Related papers: Monotone recursive types and recursive data repres…
We introduce the notion of identity coercions between non-indexed and indexed variants of inductive datatypes, such as lists and vectors. An identity coercion translates one type to another such that the coercion function definitionally…
We extend results from an earlier paper giving reconstruction results for the endomorphism monoid of the rational numbers under the strict and reflexive relations to the first order reducts of the rationals and the corresponding…
It is well known that general recursion cannot be expressed within Martin-Loef's type theory and various approaches have been proposed to overcome this problem still maintaining the termination of the computation of the typable terms. In…
Type systems certify program properties in a compositional way. From a bigger program one can abstract out a part and certify the properties of the resulting abstract program by just using the type of the part that was abstracted away.…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
We construct a realizability model of linear dependent type theory from a linear combinatory algebra. Our model motivates a number of additions to the type theory. In particular, we add a universe with two decoding operations: one takes…
We introduce a continuous-order integral analog of the Maclaurin expansion that reconstructs analytic functions from fractional derivative data. The operator integrates over continuous order, replacing the discrete sum of integer…
We show that restricting the elimination principle of the natural numbers type in Martin-L\"of Type Theory (MLTT) to a universe of types not containing $\Pi$-types ensures that all definable functions are primitive recursive. This extends…
We present an extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, and…
Temporal causal representation learning is a powerful tool for uncovering complex patterns in observational studies, which are often represented as low-dimensional time series. However, in many real-world applications, data are…
Monotone operator theory and fixed point theory for nonexpansive mappings are central areas in modern nonlinear analysis and optimization. Although these areas are fairly well developed, almost all examples published are based on…
We define the syntax and reduction relation of a recursively typed lambda calculus with a parallel case-function (a parallel conditional). The reduction is shown to be confluent. We interpret the recursive types as information systems in a…
We propose to use Church encodings in typed lambda-calculi as the basis for an automata-theoretic counterpart of implicit computational complexity, in the same way that monadic second-order logic provides a counterpart to descriptive…
Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with…
Higher-order recursion schemes are recursive equations defining new operations from given ones called "terminals". Every such recursion scheme is proved to have a least interpreted semantics in every Scott's model of \lambda-calculus in…
One may formulate the dependent product types of Martin-L\"of type theory either in terms of abstraction and application operators like those for the lambda-calculus; or in terms of introduction and elimination rules like those for the…
We investigate causal computations taking sequences of inputs to sequences of outputs where the $n$th output depends on the first $n$ inputs only. We model these in category theory via a construction taking a Cartesian category $C$ to…
We propose abstract compilation for precise static type analysis of object-oriented languages based on coinductive logic programming. Source code is translated to a logic program, then type-checking and inference problems amount to queries…
This paper is concerned with the foundations of the Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions by inductive data types. CAC generalizes inductive types equipped with higher-order primitive…
This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this…