one of “mixed-decimal numbers”, eliminating this way the main problems in the understanding of the Old Akkadian system of number notations.

Paradigms of Denotational Mathematics for Cognitive Informatics and Cognitive Computing

The abstract, rigorous, and expressive needs in cognitive informatics, intelligence science, software science, and knowledge science lead to new forms of mathematics collectively known as denotational mathematics. Denotational mathematics is a category of expressive mathematical structures that deals with high level mathematical entities beyond numbers and sets, such as abstract objects, complex relations, behavioral information, concepts, knowledge, processes, and systems. Denotational mathematics is usually in the form of abstract algebra that is a branch of mathematics in which a system of abstract notations is adopted to denote relations of abstract mathematical entities and their algebraic operations based on given axioms and laws. Four paradigms of denotational mathematics, known as concept algebra, system algebra, Real-Time Process Algebra (RTPA), and Visual Semantic Algebra (VSA), are introduced in this paper. Applications of denotational mathematics in cognitive informatics and computational intelligence are elaborated. Denotational mathematics is widely applicable to model and manipulate complex architectures and behaviors of both humans and intelligent systems, as well as long chains of inference processes.

Assignment of ordinals to patterns of resemblance

AbstractIn [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ1-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ1-elementarily as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective order isomorphism between the classical and the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization.

The power of notation

A system of notations for tracing programs devised by the author is shown to be of use in modeling operations on pointers. Typical examples using linked lists are provided.

A comparison of two systems of ordinal notations

The standard method of generating countable ordinals from uncountable ordinals can be replaced by a use of fixed point extractors available in the term calculus of Howard’s system. This gives a notion of the intrinsic complexity of an ordinal analogous to the intrinsic complexity of a function described in Godel’s T.

A jump operator for subrecursion theories

In classical recursion theory, the jump operator is an important concept fundamental in the study of degrees. In particular, it provides a way of defining the hyperarithmetic hierarchy by iterating over Kleene's O. In subrecursion theories, hierarchies (variants of the fast growing hierarchy) are also important in underlying the central concepts, e.g. in classifying provably recursive functions and associated independence results for given theories (see, e.g. [BW87], [HW96], [R84] and [Z77]). A major difference with the hyperarithmetic hierarchy is in the way each level of a subrecursive hierarchy is crucially dependent upon the system of ordinal notations used (see [F62]). This has been perhaps the major stumbling block in finding a classification of all general recursive functions using such hierarchies.Here we present a natural subrecursive analogue of the jump operator and prove that the hierarchy based on the ”subrecursive jump” is elementarily equivalent to the fast growing hierarchy.The paper is organised as follows. First the preliminary definitions are given together with a statement of the main theorem and a brief outline of its proof. The proof of the theorem is then given, with the more technical parts separated out as facts which are proven afterwards.We let {e}g(x) denote computation of the e-th partial recursion with oracle g, on input x. Furthermore [e] denotes the e-th elementary recursive function, defined so thatSimilarly, for a given oracle g the e-th relativized elementary function is denoted by [e]g.

Ordinals II: Some applications and a functorial approach

The first part of this paper is a presentation of some common applications of ordinals: definition of a system of ordinal notations for ordinals less than Γ0, direct connection between Kruskal's theorem and Γ0, consistency proofs in proof theory (such as the consistency of Peano arithmetic by means of transfinite induction up to e0). In the second part of the paper, a functorial construction of ordinals and in particular of the Veblen hierarchy is explained. This approach, introduced by Girard (theory of dilators), allows the construction of ordinals greater than Γ0 to be pursued in a more natural way than if the Bachmann hierarchy is used.

Elementary descent recursion and proof theory

We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.

The CHILDES project: tools for analyzing talk

Language research thrives on data collected from spontaneous interactions in naturally occurring situations. However, the process of collecting, transcribing, and analyzing naturalistic data can be extremely time-consuming and often unreliable. This book describes three basic tools for language analysis of transcript data by computer that have been developed in the context of the Child Language Data Exchange System (CHILDES) project. These are: the CHAT transcription and coding format, the CLAN package of analysis programs, and the database. These tools have brought about significant changes in the way research is conducted in the child language field. They are being used with great success by researchers working with second language learning, adult conversational interactions, sociological content analyses, and language recovery in aphasia, as well as by students of child language development. tools are widely applicable, although this book concentrates on their use in the child language field, believing that researchers from other areas can make the necessary analogies to their own topics. This thoroughly revised 2nd edition includes documentation on a dozen new computer programs that have been added to the basic system for transcript analysis. most important of these new programs is the Text Editor (CED) which can be used for a wide variety of purposes, including editing non-Roman orthographies, systematically adding codes to transcripts, checking the files for correct use of CHAT, and linking the files to digitized audio and videotape. In addition to information on the new computer programs, the manual documents changed the shape of the CHILDES/BIB system--given a major update in 1994--which now uses a new computer database system. documentation for the transcript database has been updated to include new information on old corpora and information on more than a dozen new corpora from many different languages. Finally, the system of CHAT notations for file transcript have been clarified to emphasize the ways in which the codes are used by particular CLAN programs. new edition concludes with a discussion of new directions in transcript analysis and links between the database and other developments in multimedia computing and global networking. It also includes complete references organized by research topic area for the more than 300 published articles that have made use of the database and/or the CLAN programs. LEA also distributes the CLAN programs and the complete Database--including corpora from several languages and discourse situations--described in The CHILDES Project. Be sure to choose the correct platform (IBM or Macintosh) for the CLAN programs; the Database CD-ROM runs on both platforms.

A NOTATION SYSTEM FOR ORDINAL USING Ψ‐FUNCTIONS ON INACCESSIBLE MAHLO NUMBERS

AbstractG. Jäger gave in Arch. Math. Logik Grundlagenforsch. 24 (1984), 49‐62, a recursive notation system on a basis of a hierarchy Iαß of α‐inaccessible regular ordinals using collapsing functions following W. Buchholz in Ann. Pure Appl. Logic 32 (1986), 195‐207. Jäger's system stops, when ordinals α with Iα0 = α enter. This border is now overcome by introducing additional a hierarchy Jαß of weakly inaccessible Mahlo numbers, which is defined similarly to the Jäger hierarchy. An ordinal μ is called Mahlo, if every normal‐function f : μ → μ has regular fixpoints. Collapsing is defined for both Mahlo and simply regular ordinals such that for every Mahlo ordinal μ out of the J‐hierarchy Ψμα is a regular σ such that Iσ0 = σ. For these regular σ again collapsing functions Ψσ are defined. To get a proper systematical order into the collapsing procedure, a pair of ordinals is associated to σ and α, and the definition of Ψσα is given by recursion on a suitable well‐ordering of these pairs. Thus a fairly large system of ordinal notations can be established. It seems rather straightforward, how to extend this setting further.

Systems of iterated projective ordinal notations and combinatorial statements about binary labeled trees

We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding “impredicative” Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our ordinal notations arearbitrary terms in the appropriate simple term algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ( ), for any primitive recursive limit wellordering . In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ( ) the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT( ) is a proper subsystem ofPRJ( ), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID< λ, for arbitrarily large limit ordinalλ. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]).