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We classify all apartness relations definable in propositional logics extending intuitionistic logic using Heyting algebra semantics. We show that every Heyting algebra which contains a non‐trivial apartness term satisfies the weak law of excluded middle, and every Heyting algebra which contains a tight apartness term is in fact a Boolean algebra. This answers a question of Rijke regarding the correct notion of apartness for propositions, and yields a short classification of apartness terms that can occur in a Heyting algebra. We also show that Martin‐Löf Type Theory is not able to construct non‐trivial apartness relations between propositions. [ABSTRACT FROM AUTHOR]

Search Based Software Engineering techniques are emerging as important tools for software maintenance. Foremost among these is Genetic Improvement, which has historically applied the stochastic techniques of Genetic Programming to optimize pre-existing program code. Previous work in this area has not generally preserved program semantics and this article describes an alternative to the traditional mutation operators used, employing deterministic proof search in the sequent calculus to yield semantics-preserving transformations on algebraic data types. Two case studies are described, both of which are applicable to the recently-introduced 'grow and graft' technique of Genetic Improvement: the first extends the expressiveness of the 'grafting' phase and the second transforms the representation of a list data type to yield an asymptotic efficiency improvement. [ABSTRACT FROM AUTHOR]

Search Based Software Engineering techniques are emerging as important tools for software maintenance. Foremost among these is Genetic Improvement, which has historically applied the stochastic techniques of Genetic Programming to optimize pre-existing program code. Previous work in this area has not generally preserved program semantics and this article describes an alternative to the traditional mutation operators used, employing deterministic proof search in the sequent calculus to yield semantics-preserving transformations on algebraic data types. Two case studies are described, both of which are applicable to the recently-introduced 'grow and graft' technique of Genetic Improvement: the first extends the expressiveness of the 'grafting' phase and the second transforms the representation of a list data type to yield an asymptotic efficiency improvement.