Related papers: Arithmetic as a theory modulo
Architectural Design Rewriting (ADR, for short) is a rule-based formal framework for modelling the evolution of architectures of distributed systems. Rules allow ADR graphs to be refined. After equipping ADR with a simple logic, we equip…
We introduce an operational rewriting-based semantics for strictly positive nested higher-order (co)inductive types. The semantics takes into account the "limits" of infinite reduction sequences. This may be seen as a refinement and…
We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.
Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…
An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are…
Dismal arithmetic is just like the arithmetic you learned in school, only simpler: there are no carries, when you add digits you just take the largest, and when you multiply digits you take the smallest. This paper studies basic number…
We generalise classical reconstruction results in algebra, using the language of monads, monoidal categories, module categories, as well as various notions of duality, such as closedness, Grothendieck--Verdier duality (also known as…
We provide a complete axiomatization of modal inclusion logic - team-based modal logic extended with inclusion atoms. We review and refine an expressive completeness and normal form theorem for the logic, define a natural deduction proof…
We describe several technical tools that prove to be efficient for investigating the rewrite systems associated with a family of algebraic laws, and might be useful for more general rewrite systems. These tools consist in introducing a…
On classes of functions defined on R^2n we introduce abstract composition laws modelled after the pseudodifferential product of symbols. We attach to these composition laws modulation mappings and spaces with useful algebraic and…
We show that numerous distinctive concepts of constructive mathematics arise automatically from an "antithesis" translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations,…
This paper proposes new derivations of three well-known sorting algorithms, in their functional formulation. The approach we use is based on three main ingredients: first, the algorithms are derived from a simpler algorithm, i.e. the…
We go into the need for, and the requirements on, a formal theory of budgets. We present a simple algebraic theory of rational budgets, i.e., budgets in which amounts of money are specified by functions on the rational numbers. This theory…
This paper proposes a modal typing system that enables us to handle self-referential formulae, including ones with negative self-references, which on one hand, would introduce a logical contradiction, namely Russell's paradox, in the…
We introduce a new formulation of the axiom of dependent choice that can be viewed as an abstract termination principle, which generalises the recursive path orderings used to establish termination of rewrite systems. We consider several…
This paper introduces Probabilistic Deduction (PD) as an approach to probabilistic structured argumentation. A PD framework is composed of probabilistic rules (p-rules). As rules in classical structured argumentation frameworks, p-rules…
The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with dependent types where beta-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type…
We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing…
We analyse the relationship between nominal algebra and nominal rewriting, giving a new and concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which…
The theory of modular deformations is generalized for the category of complex analytic polyhedra which includes germs of complex space as well as any compact complex analytic space. The objective of the theory is a construction of fine…