Search Results (5)

This article presents an optimized method for verifying the comparison of two binary numbers using the rank-1 constraint system (R1CS) representation, a standard framework for verifiable computation systems. In particular, we analyze different strategies for implementing strict comparisons of the form $t>K$, where K is a known constant and t is an integer input to the comparison. We first analyze a lexicographic approach that, although conceptually straightforward, results in a large number of constraints due to its branching logic. To address this inefficiency, we introduce a weighted-accumulation method that computes an accumulator whose sign determines the comparison outcome. By assigning position-dependent weights to bit pairs and formulating the computation through degree-2 constraints, this method eliminates branching and significantly reduces the total number of constraints. In order to validate our designs, we implemented the described comparison algorithms in an R1CS compiler called circom, allowing us to generate and analyze the corresponding R1CS constraint systems in practice. Overall, the presented design not only ensures correctness but also demonstrates how careful exploitation of the R1CS structure can lead to efficient constraint settings. Full article

Chipman contended, in stark contrast to the conventional view, that, utility is not a real number but a vector, and that it is inherently lexicographic in nature. On the other hand, in recent years continuous multi-utility representations of a preorder on a topological space, which proved to be the best kind of continuous representation, have been deeply studied. In this paper, we first state a general result, which guarantees, for every preordered topological space, the existence of a lexicographic order-embedding of the Chipman type. Then, we combine the Chipman approach and the continuous multi-utility approach, by stating a theorem that guarantees, under certain general conditions, the coexistence of these two kinds of continuous representations. Full article

This paper aims at documenting and reconstructing the linguistic processes generating and substantiating the use of number systems, numbers in general, elementary arithmetic, and the related concepts and notions among the Kula people from Alor Island, Southeastern Indonesia. The Kula is a Papuan population from the Alor–Pantar Archipelago (Timor area). The name of their language, Kula (or Kola), corresponds to the ethnonym. The language is, currently, endangered and not completely documented. At the level of linguistic features, numeral systems and the terms for numerals from Eastern Alor exhibit, to some extent, unique characteristics, if compared to other languages spoken in other sectors of the island. Therefore, the Kula numbering system is not only significant at the lexicological and lexicographic level, but also represents the essential role of cognitive strategies (e.g., the choice of the base for the numbering systems and the visual representation of counting with the aid of actual ‘objects’, like hands and fingers) in the coinage of numerical terms among the local speakers. Indeed, the development of numeral systems reflects the evolution of human language and the ability of humans to construct abstract numerical concepts. The way numerals are encoded and expressed in a language can impact the patterns according to which numerical notions are conceptualized and understood. Different numeral systems can indicate variations in cognitive processes involving notions of quantities and measurements. Therefore, the structure and characteristics of a numeral system may affect how numeral concepts are mentally represented and developed. This paper focuses on the number system of the Kula people and the lexical units used by the local speakers to indicate (and to explain) the numbers, with the related concepts, notions, and symbolism. The investigation delves into the degrees of abstraction of the Kula numeral system and tries to ascertain its origins and reconstruct it. Moreover, the article applies to the analysis a comparative approach, which takes into account several Papuan and Austronesian languages from Alor Island and Eastern Timor, with the dual aim of investigating, at a preliminary level, a possible common evolution and/or divergent naming processes in local numbering systems and their historical–linguistic and etymological origins. Full article

A symbolic analysis of Archimedes’s periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci’s Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics. Full article

This work presents a generalized implementation of the infeasible primal-dual interior point method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here the non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to a global optimum and to be able to manage infeasibility and unboundedness transparently, i.e., without considering them as corner cases: by means of a mild embedding (addition of two variables and one constraint), the NA-IPM implicitly and transparently manages their possible presence. Moreover, the new algorithm is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since the NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. To support the theoretical properties of the NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that the NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool, paving the way for its use in real-world problems in the near future. Full article