Logical approach to quantum mechanics and contextuality
Final Report Abstract
The project explored probabilistic team semantics, a logical framework for studying probability distributions and dependencies, with a particular focus on applications in quantum information theory. The main objective was to enhance our understanding of quantum information theory by formalising key properties within a structured logical system. By developing new logical tools, the research provided a way to describe, and do inference with, dependencies in probabilistic systems, such as conditional independence, which are crucial for understanding quantum phenomena. The technical contributions of the project relate to building logical formalisms and tools, including deductive proof systems, to the study of expressiveness, and computational complexity of systems dealing with probability distributions. A major contribution of the project was the development of logical and complexity theoretic tools that can be used to study logical foundations of quantum mechanics. By modelling empirical probability distributions, the research shed light on key quantum concepts, including quantum contextuality and hidden-variable models. Essential properties of quantum information theory, such as outcome independence and weak determinism, can be effectively expressed using the developed logical framework. Additionally, the study established connections between logical languages and computational complexity, offering insights into the classification of logical expressiveness and the computational power required to analyse probability distributions. In particular, we identified strong connections between complexity theory utilising real number arithmetic, logics with probabilistic team semantics, and logics that can express real arithmetic operations. Beyond the applications mentioned above, the project also contributed to the study of causal reasoning in connection to dependencies that arise from probabilistic data. Here we focussed on the study of expressivity and logical inference methods utilising interventionist counterfactuals in probabilistic causal models. Finally, we developed a novel framework for modelling quantum contextuality by the use of so-called probabilistic polyteams. The fundamental theory developed in the project related to quantitative logics and complexity theory has already provided insights and theoretical results in the fields of machine learning and database theory. In particular, they provided new insights into the complexity of training and expressive power of neural networks, and inspired development of new frameworks and techniques for consistent query answering in the setting of diversely annotated inconsistent data.
Publications
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Tractability Frontiers in Probabilistic Team Semantics and Existential Second-Order Logic over the Reals. Lecture Notes in Computer Science, 262-278. Springer International Publishing.
Hannula, Miika & Virtema, Jonni
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Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals. Annals of Pure and Applied Logic, 173(10), 103108.
Hannula, Miika & Virtema, Jonni
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Logics with Probabilistic Team Semantics and the Boolean Negation. Lecture Notes in Computer Science, 665-680. Springer Nature Switzerland.
Hannula, Miika; Hirvonen, Minna; Kontinen, Juha; Mahmood, Yasir; Meier, Arne & Virtema, Jonni
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Strongly Complete Axiomatization for a Logic with Probabilistic Interventionist Counterfactuals. Lecture Notes in Computer Science, 649-664. Springer Nature Switzerland.
Barbero, Fausto & Virtema, Jonni
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Unified Foundations of Team Semantics via Semirings. Proceedings of the Twentieth International Conference on Principles of Knowledge Representation and Reasoning, 75-85. International Joint Conferences on Artificial Intelligence Organization.
Barlag, Timon; Hannula, Miika; Kontinen, Juha; Pardal, Nina & Virtema, Jonni
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Expressivity Landscape for Logics with Probabilistic Interventionist Counterfactuals. In: 32nd EACSL Annual Conference on Computer Science Logic, CSL 2024, February 19-23, 2024, Naples, Italy
Fausto Barbero & Jonni Virtema
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Logics for Dependence and Independence: Expressivity and Complexity (Dagstuhl Seminar 24111). In: Dagstuhl Reports 14.3 (2024), pp. 31–51
Juha Kontinen, Jonni Virtema, Heribert Vollmer, Fan Yang & Nicolas Fröhlich
