Related papers: A note on arithmetic in finite types
We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known non-deterministic and algebraic lambda-calculi.
Continuous reducibilities are a proven tool in computable analysis, and have applications in other fields such as constructive mathematics or reverse mathematics. We study the order-theoretic properties of several variants of the two most…
We describe a way to represent computable functions between coinductive types as particular transducers in type theory. This generalizes earlier work on functions between streams by P. Hancock to a much richer class of coinductive types.…
The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation…
The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward…
We define a model of predicate logic in which every term and predicate, open or closed, has an absolute denotation independently of a valuation of the variables. For each variable a, the domain of the model contains an element [[a]] which…
Propositional formulas that are equivalent in intuitionistic logic, or in its extension known as the logic of here-and-there, have the same stable models. We extend this theorem to propositional formulas with infinitely long conjunctions…
We show that diagrammatic sets, a topologically sound alternative to polygraphs and strict $\omega$-categories, admit an internal notion of equivalence in the sense of coinductive weak invertibility. We prove that equivalences have the…
This paper enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: {\sf All $x$ are $y$} and {\sf Some $x$ are…
We set up a parametrised monadic translation for a class of call-by-value functional languages, and prove a corresponding soundness theorem. We then present a series of concrete instantiations of our translation, demonstrating that a number…
Simple type theory is formulated for use with the generic theorem prover Isabelle. This requires explicit type inference rules. There are function, product, and subset types, which may be empty. Descriptions (the eta-operator) introduce the…
Abstract interpretation is a method to automatically find invariants of programs or pieces of code whose semantics is given via least fixed-points. Up-to techniques have been introduced as enhancements of coinduction, an abstract principle…
In this note, we presented a new decomposition of elements of finite fields of even order and illustrated that it is an effective tool in evaluation of some specific exponential sums over finite fields, the explicit value of some…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it.…
Motivated by Ziegler's computability-theoretic characterisation of finite absolute presentability between groups, we prove an analogous theorem in symbolic dynamics. We introduce the notion of one subshift being finitely determined over…
A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive…
The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define…
We develop a class of algebraic interpretations for many-sorted and higher-order term rewriting systems that takes type information into account. Specifically, base-type terms are mapped to \emph{tuples} of natural numbers and higher-order…
This paper proposes a definition of what it means for one system description language to encode another one, thereby enabling an ordering of system description languages with respect to expressive power. I compare the proposed definition…