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1. Testing a Unified Model of Arithmetic

   Braithwaite, David W.; Siegler, Robert S. (January 2022, Proceedings of the Annual Conference of the Cognitive Science Society)

   We describe UMA (Unified Model of Arithmetic), a theory of children’s arithmetic implemented as a computational model. UMA extends a theory of fraction arithmetic (Braithwaite et al., 2017) to include arithmetic with whole numbers and decimals. We evaluated UMA in the domain of decimal arithmetic by training the model on problems from a math textbook series, then testing it on decimal arithmetic problems that were solved by 6th and 8th graders in a previous study. UMA’s test performance closely matched that of children, supporting three assumptions of the theory: (1) most errors reflect small deviations from standard procedures, (2) between-problem variations in error rates reflect the distribution of input that learners receive, and (3) individual differences in strategy use reflect underlying variation in learning parameters. 

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2. Testing a unified model of arithmetic

   Braithwaite, David W.; Siegler, Robert S. (January 2022, Proceedings of the 44th Annual Conference of the Cognitive Science Society)

   Culbertson, J.; Perfors, A.; Rabagliati, H.; Ramenzoni, V. (Ed.)

   We describe UMA (Unified Model of Arithmetic), a theory of children’s arithmetic implemented as a computational model. UMA extends a theory of fraction arithmetic (Braithwaite et al., 2017) to include arithmetic with whole numbers and decimals. We evaluated UMA in the domain of decimal arithmetic by training the model on problems from a math textbook series, then testing it on decimal arithmetic problems that were solved by 6th and 8th graders in a previous study. UMA’s test performance closely matched that of children, supporting three assumptions of the theory: (1) most errors reflect small deviations from standard procedures, (2) between-problem variations in error rates reflect the distribution of input that learners receive, and (3) individual differences in strategy use reflect underlying variation in learning parameters. 

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3. Arithmetic diagonal cycles on unitary Shimura varieties

   https://doi.org/10.1112/S0010437X20007289  

   Rapoport, M.; Smithling, B.; Zhang, W. (September 2020, Compositio Mathematica)

   null (Ed.)

   We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension. 

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4. UltraPrecise: A GPU-Based Framework for Arbitrary-Precision Arithmetic in Database Systems

   https://doi.org/10.1109/ICDE60146.2024.00294  

   Li, Xin; Xiao, Mengbai; Yu, Dongxiao; Lee, Rubao; Zhang, Xiaodong (May 2024, IEEE)

   Fixed-point decimal operations in databases with arbitrary-precision arithmetic refer to the ability to store and operate decimal fraction numbers with an arbitrary length of digits. This type of operation has become a requirement for many applications, including scientific databases, financial data processing, geometric data processing, and cryptography. However, the state-of-the-art fixed-point decimal technology either provides high performance for low-precision operations or supports arbitrary-precision arithmetic operations at low performance. In this paper, we present a design and implementation of a framework called UltraPrecise which supports arbitraryprecision arithmetic for databases on GPU, aiming to gain high performance for arbitrary-precision arithmetic operations. We build our framework based on the just-in-time compilation technique and optimize its performance via data representation design, PTX acceleration, and expression scheduling. UltraPrecise achieves comparable performance to other high-performance databases for low-precision arithmetic operations. For highprecision, we show that UltraPrecise consistently outperforms existing databases by two orders of magnitude, including workloads of RSA encryption and trigonometric function approximation. 

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5. Arithmetic Sketching

   Boneh, Dan; Boyle, Elette; Corrigan-Gibbs, Henry; Gilboa, Niv; Ishai, Yuval (January 2023, IACR International Cryptology Conference (CRYPTO))

   This paper introduces arithmetic sketching, an abstraction of a primitive that several previous works use to achieve lightweight, low-communication zero-knowledge verification of secret-shared vectors. An arithmetic sketching scheme for a language L ⊆ F^n consists of (1) a randomized linear function compressing a long input x to a short “sketch,” and (2) a small arithmetic circuit that accepts the sketch if and only if x ∈ L, up to some small error. If the language L has an arithmetic sketching scheme with short sketches, then it is possible to test membership in L using an arithmetic circuit with few multiplication gates. Since multiplications are the dominant cost in protocols for computation on secret-shared, encrypted, and committed data, arithmetic sketching schemes give rise to lightweight protocols in each of these settings. Beyond the formalization of arithmetic sketching, our contributions are: – A general framework for constructing arithmetic sketching schemes from algebraic varieties. This framework unifies schemes from prior work and gives rise to schemes for useful new languages and with improved soundness error. – The first arithmetic sketching schemes for languages of sparse vectors: vectors with bounded Hamming weight, bounded L1 norm, and vectors whose few non-zero values satisfy a given predicate. – A method for “compiling” any arithmetic sketching scheme for a language L into a low-communication malicious-secure multi-server protocol for securely testing that a client-provided secret-shared vector is in L. We also prove the first nontrivial lower bounds showing limits on the sketch size for certain languages (e.g., vectors of Hamming-weight one) and proving the non-existence of arithmetic sketching schemes for others (e.g., the language of all vectors that contain a specific value). 

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