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11
Verified Real Number Calculations: A Library for Interval Arithmetic
, 2007
"... Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally est ..."
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Cited by 6 (1 self)
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Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theoremprover interaction.
Coinduction for Exact Real Number Computation
, 2007
"... This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the corr ..."
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Cited by 5 (4 self)
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This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the correctness of various corecursive exact real number algorithms. The basic framework is the classical theory of coinductive sets as greatest fixed points of monotone operators and hence is different from (though related to) the type theoretic approach by Ciaffaglione and Gianantonio. Key words: Exact real number computation, coinduction, corecursion, signed digit streams. 1
Proofs, programs, processes
"... Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming la ..."
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Cited by 3 (2 self)
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Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t. to the binary signed digit representation. 1
Coinductive Field of Exact Real Numbers and General Corecursion
, 2006
"... In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be se ..."
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Cited by 2 (0 self)
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In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general corecursive functions. We use the machinery of Coq proof assistant for coinductive types to present the formalisation.
Realisability and adequacy for (co)induction
"... Abstract. We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in th ..."
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Cited by 1 (1 self)
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Abstract. We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in the area of exact real number computation, and hint at further nontrivial applications in computable analysis. 1
Minlog A Tool for Program Extraction Supporting Algebras and Coalgebras
"... Abstract. Minlog is an interactive system which implements prooftheoretic methods and applies them to verification and program extraction. We give an overview of Minlog and demonstrate how it can be used to exploit the computational content in (co)algebraic proofs and to develop correct and efficien ..."
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Abstract. Minlog is an interactive system which implements prooftheoretic methods and applies them to verification and program extraction. We give an overview of Minlog and demonstrate how it can be used to exploit the computational content in (co)algebraic proofs and to develop correct and efficient programs. We illustrate this by means of two examples: one about parsing, the other about exact real numbers in signed digit representation. 1
PreProceedings of the Ninth International Workshop on Automated Verification of Critical Systems
"... AVOCS, the workshop on Automated Verification of Critical Systems, is an annual meeting that brings together researchers and practitioners to exchange new results on tools and techniques for the verification of critical systems. Topics of interest cover all aspects of automated verification, includi ..."
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AVOCS, the workshop on Automated Verification of Critical Systems, is an annual meeting that brings together researchers and practitioners to exchange new results on tools and techniques for the verification of critical systems. Topics of interest cover all aspects of automated verification, including model checking, theorem proving, abstract interpretation, and refinement pertaining to various types of critical systems (safetycritical, securitycritical, businesscritical, performancecritical, etc.). Contributions that describe different techniques, or industrial case studies are encouraged.
In Pursuit of Real Answers ∗
"... Digital computers permeate our physical world. This phenomenon creates a pressing need for tools that help us understand a priori how digital computers can affect their physical environment. In principle, simulation can be a powerful tool for animating models of the world. Today, however, there is n ..."
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Digital computers permeate our physical world. This phenomenon creates a pressing need for tools that help us understand a priori how digital computers can affect their physical environment. In principle, simulation can be a powerful tool for animating models of the world. Today, however, there is not a single simulation environment that comes with a guarantee that the results of the simulation are determined purely by a realvalued model and not by artifacts of the digitized implementation. As such, simulation with guaranteed fidelity does not yet exist. Towards addressing this problem, we offer an expository account of what is known about exact real arithmetic. We argue that this technology, which has roots that are over 200 years old, bears significant promise as offering exactly the right technology to build simulation environments with guaranteed fidelity. And while it has only been sparsely studied in this large span of time, there are reasons to believe that the time is right to accelerate research in this direction. ∗ This research was sponsored by the NSF under Award 0439017,
Coinductive Proofs for Basic Real Computation Tie Hou
"... Abstract. We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof ..."
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Abstract. We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof for the correctness of a corecursive program for the average function with regard to the signed digit stream representation. We implemented this proof in the interactive proof system Minlog. Thus, reliable, corecursive functions for real computation can be guaranteed, which is very helpful in formal software development for real numbers.
Coinductive Pearl: Modular FirstOrder Logic Completeness
"... Codatatypes are regrettably absent from many programming languages and proof assistants. We make a case for their usefulness by revisiting a classic result: the completeness theorem for firstorder logic established through a Gentzen system. Codatatypes help capture the essence of the proof, which e ..."
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Codatatypes are regrettably absent from many programming languages and proof assistants. We make a case for their usefulness by revisiting a classic result: the completeness theorem for firstorder logic established through a Gentzen system. Codatatypes help capture the essence of the proof, which establishes an abstract property of derivation trees independently of the concrete syntax or inference rules. This separation of concerns simplifies the presentation, especially for readers acquainted with lazy data structures. The proof is formalized in Isabelle/HOL and demonstrates the recently introduced definitional package for codatatypes and its integration with Isabelle’s Haskell code generator.