Lazy algorithms for exact real arithmetic

In this article we propose a new representation for the real numbers. This representation can be conveniently used to implement exact real number computation with a lazy programming languages. In fact the new representation permits the exploitation of hardware implementation of arithmetic functions without generating the granularity problem. Moreover we present a variation of the Karatsuba algorithm for multiplication of integers. The new algorithm performs exact real number multiplication in a lazy way and has a lower complexity than the standard algorithm. \ua9 2004 Elsevier B.V. All rights reserved

A definitional implementation of the LAX logical framework LLFP in CoQ, for supporting fast and loose reasoning

The Lax Logical Framework, LLFP, was introduced, by a team including the last two authors, to provide a conceptual framework for integrating different proof development tools, thus allowing for external evidence and for postponing, delegating, or factoring-out side conditions. In particular, LLFP allows for reducing the number of times a proof-irrelevant check is performed. In this paper we give a shallow, actually definitional, implementation of LLFP in Coq, i.e. we use Coq both as host framework and oracle for LLFP. This illuminates the principles underpinning the mechanism of Lock-types and also suggests how to possibly extend Coq with the features of LLFP. The derived proof editor is then put to use for developing case-studies on an emerging paradigm, both at logical and implementation level, which we call fast and loose reasoning following Danielsson et alii [6]. This paradigm trades off efficiency for correctness and amounts to postponing, or running in parallel, tedious or computationally demanding checks, until we are really sure that the intended goal can be achieved. Typical examples are branch-prediction in CPUs and optimistic concurrency control

LF+ in Coq for fast-and-loose reasoning

We develop the metatheory and the implementation, in Coq, of the novel logical framework LF+ and discuss several of its applications. LF+ generalises research work, carried out by the authors over more than a decade, on Logical Frameworks conservatively extending LF and featuring lock-type constructors L-P(N:sigma)[center dot]. Lock-types capture monadically the concept of inhabitability up-to. They were originally introduced for factoring-out, postponing, or delegating to external tools the verification of time-consuming judgments, which are morally proof-irrelevant, thus allowing for integrating different sources of epistemic evidence in a unique Logical Framework. Besides introducing LF+ and its "shallow" implementation in Coq, the main novelty of the paper is to show that lock-types are also a very flexible tool for expressing in Type Theory several diverse cognitive attitudes and mental strategies used in ordinary reasoning, which essentially amount to reasoning up-to, as in e.g. Typical Ambiguity provisos or co-inductive Coq proofs. In particular we address the encoding of the emerging paradigm of fast-and-loose reasoning, which trades off efficiency for correctness. This paradigm, implicitly used normally in naive Set Theory, is producing considerable impact also in computer architecture and distributed systems, when branch prediction and optimistic concurrency control are implemented

Abstract Datatypes for Real Numbers in Type Theory

Abstract. We propose an abstract datatype for a closed interval of real numbers to type theory, providing a representation-independent approach to programming with real numbers. The abstract datatype requires only function types and a natural numbers type for its formulation, and so can be added to any type theory that extends Gödel’s System datatype is equivalent in power to programming intensionally with representations of real numbers. We also consider representing arbitrary real numbers using a mantissa-exponent representation in which the mantissa is taken from the abstract interval.

Variable binding, symmetric monoidal closed theories, and bigraphs

This paper investigates the use of symmetric monoidal closed (SMC) structure for representing syntax with variable binding, in particular for languages with linear aspects. In our setting, one first specifies an SMC theory T, which may express binding operations, in a way reminiscent from higher-order abstract syntax. This theory generates an SMC category S(T) whose morphisms are, in a sense, terms in the desired syntax. We apply our approach to Jensen and Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This leads to an alternative category of bigraphs, which we compare to the original.Comment: An introduction to two more technical previous preprints. Accepted at Concur '0

Differential calculus with imprecise input and its logical framework

We develop a domain-theoretic Differential Calculus for locally Lipschitz functions on finite dimensional real spaces with imprecise input/output. The inputs to these functions are hyper-rectangles and the outputs are compact real intervals. This extends the domain of application of Interval Analysis and exact arithmetic to the derivative. A new notion of a tie for these functions is introduced, which in one dimension represents a modification of the notion previously used in the one-dimensional framework. A Scott continuous sub-differential for these functions is then constructed, which satisfies a weaker form of calculus compared to that of the Clarke sub-gradient. We then adopt a Program Logic viewpoint using the equivalence of the category of stably locally compact spaces with that of semi-strong proximity lattices. We show that given a localic approximable mapping representing a locally Lipschitz map with imprecise input/output, a localic approximable mapping for its sub-differential can be constructed, which provides a logical formulation of the sub-differential operator

A coinductive semantics of the Unlimited Register Machine

We exploit (co)inductive specifications and proofs to approach the evaluation of low-level programs for the Unlimited Register Machine (URM) within the Coq system, a proof assistant based on the Calculus of (Co)Inductive Constructions type theory. Our formalization allows us to certify the implementation of partial functions, thus it can be regarded as a first step towards the development of a workbench for the formal analysis and verification of both converging and diverging computations

Propylthiouracil Is Teratogenic in Murine Embryos

Background: Hyperthyroidism during pregnancy is treated with the antithyroid drugs (ATD) propylthiouracil (PTU) and methimazole (MMI). PTU currently is recommended as the drug of choice during early pregnancy. Yet, despite widespread ATD use in pregnancy, formal studies of ATD teratogenic effects have not been performed. Methods: We examined the teratogenic effects of PTU and MMI during embryogenesis in mice. To span different periods of embryogenesis, dams were treated with compounds or vehicle daily from embryonic day (E) 7.5 to 9.5 or from E3.5 to E7.5. Embryos were examined for gross malformations at E10.5 or E18.5 followed by histological and micro-CT analysis. Influences of PTU on gene expression levels were examined by RNA microarray analysis. Results: When dams were treated from E7.5 to E9.5 with PTU, neural tube and cardiac abnormalities were observed at E10.5. Cranial neural tube defects were significantly more common among the PTU-exposed embryos than those exposed to MMI or vehicle. Blood in the pericardial sac, which is a feature indicative of abnormal cardiac function and/or abnormal vasculature, was observed more frequently in PTU-treated than MMI-treated or vehicle-treated embryos. Following PTU treatment, a total of 134 differentially expressed genes were identified. Disrupted genetic pathways were those associated with cytoskeleton remodeling and keratin filaments. At E 18.5, no gross malformations were evident in either ATD group, but the number of viable PTU embryos per dam at E18.5 was significantly lower from those at E10.5, indicating loss o