# Blog Archives

# Topic Archive: Sigma definability

# Σ-Presentable structures over *HF(R)*

If we replace in the definition of computable structures the concept of classical computability over ω with Σ-definability over the structure ** HF(R)** (the hereditarily finite superstructure over the ordered field

**of reals), we obtain its generalization, namely, the notion of Σ-presentable structures over**

*R***. This generalization could correspond to the hypothetical situation when we have an opportunity to use algorithms written in a powerful programming language with “real” reals, elements of**

*HF(R)***, not approximations, where we in addition have opportunity to find roots of polynomials and use them in further computations.**

*R*In the talk, a survey will be given of the results on the existence of Σ-presentations for various structures, on the number of non Σ-isomorphic presentations, and on the existence of Σ-parameterizations for classes of presentations.