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Formally Verified Computation of Enclosures of Solutions of Ordinary Differential Equations
, 2014
"... Ordinary differential equations (ODEs) are ubiquitous when modeling continuous dynamics. Classical numerical methods compute approximations of the solution, however without any guarantees on the quality of the approximation. Nevertheless, methods have been developed that are supposed to compute e ..."
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Ordinary differential equations (ODEs) are ubiquitous when modeling continuous dynamics. Classical numerical methods compute approximations of the solution, however without any guarantees on the quality of the approximation. Nevertheless, methods have been developed that are supposed to compute enclosures of the solution. In this paper, we demonstrate that enclosures of the solution can be verified with a high level of rigor: We implement a functional algorithm that computes enclosures of solutions of ODEs in the interactive theorem prover Isabelle/HOL, where we formally verify (and have mechanically checked) the safety of the enclosures against the existing theory of ODEs in Isabelle/HOL. Our algorithm works with dyadic rational numbers with statically fixed precision and is based on the wellknown Euler method. We abstract discretization and roundoff errors in the domain of affine forms.
Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs?
"... Abstract. Monotone algebras are frequently used to generate reduction orders in automated termination and complexity proofs. To be able to certify these proofs, we formalized several kinds of interpretations in the proof assistant Isabelle/HOL. We report on our integration of matrix interpretations, ..."
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Abstract. Monotone algebras are frequently used to generate reduction orders in automated termination and complexity proofs. To be able to certify these proofs, we formalized several kinds of interpretations in the proof assistant Isabelle/HOL. We report on our integration of matrix interpretations, arctic interpretations, and nonlinear polynomial interpretations over various domains, including the reals. 1
Unified decision procedures for regular expression equivalence. http://www.in.tum.de/∼nipkow/pubs/regex equiv. pdf
, 2014
"... Abstract. We formalize a unified framework for verified decision procedures for regular expression equivalence. Five recently published formalizations of such decision procedures (three based on derivatives, two on marked regular expressions) can be obtained as instances of the framework. We discov ..."
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Abstract. We formalize a unified framework for verified decision procedures for regular expression equivalence. Five recently published formalizations of such decision procedures (three based on derivatives, two on marked regular expressions) can be obtained as instances of the framework. We discover that the two approaches based on marked regular expressions, which were previously thought to be the same, are different, and we prove a quotient relation between the automata produced by them. The common framework makes it possible to compare the performance of the different decision procedures in a meaningful way. 1
Contents
, 2013
"... We implement the Babylonian method [1] to compute square roots of numbers. We provide precise algorithms for naturals, integers and rationals, and offer an approximation algorithm for linear ordered fields. ..."
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We implement the Babylonian method [1] to compute square roots of numbers. We provide precise algorithms for naturals, integers and rationals, and offer an approximation algorithm for linear ordered fields.