World Logic Day in Aveiro

Department of Mathematics, University of Aveiro, 14th Jan, 2021

Part of 3rd WORLD LOGIC DAY

Schedule

16:00-16:30 - Eros Martinelli
16:30-17:00 - Leandro Gomes
17:00-18:00 - Rui Soares Barbosa

Invited Talk

Partial Boolean algebras: the logic of contextuality

ABSTRACT: Contextually is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum computational advantage. Kochen and Specker’s seminal work on contextuality contains elements of a logical flavour that have largely been overlooked in subsequent literature on the topic. In particular, it introduced the notion of partial Boolean algebra, which provides a natural (algebraic) logical setting for studying contextual systems. It contrasts with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, being only defined in the domain where they are physically meaningful. In the key example of the projectors on a Hilbert space, the operations are only defined for commuting projectors, which correspond to properties of a quantum system that are commensurable, \ie can be measured simultaneously.
In this talk, we will give an introduction to partial Boolean algebras and discuss various topics arising in our recent work, including:
– the formulation of contextuality properties in this setting, including Kochen–Specker paradoxes, logically contradictory statements that are validated by a partial Boolean algebra;
– the Logical Exclusivity Principle and its relation to Probabilistic Exclusivity widely studied in the quantum foundations literature as a step to closing in on the set of quantum-realisable correlations;
– work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.
A central role in this is played by a universal construction that freely extends the commeasurability relation on a partial Boolean algebra. This is given through a concrete inductive presentation by generators and relations.
This is joint work with Samson Abramsky, and can be found at arXiv:2011.03064 [quant-ph].

Contributed Talks

Eros Martinelli

Injective Hulls of Quantale-Enriched Multicategories

ABSTRACT:

It is well known that there is a connection between the Dedekind-MacNeille completion of an ordered set and its injective hull. Namely, given a poset $X$, its injective hull (with respect to embeddings) is the Dedekind-MacNeille completion of $X$. This construction can be generalize to the realm of quantale-enriched categories where, in a similar way, one can build injective hulls as algebras for the monad that arises from the Isbell adjuction.\\ In this talk we study this problem in the realm of quantale-enriched multicategories, a generalization of promonoidal categories. This kind of categories naturally appear when one wants to "mix" cocompleteness with monoidal completions. The classical example is the construction of the free quantale $Q$ starting from an ordered set $X$. First one has to generate an ordered monoid out of $X$ by taking finite lists, then one has to add all possible suprema in order to make it complete. In this way, every element of $Q$ is a suprema of lists of element of $X$; this is a particular example of "monoidal" colimit, which naturally arises when one study colimits for quantale-enriched multicategories. Unfortunately, the situation is not so smooth as in the "classical" case. In order to be able to construct injective hulls, we have to make a detour to the category of the so-called quantum B-algebras, representable promonoidal categories. For this category we will be able to mimic all the constructions done for quantale-enriched categories and build injective hulls as algebras for a lax-monoidal monad which resembles the one induced by the Isbell adjuction. Luckily, the restriction to quantum B-algebras does not prevent us from constructing an injective hull for every multicategory, by embedding every multicategory in a (suitable) quantum B-algebra we will provide the injective hull we were searching for.

Leandro Gomes

Semantics and dynamic logics for fuzzy programs

ABSTRACT:

References:

Local Organisers

Manuel A. Martins

Alexandre Madeira