Aaron Stump’s research while affiliated with University of Iowa and other places

We introduce dual counterpart intuitionistic logic (or DCInt): a constructive logic that is a conservative extension of intuitionistic logic, a sublogic of bi-intuitionistic logic, has the logical duality property of classical logic, and also retains the modal character of its interpretation of the connective dual to intuitionistic implication. We define its Kripke semantics along with the corresponding notion of a bisimulation, and then prove that it has both the disjunction property and (its dual) the constructible falsity property. Also, for any class ${\mathcal{C}}$ of Kripke frames from our semantics, we identify a condition such that ${\mathcal{C}}$ will have the disjunction property if it satisfies the condition. This provides a method for generating extensions of DCInt that retain the disjunction property.

Cedille is a dependently typed programming language known for expressive and efficient impredicative encodings. In this work, we show that encodings of induction-induction are also possible by employing a standard technique from other encodings in Cedille, where a type representing the shape of data is intersected with a predicate that further constrains. Thus, just as with indexed inductive data, Cedille can encode a notion that is often axiomatically postulated or directly implemented in other dependent type theories without sacrificing efficiency.KeywordsImpredicative EncodingInduction-InductionCedille

This paper proposes a new approach to writing and verifying divide-and-conquer programs in Coq. Extending the rich line of previous work on algebraic approaches to recursion schemes, we present an algebraic approach to divide-and-conquer recursion: recursions are represented as a form of algebra, and from outer recursions, one may initiate inner recursions that can construct data upon which the outer recursions may legally recurse. Termination is enforced entirely by the typing discipline of our recursion schemes. Despite this, our approach requires little from the underlying type system, and can be implemented in System F ω plus a limited form of positive-recursive types. Our implementation of the method in Coq does not rely on structural recursion or on dependent types. The method is demonstrated on several examples, including mergesort, quicksort, Harper’s regular-expression matcher, and others. An indexed version is also derived, implementing a form of divide-and-conquer induction that can be used to reason about functions defined via our method.

Large eliminations provide an expressive mechanism for arity- and type-generic programming. However, as large eliminations are closely tied to a type theory's primitive notion of inductive type, this expressivity is not expected within polymorphic lambda calculi in which datatypes are encoded using impredicative quantification. We report progress on simulating large eliminations for datatype encodings in one such type theory, the calculus of dependent lambda eliminations (CDLE). Specifically, we show that the expected computation rules for large eliminations, expressed using a derived type of extensional equality of types, can be proven within CDLE. We present several case studies, demonstrating the adequacy of this simulation for a variety of generic programming tasks, and a generic formulation of the simulation allowing its use for any datatype. All results have been mechanically checked by Cedille, an implementation of CDLE.

Guided by Tarksi’s fixpoint theorem in order theory, we show how to derive monotone recursive types with constant-time roll and unroll operations within Cedille, an impredicative, constructive, and logically consistent pure typed lambda calculus. This derivation takes place within the preorder on Cedille types induced by type inclusions , a notion which is expressible within the theory itself. As applications, we use monotone recursive types to generically derive two recursive representations of data in lambda calculus, the Parigot and Scott encoding. For both encodings, we prove induction and examine the computational and extensional properties of their destructor, iterator, and primitive recursor in Cedille. For our Scott encoding in particular, we translate into Cedille a construction due to Lepigre and Raffalli (2019) that equips Scott naturals with primitive recursion, then extend this construction to derive a generic induction principle. This allows us to give efficient and provably unique (up to function extensionality) solutions for the iteration and primitive recursion schemes for Scott-encoded data.

This paper introduces Relational Type Theory (RelTT), a new approach to type theory with extensionality principles, based on a relational semantics for types. The type constructs of the theory are those of System F plus relational composition, converse, and promotion of application of a term to a relation. A concise realizability semantics is presented for these types. The paper shows how a number of constructions of traditional interest in type theory are possible in RelTT, including eta-laws for basic types, inductive types with their induction principles, and positive-recursive types. A crucial role is played by a lemma called Identity Inclusion, which refines the Identity Extension property familiar from the semantics of parametric polymorphism. The paper concludes with a type system for RelTT, paving the way for implementation.

This paper describes an implementation of Harper's continuation-based regular-expression matcher as a strong functional program in Cedille; i.e., Cedille statically confirms termination of the program on all inputs. The approach uses neither dependent types nor termination proofs. Instead, a particular interface dubbed a recursion universe is provided by Cedille, and the language ensures that all programs written against this interface terminate. Standard polymorphic typing is all that is needed to check the code against the interface. This answers a challenge posed by Bove, Krauss, and Sozeau.

We present a simple characterization of definable quotient types as being induced by idempotent functions, and an encoding of this in Cedille (a dependently typed programming language) in which both equational constraints and the packaging that associates these with elements of the carrier type are irrelevant, facilitating equational reasoning in proofs. We provide several concrete examples of definable quotients using this encoding and give combinators for function lifting (with one variant having zero run-time cost).