arXiv subjects

Niels Voorneveld

Publications and source records attributed to Niels Voorneveld.

9 recordsLinked to original sources

arXiv2026

A Simple Categorical Calculus of Interacting Processes

We present a calculus that models a simple sort of process interaction. Our calculus consists of a collection of terms together with a rewrite relation, parameterised by an arbitrary multicategory whose morphisms we understand as non-interactive processes. We show that our calculus is confluent and terminating, and that terms modulo the induced convertibility relation form a virtual double category. We relate our calculus to the free cornering of a monoidal category, which is a double-categorical model of process interaction that is similar in spirit to the calculus presented herein. Precisely, we construct a functor from the virtual double category given by our calculus into the underlying virtual double category of the free cornering of the free monoidal category on the multicategory of non-interacting processes. If we think of the terms of our calculus as programs and the rewriting system as an operational semantics for these programs, this functor gives a sound denotational semantics for our calculus in terms of the free cornering.

math.CT↗

arXiv2026

Modal Logic for Distributed Trust

We propose a method for reasoning about trust in multi-agent systems, specifying a language for describing communication protocols and making trust assumptions and derivations. This is given an interpretation in a modal logic for describing the beliefs and communications of agents in a network. We define how information in the network can be shared via forwarding, and how trust between agents can be generalized to trust across networks. We give specifications for the modal logic which can be readily adapted into a lambda calculus of proofs. We show that by nesting modalities, we can describe chains of communication between agents, and establish suitable notions of trust for such chains. We see how this can be applied to trust models in public key infrastructures, as well as other interaction protocols in distributed systems.

cs.LO↗

arXiv2026

Dynamics and Coherence for the Free Cornering with Protocol Choice

We present a term rewriting system that models the dynamic aspects of the free cornering with protocol choice of a monoidal category, which has been proposed as a categorical model of process interaction. This term rewriting system is confluent and terminating in an appropriate sense. We use this machinery to prove a coherence theorem for the free cornering with protocol choice.

math.CT↗

arXiv2025

Parametric Iteration in Resource Theories

Many algorithms are specified with respect to a fixed but unspecified parameter. Examples of this are especially common in cryptography, where protocols often feature a security parameter such as the bit length of a secret key. Our aim is to capture this phenomenon in a more abstract setting. We focus on resource theories -- general calculi of processes with a string diagrammatic syntax -- introducing a general parametric iteration construction. By instantiating this construction within the Markov category of probabilistic Boolean circuits and equipping it with a suitable metric, we are able to capture the notion of negligibility via asymptotic equivalence, in a compositional way. This allows us to use diagrammatic reasoning to prove simple cryptographic theorems -- for instance, proving that guessing a randomly generated key has negligible success.

cs.LO↗

arXiv2024

Protocol Choice and Iteration for the Free Cornering

We extend the free cornering of a symmetric monoidal category, a double categorical model of concurrent interaction, to support branching communication protocols and iterated communication protocols. We validate our constructions by showing that they inherit significant categorical structure from the free cornering, including that they form monoidal double categories. We also establish some elementary properties of the novel structure they contain. Further, we give a model of the free cornering in terms of strong functors and strong natural transformations, inspired by the literature on computational effects.

math.CT↗

arXiv2020

From Equations to Distinctions: Two Interpretations of Effectful Computations

There are several ways to define program equivalence for functional programs with algebraic effects. We consider two complementing ways to specify behavioural equivalence. One way is to specify a set of axiomatic equations, and allow proof methods to show that two programs are equivalent. Another way is to specify an Eilenberg-Moore algebra, which generate tests that could distinguish programs. These two methods are said to complement each other if any two programs can be shown to be equivalent if and only if there is no test to distinguish them. In this paper, we study a generic method to formulate from a set of axiomatic equations an Eilenberg-Moore algebra which complements it. We will look at an additional condition which must be satisfied for this to work. We then apply this method to a handful of examples of effects, including probability and global store, and show they coincide with the usual algebras from the literature. We will moreover study whether or not it is possible to specify a set of unary Boolean modalities which could function as distinction-tests complementing the equational theory.

cs.LO↗

arXiv2019

Behavioural Equivalence via Modalities for Algebraic Effects

The paper investigates behavioural equivalence between programs in a call-by-value functional language extended with a signature of (algebraic) effect-triggering operations. Two programs are considered as being behaviourally equivalent if they enjoy the same behavioural properties. To formulate this, we define a logic whose formulas specify behavioural properties. A crucial ingredient is a collection of modalities expressing effect-specific aspects of behaviour. We give a general theory of such modalities. If two conditions, openness and decomposability, are satisfied by the modalities then the logically specified behavioural equivalence coincides with a modality-defined notion of applicative bisimilarity, which can be proven to be a congruence by a generalisation of Howe's method. We show that the openness and decomposability conditions hold for several examples of algebraic effects: nondeterminism, probabilistic choice, global store and input/output.

cs.LO↗

arXiv2019

Quantitative Logics for Equivalence of Effectful Programs

In order to reason about effects, we can define quantitative formulas to describe behavioural aspects of effectful programs. These formulas can for example express probabilities that (or sets of correct starting states for which) a program satisfies a property. Fundamental to this approach is the notion of quantitative modality, which is used to lift a property on values to a property on computations. Taking all formulas together, we say that two terms are equivalent if they satisfy all formulas to the same quantitative degree. Under sufficient conditions on the quantitative modalities, this equivalence is equal to a notion of Abramsky's applicative bisimilarity, and is moreover a congruence. We investigate these results in the context of Levy's call-by-push-value with general recursion and algebraic effects. In particular, the results apply to (combinations of) nondeterministic choice, probabilistic choice, and global store.

cs.LO↗

arXiv2016

Extensions of Scott's Graph Model and Kleene's Second Algebra

We use a way to extend partial combinatory algebras (pcas) by forcing them to represent certain functions. In the case of Scott's Graph model, equality is computable relative to the complement function. However, the converse is not true. This creates a hierarchy of pcas which relates to similar structures of extensions on other pcas. We study one such structure on Kleene's second model and one on a pca equivalent but not isomorphic to it. For the recursively enumerable sub pca of the Graph model, results differ as we can compute the (partial) complement function using the equality.

math.LO↗