Periodic Reporting for period 2 - TEAMDEP (Team semantics and dependence logic)
Periodo di rendicontazione: 2022-12-01 al 2024-05-31
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.
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, resulting in a Principles of Knowledge Representation and Reasoning 2023 Conference publication. Furthermore, we have examined conditional independence in semiring teams and algorithmic facets of entropic inequalities, leading to two single-authored publications scheduled for International Conference on Database Theory 2024. Additionally, we examined probabilistic team semantics (European Conference on Logics in Artificial Intelligence 2023) and the complexity of training neural networks (Thirty-Eighth AAAI Conference on Artificial Intelligence).
In set theory, with respect to the set theoretic multiverse part of the project, a lot of progress was made on the question of inner models from extended logics. The proof of the Continuum Hypothesis in C(aa) under large cardinal assumptions was finalized leading to the question whether so-called fine structure can be developed for models like C(aa). In general understanding of C(aa) took leaps ahead and was presented in a plenary talk in the European Set Theory Conference 2023. For example, the full GCH was proved for C(aa), again under large cardinal assumptions. Also, C(aa) was proved to possess higher measurable cardinals. The model C(aa) was enhanced to C(aa+) which seems a more stable model. All the basic results about C(aa) were, or are being, extended to C(aa+). Moreover, a subtle argument was given for C(aa+) to contain inner models with strong cardinals, which is a great improvement from the previous knowledge that C(aa) contains inner models with many measurable cardinals.
In set theory, with respect to the set theoretic multiverse part of the project, a lot of progress was made on the question of inner models from extended logics. The proof of the Continuum Hypothesis in C(aa) under large cardinal assumptions was finalized leading to the question whether so-called fine structure can be developed for models like C(aa). In general understanding of C(aa) took leaps ahead and was presented in a plenary talk in the European Set Theory Conference 2023. For example, the full GCH was proved for C(aa), again under large cardinal assumptions. Also, C(aa) was proved to possess higher measurable cardinals. The model C(aa) was enhanced to C(aa+) which seems a more stable model. All the basic results about C(aa) were, or are being, extended to C(aa+). Moreover, a subtle argument was given for C(aa+) to contain inner models with strong cardinals, which is a great improvement from the previous knowledge that C(aa) contains inner models with many measurable cardinals.
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 plant 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 second 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.