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A suitably programmed quantum computer should act on a number x to produce another number f(x) for some specified function f. Appropriately interpreted, with an accuracy that increases with increasing k, we can treat such numbers as non-negative integers less than 2k. Each integer is represented in the quantum computer by the corresponding computational-basis state of k Qbits.
If we specify the numbers x as n-bit integers and the numbers f(x) as m-bit integers, then we shall need at least n + m Qbits: a set of n-Qbits, called the input register, to represent x, and another set of m-Qbits, called the output register, to represent f(x). Qbits being a scarce commodity, you might wonder why we need separate registers for input and output. One important reason is that if f(x) assigns the same value to different values of x, as many interesting functions do, then the computation cannot be inverted if its only effect is to transform the contents of a single register from x to f(x). Having separate registers for input and output is standard practice in the classical theory of reversible computation. Since quantum computers must operate reversibly to perform their magic (except for measurement gates), they are generally designed to operate with both input and output registers. We shall find that this dual-register architecture can also be usefully exploited by a quantum computer in some strikingly nonclassical ways.
Simon's problem (Section 2.5) starts with a subroutine that calculates a function f(x), which satisfies f(x) = f(y) for distinct x and y if and only if y = x ⊕ a, where ⊕ denotes the bitwise modulo-2 sum of the n-bit integers a and x. The number of times a classical computer must invoke the subroutine to determine a grows exponentially with n, but with a quantum computer it grows only linearly.
This is a rather artificial example, of interest primarily because it gives a simple demonstration of the remarkable computational power a quantum computer can possess. It amounts to finding the unknown period a of a function on n-bit integers that is “periodic” under bitwise modulo-2 addition. A more difficult, but much more natural problem is to find the period r of a function f on the integers that is periodic under ordinary addition, satisfying f(x) = f(y) for distinct x and y if and only if x and y differ by an integral multiple of r. Finding the period of such a periodic function turns out to be the key to factoring products of large prime numbers, a mathematically natural problem with quite practical applications.
We illustrate here the mathematics of the final (post-quantum-computational) stage of Shor's period-finding procedure. The final measurement produces (with high probability) an integer y that is within ½ of an integral multiple of 2n/r, where n is the number of Qbits in the input register, satisfying 2n > N2 > r2. Deducing the period r of the function f from such an integer y makes use of the theorem that if x is an estimate for the fraction j/r that differs from it by less than ½r2, then j/r will appear as one of the partial sums in the continued-fraction expansion of x. In the case of Shor's period finding algorithm x = y/2n. If j and r happen to have no factors in common, r is given by the denominator of the partial sum with the largest denominator less than N. Otherwise the continued-fraction expansion of x gives r0: r divided by whatever factor it has in common with the random integer j. If several small multiples of r0 fail to be a period of f, one repeats the whole procedure, getting a different submultiple r1 of r. There is a good chance that r will be the least common multiple of r0 and r1, or a not terribly large multiple of it. If not, one repeats the whole procedure a few more times until one succeeds in finding a period of f.
Until recently, use has been made almost exclusively of text-based concordancers in the analysis of spoken corpora. This article discusses research being carried out on Padua University's Multimedia English Corpus (Padova MEC) using the multimodal concordancer MCA (Multimodal Corpus Authoring System, Baldry, 2005). This highly innovative concordancer enables the retrieval of parts of video and audio from a tagged corpus and access to examples of language in context, thereby providing non-verbal information about the environment, the participants and their moods, details that can be gleaned from a combination of word, sound, image and movement. This is of use to language learners of all levels because if “communication is to be successful, a relevant context has to be constructed by the discourse participants” (Braun, 2005: 52). In other words, transcripts alone are not sufficient if learners are to have anything like participant knowledge and comprehend spoken language. In the article it will be demonstrated how language functions expressed in the multimedia corpus of spoken English are retrieved using MCA. Online learning materials based on the multimodal concordances take into consideration not only language, but also the way in which it co-patterns with other semiotic resources, thereby raising the issue of the importance of learner awareness of the multimodal nature of communication.
This paper reports on an empirical case study conducted to investigate the overall conditions and challenges of integrating corpus materials and corpus-based learning activities into English-language classes at a secondary school in Germany. Starting from the observation that in spite of the large amount of research into corpus-based language learning, hands-on work with corpora has remained an exception in secondary schools, the paper starts by outlining a set of pedagogical requirements for corpus integration and the approach which has formed the basis for designing the case study. Then the findings of the study are reported and discussed. As a result of the methodological challenges identified in the study, the author argues for a move from ‘data-driven learning’ to needs-driven corpora, corpus activities and corpus methodologies.
This study introduces a new computer-based methodology, ‘concgramming’, that has as its primary aim the automatic identification of the phraseological profile and hence the ‘aboutness’, of a text or corpus. It is argued that this methodology can be employed by language learners and teachers to raise awareness of the importance of the phraseological tendency in language. The methodology is outlined, and examples of its potential for use by language learners in a data-driven learning mode are described. The wider implications of concgramming, and the concgrams so generated, are also discussed with regard to CALL.
Corpora and concordancing have become much more widely available as researchers recognise that they can significantly enrich the language learning environment. There is still, however, a strong resistance towards corpus use by teachers and learners (Römer, 2006:122). An understanding of the implications and relevance of corpus use for pedagogy may help teachers and learners overcome this resistance, and hence accelerate the process of “percolation” (McEnery & Wilson, 1997:5) or the “trickle down” (Leech, 1997:2) of corpus research to language teaching and learning. The pedagogical context in which learners' consultation of corpora (corpus consultation literacy) can be developed is fundamental in understanding this new literacy and developing it so that it leads to successful language teaching and learning. This paper seeks to investigate the role which corpus consultation literacy plays in enhancing the language learning process and, consequently, aims to establish whether this new literacy can contribute to a process-oriented approach to language learning. Firstly, a theoretical overview of a process-oriented approach to language learning will be outlined, before investigating if corpus consultation can potentially enhance such an approach. This will be supported by evidence from a number of published empirical studies, covering aspects such as learning within a constructivist framework, and the development of cognitive and metacognitive skills through the use of cognitive and developmental tools. Learners' comments from related studies, namely Chambers and O'Sullivan (2004), O'Sullivan (2006), and O'Sullivan and Chambers (2006), which pertain to the learning process and the influence of corpus consultation literacy on this same process, will also be considered. The hypothesis presented here is that corpus consultation literacy can enhance a process-oriented approach to language teaching and learning. It is envisaged that this research will contribute towards the establishment of a sound theoretical and pedagogical foundation for the integration of corpus consultation literacy into language teaching and learning.
Corpora have been used for pedagogical purposes for more than two decades but empirical studies are relatively rare, particularly in the context of grammar teaching. The present study focuses on students' attitudes towards grammar and how these attitudes are affected by the introduction of concordancing. The principal aims of the project were to increase the students' motivation by showing them that English grammar is more than a set of rules in a book and to enable them to assume more responsibility for their own learning. The idea was to introduce the use of language corpora into the curriculum for first-semester English at Växjö University in Sweden, as a complement to grammar textbooks and ordinary exercise materials. Between classes, the students worked with problem-solving assignments that involved formulating their own grammar rules based on the examples they found in the corpus. In the classroom, a system of peer teaching was applied, where the students took turns at explaining grammatical rules to each other. Besides presenting a new way of working with grammar, we also provided the students with a tool for checking questions of usage when writing English texts in the future, since the corpus we use is free of charge and available to all. The work with corpora and peer teaching was evaluated by means of questionnaires and interviews. This article describes and evaluates this initiative and presents insights gained in the process. One important conclusion is that using corpora with students requires a large amount of introduction and support. It takes time and practice to get students to become independent corpus users, knowing how to formulate relevant corpus queries and interpret the results. Working with corpora is a method that some students appreciate while others, especially weak students, find it difficult or boring. Several of the students did not find corpora very useful for learning about grammatical rules, but realized the potential of using corpora when writing texts in English.
Learner corpora, electronic collections of spoken or written data from foreign language learners, offer unparalleled access to many hitherto uncovered aspects of learner language, particularly in their error-tagged format. This article aims to demonstrate the role that the learner corpus can play in CALL, particularly when used in conjunction with web-based interfaces which provide flexible access to error-tagged corpora that have been enhanced with simple NLP techniques such as POS-tagging or lemmatization and linked to a wide range of learner and task variables such as mother tongue background or activity type. This new resource is of interest to three main types of users: teachers wishing to prepare pedagogical materials that target learners' attested difficulties; learners themselves for editing or language awareness purposes and NLP researchers, for whom it serves as a benchmark for testing automatic error detection systems.