The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the $\Sigma_1$ -fragment of the theory Th(R) is decidable.

We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is ${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is $0''$. We also prove that, with a bit of work, the latter result can be pushed beyond $\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits.

As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.

The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

This paper presents two extensions of the second order polymorphiclambda calculus, system F, with monotone (co)inductive types supporting(co)iteration, primitive (co)recursion and inversion principles asprimitives. One extension is inspired by the usual categoricalapproach to programming by means of initial algebras and finalcoalgebras; whereas the other models dialgebras, and can be seen as an extension of Hagino'scategorical lambda calculus within the framework of parametricpolymorphism. The systems are presented in Curry-style, and are proven to be terminating andtype-preserving. Moreovertheir expressiveness is shown by means of several programmingexamples, going from usual data types to lazy codata types such as streamsor infinite trees.

We give a complete characterization of the class of functions that are the intensional behaviours of primitive recursive (PR) algorithms. This class is the set of primitive recursive functions that have a null basic case of recursion. This result is obtained using the property of ultimate unarity and a geometrical approach of sequential functions on N the set of positive integers.

We study five extensions of the polymorphically typed lambda-calculus (system F) by type constructs intended to model fixed-points of monotone operators. Building on work by Geuvers concerning the relation between term rewrite systems for least pre-fixed-points and greatest post-fixed-points of positive type schemes (i.e., non-nested positive inductive and coinductive types) and so-called retract types, we show that there are reduction-preserving embeddings even between systems of monotone (co)inductive types and non-inter leav ing positive fixed-point types (which are essentially those retract types). The reduction relation considered is β- and η-reduction for system FF plus either (full) primitive recursion on the inductive types or (full) primitive corecursion on the coinductive types or an extremely simple rule for the fixed-point types. Monotonicity is not confined to the syntactic restriction on type formation of having only positive occurrences of the type variable α in ρ for the inductive type µαρ or the coinductive type ναρ. Instead of that only a “monotonicity witness” which is a term of type ∀α∀β.(α → β) → ρ → ρ[α:=β] is required. This term may already use (co)recursion such that our monotone (co)inductive types may even be “interleaved” and not only nested.