This paper considers parametricity and its consequent free theorems for nested data types. Rather than representing nested types via their Church encodings in a higher-kinded or dependently typed extension of System F, we adopt a functional programming perspective and design a Hindley-Milner-style calculus with primitives for constructing nested types directly as fixpoints. Our calculus can express all nested types appearing in the literature, including truly nested types. At the level of terms, it supports primitive pattern matching, map functions, and fold combinators for nested types. Our main contribution is the construction of a parametric model for our calculus. This is both delicate and challenging. In particular, to ensure the existence of semantic fixpoints interpreting nested types, and thus to establish a suitable Identity Extension Lemma for our calculus, our type system must explicitly track functoriality of types, and cocontinuity conditions on the functors interpreting them must be appropriately threaded throughout the model construction. We also prove that our model satisfies an appropriate Abstraction Theorem, as well as that it verifies all standard consequences of parametricity in the presence of primitive nested types. We give several concrete examples illustrating how our model can be used to derive useful free theorems, including a short cut fusion transformation, for programs over nested types. Finally, we consider generalizing our results to GADTs, and argue that no extension of our parametric model for nested types can give a functorial interpretation of GADTs in terms of left Kan extensions and still be parametric.

This paper considers parametricity and its resulting free theorems for nested data types. Rather than representing nested types via their Church encodings in a higher-kinded or dependently typed extension of System F, we adopt a functional programming perspective and design a Hindley-Milner-style calculus with primitives for constructing nested types directly as fixpoints. Our calculus can express all nested types appearing in the literature, including truly nested types. At the term level, it supports primitive pattern matching, map functions, and fold combinators for nested types. Our main contribution is the construction of a parametric model for our calculus. This is both delicate and challenging: to ensure the existence of semantic fixpoints interpreting nested types, and thus to establish a suitable Identity Extension Lemma for our calculus, our type system must explicitly track functoriality of types, and cocontinuity conditions on the functors interpreting them must be appropriately threaded throughout the model construction. We prove that our model satisfies an appropriate Abstraction Theorem and verifies all standard consequences of parametricity for primitive nested types.

Deep data types are those that are constructed from other data types, including, possibly, themselves. In this case, they are said to be truly nested. Deep induction is an extension of structural induction that traverses all of the structure in a deep data type, propagating predicates on its primitive data throughout the entire structure. Deep induction can be used to prove properties of nested types, including truly nested types, that cannot be proved via structural induction. In this paper we show how to extend deep induction to GADTs that are not truly nested GADTs. This opens the way to incorporating automatic generation of (deep) induction rules for them into proof assistants. We also show that the techniques developed in this paper do not suffice for extending deep induction to truly nested GADTs, so more sophisticated techniques are needed to derive deep induction rules for them.

We introduce categories M and S internal in the tricategory Bicat 3 of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory V. Their horizontal tricategories are the tricategories of matrices and spans in V. Both the internal and the enriched constructions are tricategorifications of the corresponding constructions in 1-categories. Following [11] we introduce monads and their vertical morphisms in categories internal in tricategories. We prove an equivalent condition for when the internal categories for matrices M and spans S in a 1-strict tricategory V are equivalent, and deduce that in that case their corresponding categories of (strict) monads and vertical monad morphisms are equivalent, too. We prove that the latter categories are isomorphic to those of categories enriched and discretely internal in V, respectively. As a byproduct of our tricategorical constructions we recover some results from [10]. Truncating to 1-categories we recover results from [6] and [9] on the equivalence of enriched and discretely internal 1-categories.

The problem of passive learning of linear temporal logic formulae consists in finding the best explanation for how two sets of execution traces differ, in the form of the shortest formula that separates the two sets. We approach the problem by implementing an exhaustive search algorithm optimized for execution speed. We apply it to the use-case of a robot moving in an unstructured environment as its battery discharges, both in simulation and in the real world. The results of our experiments confirm that our approach can learn temporal formulas explaining task failures in a case of practical interest.