Section: New Results

Computing with Infinite Objects

Participant : Mathieu Hoyrup.

• Decidable properties of subrecursive functions

  We have studied the following problem : given a subrecursive class (like the primitive recursive functions, the polynomial-time computable functions, etc.) and a sound and complete programming language for that class, what are the properties of functions that are decidable (by a Turing machine), given a program for that function in the restricted language? We give a complete characterization of these properties. We show that they can be expressed as unions of elementary properties of being compressible. If $h:\mathbb{N}\to \mathbb{N}$ is a computable increasing unbounded function (like $log\left(n\right)$ or $2^n$), we say that a function $f:\mathbb{N}\to \mathbb{N}$ is $h$-compressible if for each $n$ there is a program (in the restricted language) of size at most $h\left(n\right)$ computing a function that coincides with $f$ on entries $0,1,...,n$. Whether $f$ is $h$-compressible is decidable given a program for $f$, and every decidable property can be obtained as a combination of such elementary properties.

  We also prove that such a characterization does not hold for the whole class of total recursive functions, and leave the problem open for that class.

  The results appears in an article presented at ICALP 2016 [19].

• Baire category and computability theory

  Baire category is a very powerful tool in mathematical analysis to prove existence of objects with prescribed properties without having to explicitly build them, but showing instead that the class of objects with these properties is large in some sense. In Computability theory one often builds objects with very specific properties, notably to separate classes, and the proofs are often very involved. We show how Baire category can be adapted in order to be applied to computability theory, to prove existence results without the need of an explicit construction. We review notions that we introduced in the last years and provide new results in an invited paper at CiE 2016 [14].