Computational Logic Seminar

General Information

This is the website of the research seminar of the Computational Logic Group at the Institute of Discrete Mathematics and Geometry of TU Wien. The seminar usually takes places on Wednesdays from 10:00 to 11:00 in the seminar room DC red 07. The seminar is organised by J. Aguilera, E. Fokina and S. Hetzl.

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January 14, 2026: Iosif Petrakis
title: The "complemented subsets" point of view
abstract:
Bishop and Cheng introduced complemented subsets as positive and strong counterparts to subsets in their constructive development of the Daniell approach to measure theory. A complemented subset is a pair of subsets (A¹, A⁰), where A¹ and A⁰ are disjoint in a positive and strong sense. While constructively the weak and strong complements of a subset have a poor algebraic behaviour, the swapped pair (A⁰, A¹) is a well-behaved notion of a constructive complement of (A¹, A⁰). In this talk we give an overview of recent developments within the "complemented subsets" point of view.
The abstract algebraic properties of the complemented powerset define the notion of a swap algebra, a generalisation of a Boolean algebra, while the abstract properties of partial, Boolean-valued functions define the notion of a swap ring, a generalisation of a Boolean ring. An orthocomplemented subspace of a Hilbert space H is a pair (L¹, L⁰) of orthogonal, closed subspaces L¹ and L⁰ of H. Orthocomplemented subspaces correspond to partial projections on H, and provide new models of constructive quantum logic. Topologies of open complemented subsets constitute a new approach to constructive point-set topology, while in constructive computability theory a recursive complemented set is a pair (A¹, A⁰), where A¹ and A⁰ are recursively enumerable subsets of N.