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From coinductive proofs to exact real arithmetic
"... Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresp ..."
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Cited by 8 (6 self)
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Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching nonwellfounded trees describing when the algorithm writes and reads digits. This is a pilot study in using prooftheoretic methods for obtaining certified algorithms in exact real arithmetic. 1
RZ: A tool for bringing constructive and computable mathematics closer to programming practice
 CiE 2007: Computation and Logic in the Real World, volume 4497 of LNCS
, 2007
"... Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Obje ..."
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Cited by 5 (2 self)
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Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 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
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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Cited by 1 (0 self)
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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.
A Denotional Semantics for . . .
, 2007
"... We provide a denotational model for a functional programming language for exact real number computation. A well known difficulty in real number computation is that the tests x = y and x ≤ y are undecidable and hence cannot be used to control the execution flow of programs. One solution, proposed by ..."
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We provide a denotational model for a functional programming language for exact real number computation. A well known difficulty in real number computation is that the tests x = y and x ≤ y are undecidable and hence cannot be used to control the execution flow of programs. One solution, proposed by Boehm and Cartwright, is to use a nondeterministic test. For any two rational numbers p < q and any real number x, at least one of the relations p < x or x < q can be determined to hold; thus, an operator rtest is used, whose evaluation never diverges when x is a real number: 1. rtestp,q(x) evaluates to true or to false, 2. rtestp,q(x) may evaluate to true iff x < q and 3. rtestp,q(x) may evaluate to false iff p < x. Since a program can in general produce different results in different runs, Escardó and MarcialRomero took the view in previous work that programs of realnumber type denote sets of real numbers, and the question arose as to which power domains would be suitable for modelling the behaviour of rtest. It was shown that, among the known power domains,
1 Complex CoEvolutionary Dynamics – Structural Stability and Finite Population Effects
"... Abstract—Unlike evolutionary dynamics, coevolutionary dynamics can exhibit a wide variety of complex regimes. This has been confirmed by numerical studies e.g. in the context of Evolutionary Game Theory (EGT) and population dynamics of simple twostrategy games with various types of replication and ..."
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Abstract—Unlike evolutionary dynamics, coevolutionary dynamics can exhibit a wide variety of complex regimes. This has been confirmed by numerical studies e.g. in the context of Evolutionary Game Theory (EGT) and population dynamics of simple twostrategy games with various types of replication and selection mechanisms. Using the framework of shadowing lemma we study to what degree can such infinite population dynamics (1) be reliably simulated on finite precision computers and (2) be trusted to represent coevolutionary dynamics of possibly very large, but finite populations. In a simple EGT setting of twoplayer symmetric games with two pure strategies and a polymorphic equilibrium we prove that for (µ, λ), truncation, sequential tournament, bestofgroup tournament and linear ranking selections, the coevolutionary dynamics do not possess the shadowing property. In other words, infinite population simulations cannot be guaranteed to represent real trajectories or to be representative of coevolutionary dynamics of potentially very large, but finite populations. I.
DOI 10.1007/s0022401193258 Proofs, Programs, Processes
, 2011
"... Abstract The objective of this paper is to provide a theoretical foundation for program extraction from inductive and coinductive proofs geared to practical applications. The novelties consist in the addition of inductive and coinductive definitions to a realizability interpretation for firstorder ..."
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Abstract The objective of this paper is to provide a theoretical foundation for program extraction from inductive and coinductive proofs geared to practical applications. The novelties consist in the addition of inductive and coinductive definitions to a realizability interpretation for firstorder proofs, a soundness proof for this system, and applications to the synthesis of nontrivial provably correct programs in the area of exact real number computation. We show that realizers, although per se untyped, can be assigned polymorphic recursive types and hence represent valid 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. the binary signed digit representation.
Functional first order definability of LRTp
"... Abstract. The language LRTp is a nondeterministic language for exact real number computation. It has been shown that all computable first order relations in the sense of Brattka are definable in the language. If we restrict the language to singlevalued total relations (e.g. functions), all polynom ..."
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Abstract. The language LRTp is a nondeterministic language for exact real number computation. It has been shown that all computable first order relations in the sense of Brattka are definable in the language. If we restrict the language to singlevalued total relations (e.g. functions), all polynomials are definable in the language. In this paper we show that the nondeterministic version of the limit operator, which allows to define all computable first order relations, when restricted to singlevalued total inputs, produces singlevalued total outputs. This implies that not only the polynomials are definable in the language but also all computable first order functions.
3.1 Classical Approach.......................... 7 3.2 Unit Floor Model Checking Approach................ 7
, 2008
"... University of Sheffield, RollsRoyce plc., Goodrich Engine Control Systems and Jaguar Cars Limited. This project is cofunded by the Technology Strategy Board’s Collaborative Research and Development programme, following an open competition. The Technology Strategy Board is an executive body establi ..."
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University of Sheffield, RollsRoyce plc., Goodrich Engine Control Systems and Jaguar Cars Limited. This project is cofunded by the Technology Strategy Board’s Collaborative Research and Development programme, following an open competition. The Technology Strategy Board is an executive body established by the Government to drive innovation. It promotes and invests in research, development and the exploitation of science, technology and new ideas for the benefit of business —