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We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of…

This paper describes an algorithm for the computation of FIRST and FOLLOW sets for use with feature-theoretic grammars in which the value of the sets consists of pairs of feature-theoretic categories. The algorithm preserves as much…

One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…

Comparison-based algorithms are algorithms for which the execution of each operation is solely based on the outcome of a series of comparisons between elements. Comparison-based computations can be naturally represented via the following…

We present a program logic for Pitts and Stark's {\nu}-calculus, an extension of the call-by-value simply-typed {\lambda}-calculus with a mechanism for the generation of fresh names. Names can be compared for (in)-equality, producing…

This paper discusses the formalization of proofs "by diagram chasing", a standard technique for proving properties in abelian categories. We discuss how the essence of diagram chases can be captured by a simple many-sorted first-order…

Initiated by Abramsky [1994], the Proofs as Processes agenda is to establish a solid foundation for the study of concurrent languages, by researching the connection between linear logic and the $\pi$-calculus. To date, Proofs as Processes…

Tackling large approximate dynamic programming or reinforcement learning problems requires methods that can exploit regularities, or intrinsic structure, of the problem in hand. Most current methods are geared towards exploiting the…

We introduce a proper display calculus for first-order logic, of which we prove soundness, completeness, conservativity, subformula property and cut elimination via a Belnap-style metatheorem. All inference rules are closed under uniform…

In the logic programming paradigm, it is difficult to develop an elegant solution for generating distinguishing formulae that witness the failure of open-bisimilarity between two pi-calculus processes; this was unexpected because the…

We define BioScapeL, a stochastic pi-calculus in 3D-space. A novel aspect of BioScapeL is that entities have programmable locations. The programmer can specify a particular location where to place an entity, or a location relative to the…

We consider the problem of answering queries about formulas of first-order logic based on background knowledge partially represented explicitly as other formulas, and partially represented as examples independently drawn from a fixed…

In this paper, our aim is to briefly survey and articulate the logical and philosophical foundations of using (first-order) logic to represent (probabilistic) knowledge in a non-technical fashion. Our motivation is three fold. First, for…

This note discusses proofs for convergence of first-order methods based on simple potential-function arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants.

The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension…

In this paper, we enrich the pi-calculus with an operator for confidentiality (hide), whose main effect is to restrict the access to the object of the communication, thus representing confidentiality in a natural way. The hide operator is…

Reasoning under uncertainty is a fundamental challenge in Artificial Intelligence. As with most of these challenges, there is a harsh dilemma between the expressive power of the language used, and the tractability of the computational…

We define an extension of lambda-calculus with dependents types that enables us to encode transparent and opaque probabilistic programs and prove a strong normalisation result for it by a reducibility technique. While transparent…

The logic of information flows (LIF) has recently been proposed as a general framework in the field of knowledge representation. In this framework, tasks of procedural nature can still be modeled in a declarative, logic-based fashion. In…

Computability relative to a partial function $f$ on the natural numbers can be formalized using the notion of an oracle for this function $f$. This can be generalized to arbitrary partial combinatory algebras, yielding a notion of…