Abstract Interpretation, Logical Relations, and Kan Extensions

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Abstract
We develop a formalism for abstract interpretation based on logical relations. As a case study, we use this formalism to give new proofs of correctness for strictness analysis on the typed λ-calculus, and also for termination analysis. We then go on to a deeper study of the duality between safety and liveness properties, and the construction of abstraction functions which can be used to give the best possible interpretations of higher-type constants. This turns out to be a special case of the construction of Kan extensions in category theory. Necessary and sufficient conditions are given for abstraction functions to be definable over continuous type structures and for the abstraction functions themselves to be continuous.

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