Team semantics and dependence logic

Project Information

TEAMDEP

Grant agreement ID: 101020762

DOI

10.3030/101020762

Project closed

EC signature date 8 May 2021

Start date 1 June 2021

End date 31 May 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 2 496 146,25

EU contribution

€ 2 496 146,25

Coordinated by

HELSINGIN YLIOPISTO
Finland

Periodic Reporting for period 3 - TEAMDEP (Team semantics and dependence logic)

Reporting period: 2024-06-01 to 2025-11-30

Summary of the context and overall objectives of the project

The overall goal of this project is to reach a better understanding of higher order concepts in mathematics, with applications to computer science, physics, and philosophy. The main innovation is to use so-called team semantics, developed by the PI in his earlier work. A team is essentially a relation, a collection of tuples, on a fixed domain. A tuple can be thought of as a function from a fixed finite set of attributes to the domain. If the attributes are e.g. “name”, “rank” and “salary”, then examples of possible tuples are e.g. (Jones, typist, 50.000) (Brown, director, 120.000) and (Taylor, secretary, 50.000). By studying dependences and independence in such teams of tuples a surprising amount of information can be unearthed. Technically speaking anything that is existential second order, e.g. functional dependence, can be expressed. This turns out to be useful in applications to database theory. On the other hand, teams offer a technical tool to address many questions involving plurality, e.g. non-locality phenomena in Quantum Mechanics, or independence phenomena in the multiverse of set theory. More exactly the goal is to axiomatize and find tractable fragments of existential second order, which itself is seriously non-axiomatizable (Pi-3-complete) and intractable (NO-complete). On the plurality side the goal is to develop a new understanding of inner models of set theory. The importance of this project to society stems from the ubiquity of the concepts of dependence and independence in society. In this project we develop a philosophical and mathematical theory of dependence and independence concepts that covers applications from database theory to Quantum Foundations and to the multiverse of set theory.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

We have explored dependencies and team semantics using a fresh approach grounded in information theory and semiring provenance. Accordingly, we have developed a novel semiring team semantics, which unifies various forms of team semantics and dependency notions (Publ. 50, 51, and 52.) Furthermore, we have examined conditional independence in semiring teams and algorithmic facets of entropic inequalities. Additionally, we examined probabilistic team semantics and the complexity of training neural networks. We developed dimension theoretic methods for team based logics (Publ. 6 and 47). In the propositional setting, many team theoretical atoms can be defined in terms of atoms of smaller arity but complexity of the formula increases in the sense that the defining formula gets longer with more occurrences of the atoms. We quantified this increase of complexity using the arity and the number of times the atom occurs in a formula as parameters and obtained several succinctness results using dimension theoretic methods. Further steps were taken in the team semantics approach to Quantum Foundations (Research Question Q2), based on the progress on Research Question Q1, and a new version of Publ. 10 was accepted to a journal. In set theory a lot of progress was made on the question of inner models from extended logics (Research Question Q3). A new version of Publ. 4 was produced and accepted to a journal. In general, understanding of C(aa) increased substantially. It was proved that C(aa) does not contain an inner model with a Woodin cardinal, which improves the earlier result in publication 4 to the effect that there are no Woodin cardinals in C(aa). Also, C(aa) was proved to possess higher measurable cardinals. The model C(aa) was enhanced to C(aa+) (Publ. 53), obtained by incorporating to C(aa) the concept of the next admissible set. A new interesting inner model was introduced based on a fragment of second order logic (Publ. 33). A major breakthrough is Publ. 38 in which it is proved consistent, relative to the consistency of a weakly compact cardinal, that there is a maximal wide aleph-2-Aronszajn tree. This solves a well-known problem that has been open for 30 years. The result was established for aleph-1, too (Publ. 46).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

In team semantics, our semiring team semantics is a clear step beyond the state of the art. Our approach covers the kind of probabilistic team semantics that we hope to use in foundations of Quantum Mechanics and the multiteam semantics that is used in database theory. We plan to extend this by means of polyteams to cover mixed states of Quantum Mechanics and the Kochen Specker type phenomena that do not fall naturally to the ordinary probabilistic team semantics framework. With respect to the set theoretic multiverse part of the project, our progress so far already goes well beyond the state of the art, for example in our work on GCH for C(aa) and our work on C(aa+). These models are of an entirely new kind that have (essentially) not been studied in set theory before. Our plan for the third half of the grant period is to push forward to further extensions of C(aa) by moving from the almost all (aa) quantifier to more general game quantifiers thereby obtaining models with bigger large cardinals and inner models for even larger cardinals. On a different note, we will investigate connections with the multiverse approach to set theory and the semiring team semantics.

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