Datatype defining rewrite systems for naturals and integers

A datatype defining rewrite system (DDRS) is an algebraic (equational) specification intended to specify a datatype. When interpreting the equations from left-to-right, a DDRS defines a term rewriting system that must be ground-complete. First we define two DDRSs for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying). Next we specify arithmetic based on two other views: binary and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negat

An Overview of Datatype Quantization Techniques for Convolutional Neural Networks

Convolutional Neural Networks (CNNs) are becoming increasingly popular due to their superior performance in the domain of computer vision, in applications such as objection detection and recognition. However, they demand complex, power-consuming hardware which makes them unsuitable for implementation on low-power mobile and embedded devices. In this paper, a description and comparison of various techniques is presented which aim to mitigate this problem. This is primarily achieved by quantizing the floating-point weights and activations to reduce the hardware requirements, and adapting the training and inference algorithms to maintain the networks performance.

Supporting mixed-datatype matrix multiplication within the BLIS framework

We approach the problem of implementing mixed-datatype support within the general matrix multiplication (GEMM) operation of the BLIS framework, whereby each matrix operand A, B, and C may be stored as single- or double-precision real or complex values. Another factor of complexity, whereby the computation is allowed to take place in a precision different from the storage precisions of either A or B, is also included in the discussion. We first break the problem into mostly orthogonal dimensions, considering the mixing of domains separately from mixing precisions. Support for all combinations of matrix operands stored in either the real or complex domain is mapped out by enumerating the cases and describing an implementation approach for each. Supporting all combinations of storage and computation precisions is handled by typecasting the matrices at key stages of the computation---during packing and/or accumulation, as needed. Several optional optimizations are also documented. Performance results gathered on a 56-core Marvell ThunderX2 and a 52-core Intel Xeon Platinum demonstrate that high performance is mostly preserved, with modest slowdowns incurred from unavoidable typecast instructions. The mixed-datatype implementation confirms that combinatoric intractability is avoided, with the framework relying on only two assembly microkernels to implement 128 datatype combinations.

The Extended Theory of Trees and Algebraic (Co)datatypes

The first-order theory of finite and infinite trees has been studied since the eighties, especially by the logic programming community. Following Djelloul, Dao and Fruhwirth, we consider an extension of this theory with an additional predicate for finiteness of trees, which is useful for expressing properties about (not just datatypes but also) codatatypes. Based on their work, we present a simplification procedure that determines whether any given (not necessarily closed) formula is satisfiable, returning a simplified formula which enables one to read off all possible models. Our extension makes the algorithm usable for algebraic (co)datatypes, which was impossible in their original work due to restrictive assumptions. We also provide a prototype implementation of our simplification procedure and evaluate it on instances from the SMT-LIB.

Reasoning about modular datatypes with Mendler induction

In functional programming, datatypes a la carte provide a convenient modular representation of recursive datatypes, based on their initial algebra semantics. Unfortunately it is highly challenging to implement this technique in proof assistants that are based on type theory, like Coq. The reason is that it involves type definitions, such as those of type-level fixpoint operators, that are not strictly positive. The known work-around of impredicative encodings is problematic, insofar as it impedes conventional inductive reasoning. Weak induction principles can be used instead, but they considerably complicate proofs. This paper proposes a novel and simpler technique to reason inductively about impredicative encodings, based on Mendler-style induction. This technique involves dispensing with dependent induction, ensuring that datatypes can be lifted to predicates and relying on relational formulations. A case study on proving subject reduction for structural operational semantics illustrates that the approach enables modular proofs, and that these proofs are essentially similar to conventional ones.