Semantics of a sequential language for exact real-number computation

Abstract

We study a programming language with a built-in ground type for real numbers. In order for the language to be sufficiently expressive but still sequential, we consider a construction proposed by Boehm and Cartwright. The nondeterministic nature of the construction suggests the use of powerdomains in order to obtain a denotational semantics for the language. We show that the construction cannot be modelled by the Plotkin or Smyth powerdomains, but that the Hoare powerdomain gives a computationally adequate semantics. As is well known, Hoare semantics can be used in order to establish partial correctness only. Since computations on the reals are infinite, one cannot decompose total correctness into the conjunction of partial correctness and termination as it is traditionally done. We instead introduce a suitable operational notion of strong convergence and show that total correctness can be proved by establishing partial correctness (using denotational methods) and strong convergence (using operational methods). We illustrate the technique with a representative example.then non-trivial predicates are obtained, and this together with the parallel conditional allow us to define non-trivial total functions [6].This phenomenon had been anticipated by Boehm and Cartwright [3], who also proposed a solution to this problem. In this paper, we develop the proposed solution and study its operational and denotational semantics. The idea is based on the following observations. In classical mathematics, the trichotomy law "

Coinductive Formal Reasoning in Exact Real Arithmetic

“…This is in accordance with separation of termination and correctness in the method of Bove-Capretta for general recursion [BC01] or already in the Hoare logic. Moreover, this separation is also evident in the domain theoretic semantics of the real numbers [MRE07].…”

“…The trade-off between the expressibility and the existence of parallelism in these work led to some improvements on the domain theoretic semantics of the Edalat-Potts algorithm, e.g. as in the recent work in [MRE07] where a sequential language with a non-deterministic cotransitivity test is proposed. This line of research is an instance of the extensional approach to exact real arithmetic while our work in which we have direct access to the digits of the representation is a study in the intensional exact arithmetic in the sense of [BES02].…”

“…Finally, our focus on the partial productivity is related to the aforesaid domain-theoretic semantics [Esc97] where partial real numbers are denoted by interval and the strong convergence (akin to our notion of productivity) of the functions is studied [MRE07]. This relationship is not a coincidence as manifested by the original analytic proof of adequacy for the Edalat-Potts algorithm [PEE97].…”

Minlog - A Tool for Program Extraction Supporting Algebras and Coalgebras

“…Whilst such algorithms have been verified before (see eg. [CDG06,MRE07,GNSW07,BH08]), in the present paper we show by means of an example how to extract them. We extract a program which for every rational number a ∈ [−1, 1] computes a signed binary digit representation, that is, a (finite or infinite) stream of digits…”

From Coinductive Proofs to Exact Real Arithmetic

“…Most of the recent work on exact real number computation describes algorithms for functions on certain exact representations of the reals (for example streams of signed digits [MRE07,GNSW07] or linear fractional transformations [EH02]) and proves their correctness using a certain proof method (for example coinduction [CDG06,Ber07,BH07]). Our work has a similar aim, and builds on the work cited above, but there are two important differences.…”