Generalizable synthesis through unification

The generalizability of PBE solvers is the key to the empirical synthesis performance. Despite the importance of generalizability, related studies on PBE solvers are still limited. In theory, few existing solvers provide theoretical guarantees on generalizability, and in practice, there is a lack of PBE solvers with satisfactory generalizability on important domains such as conditional linear integer arithmetic (CLIA). In this paper, we adopt a concept from the computational learning theory, Occam learning, and perform a comprehensive study on the framework of synthesis through unification (STUN), a state-of-the-art framework for synthesizing programs with nested if-then-else operators. We prove that Eusolver, a state-of-the-art STUN solver, does not satisfy the condition of Occam learning, and then we design a novel STUN solver, PolyGen, of which the generalizability is theoretically guaranteed by Occam learning. We evaluate PolyGen on the domains of CLIA and demonstrate that PolyGen significantly outperforms two state-of-the-art PBE solvers on CLIA, Eusolver and Euphony, on both generalizability and efficiency.

Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra

Showing whether the longest-edge (LE) bisection of tetrahedra meshes degenerates the stability condition or not is still an open problem. Some reasons, in part, are due to the cost for achieving the computation of similarity classes of millions of tetrahedra. We prove the existence of tetrahedra where the LE bisection introduces, at most, 37 similarity classes. This family of new tetrahedra was roughly pointed out by Adler in 1983. However, as far as we know, there has been no evidence confirming its existence. We also introduce a new data structure and algorithm for computing the number of similarity tetrahedral classes based on integer arithmetic, storing only the square of edges. The algorithm lets us perform compact and efficient high-level similarity class computations with a cost that is only dependent on the number of similarity classes.

On a nonlinear relation for computing the overpartition function

In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function p (n). Computing p (n) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of p (n) that requires only the values of p (k) with k ≤ n/2. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p (n) to obtain a simple and fast computation of the value of p (n). This new method uses only (large) integer arithmetic and it is simpler to program.

LSTM Cell Implementation on FPGAs

This paper presents and discusses the implementation of an LSTM cell on an FPGA with an activation function inspired by the CORDIC algorithm. The realization is performed using both IEEE754 standard and 32-bit integer numbers. The case with floating-point arithmetic is analyzed with and without DSP blocks provided by the Xilinx design suite. The alternative implementation including the integer arithmetic was optimized for a minimal number of clock cycles. Presented implementation uses xc6slx150t-2fgg900 and achieves high calculations accuracy for both cases.

Lexicographic Unranking of Combinations Revisited

In the context of combinatorial sampling, the so-called “unranking method” can be seen as a link between a total order over the objects and an effective way to construct an object of given rank. The most classical order used in this context is the lexicographic order, which corresponds to the familiar word ordering in the dictionary. In this article, we propose a comparative study of four algorithms dedicated to the lexicographic unranking of combinations, including three algorithms that were introduced decades ago. We start the paper with the introduction of our new algorithm using a new strategy of computations based on the classical factorial numeral system (or factoradics). Then, we present, in a high level, the three other algorithms. For each case, we analyze its time complexity on average, within a uniform framework, and describe its strengths and weaknesses. For about 20 years, such algorithms have been implemented using big integer arithmetic rather than bounded integer arithmetic which makes the cost of computing some coefficients higher than previously stated. We propose improvements for all implementations, which take this fact into account, and we give a detailed complexity analysis, which is validated by an experimental analysis. Finally, we show that, even if the algorithms are based on different strategies, all are doing very similar computations. Lastly, we extend our approach to the unranking of other classical combinatorial objects such as families counted by multinomial coefficients and k-permutations.

Penggunaan Media Lego Bricks Untuk Meningkatkan Kemampuan Kognitif Matematika Pada Materi Operasi Hitung Bilangan Bulat

This study aims to improve mathematical cognitive abilities in integer arithmetic operations through the use of lego bricks for fifth grade students of SDN 02 Winongo Madiun. This type of research is a classroom action research conducted in two cycles and each cycle is carried out twice. The research subjects were all fifth grade students totaling 17 SDN 02 Winongo Madiun. The research data were collected through observation, written tests in the form of pretest and posttest and documentation. The results of this study indicate that the Lego brick media can improve the cognitive ability of mathematics in integer arithmetic operations. This increase was marked by an increase in the number of students with test scores that exceeded the KKM, with scores between 70 and 100.With an increase in percentage of 35.5% of students who scored above the KKM, from 41.17% of students who scored in the KKM to 76, 14%. Keyword: mathematical cognitive abilities, lego bricks, integer operations

PERKEMBANGAN PEMAHAMAN SISWA PADA MATERI OPERASI BILANGAN BULAT DALAM SETTING PEMBELAJARAN MATEMATIKA REALISTIK BERBANTUAN MEDIA “MOGER” DI KELAS IV

Abstrak Penelitian ini bertujuan Mendeskripsikan Perkembangan Pemahaman Siswa pada Operasi Hitung Bilangan Bulat dalam setting Pembelajaran Matematika Realistik berbantuan media “Mobil Bergerak” di kelas IV.Penelitian ini deskriptif kualitatif dan kuantitatif, penelitian yang dilakukan dengan menjelaskan atau menggambarkan variabel masa lalu dan sekarang yang bertujuan untuk membuat deskripsi, gambaran atau lukisan secara sistematis, faktual dan akurat mengenai fakta-fakta serta hubungan antara fenomena yang diselidiki. Hasil dari Tes pertama, tes kedua, dan tes ketiga siswa mengalami peningkatan ditunjukkan dengan nilai tertinggi dari 45 ke 77,5 dan ke 100, Nilai terendah dari 10 ke 25 dan 45, nilai rata-rata dari 19,17 ke 54,06 dan 74,90. Tingkat pemahaman rata-rata kelas dari sangat rendah ke sedang dan tinggi.Hasil perbandingan dari tes pertama, tes kedua, dan tes ketiga menunjukkan perubahan yang lebih baik atau maju dengan begitu dapat disimpulkan siswa mengalami perkembangan pemahannya dalam operasi hitung bilangan bulat melalui setting pembelajaran matematika realistik berbantuan media “MOGER”.Berdasarkan hasil data, dapat disimpulkan siswa mengalami perkembangan pemahaman dalam operasi hitung bilangan bulat dengan menggunakan setting pembelajaran matematika realistik berbantuan media “MOGER” di kelas IV. Kata kunci: Pemahaman Siswa pada Operasi Bilangan Bulat, Pembelajaran MatematikaRealistik, Media “MOGER”. ABSTRACTThis study aims to describe the development of understanding of students on Operation Count Integer in setting medium-aided Realistic Mathematics Education "Moving Cars" in the fourth grade.This research is descriptive qualitative and quantitative research conducted to explain or illustrate the past and present variables which aims to create a description, picture or painting in a systematic, factual and accurate information on the facts and the relationship between the phenomenon investigated. Results of the first test, a second test and third test students have shown improvement with the highest score of 45 to 77.5 and to 100, the lowest value of 10 to 25 and 45, the average value of 19.17 to 54.06 and 74.90. Level of understanding of the average grade from very low to medium and high. The comparison of the first test, a second test and third test showed changes for the better or forward so we can conclude pemahannya students progressing in integer arithmetic operations by setting realistic mathematics-assisted learning media "Moger".Based on the results of the data, it can be concluded the students to experience growth in the understanding of integer arithmetic operations using mathematical learning setting realistic media-assisted "Moger" in the fourth grade. Keywords: Students Understanding on Operations Integer, Realistic Mathematics Education, Media "Moger".

HUBUNGAN LEMBAR KERJA PESERTA DIDIK BERBASIS CAT TERHADAP KEMAMPUAN PEMAHAMAN MATEMATIS SISWA

To find out the development, effectiveness, relationship, response, enhancing the ability of mathematical understanding with the development of learner-based worksheets CAT six graders Methodist-12 field in the material integer arithmetic operations. This research is included in research and development (Research and Development). As that R&D is a research method used to produce certain products and test the effectiveness of these products. This method is used with the aim of developing CAT-based student worksheets for grade VI SD Methodist-12 Medan on integer arithmetic operations. The study found, namely (1) CAT-based Student Worksheets are feasible to be developed and good to be applied in learning, (2) CAT-based Student Worksheets have a positive and significant relationship with students' mathematical understanding abilities, (3) CAT-based Student Worksheets have a positive effect on students' mathematical understanding abilities. Teachers can create and present the teaching materials with exciting creativity, media with an attractive creativity, practice questions that are presented in accordance with the material, enhancing the creativity of teachers in designing learning media.