List of notions of "model abstraction" [lcc-0015]

The morphisms in Categorical Systems Theory

There are actually two notions of morphism between systems, here. In both cases, you have a mapping on the "state space", and some sort of mapping on the input/output interface which needs to be compatible.

In the case of lenses, you have a map forwards on the output, and a map backwards on the input - this clearly allows you to change the interface of the system, and then you require that the map on state is dynamics-preserving after changing interface.

In the case of charts, everything goes in the same direction, and you can ask for compatibility between the systems, but you don't have a change-of-interface functor.

Abstractions of causal DAG models

See eg Visual causal feature learning, Multi-level cause-effect systems, Causal feature learning: An overview, Approximate causal abstraction, Compositional abstraction error and a category of causal models.

There is a complicated literature on these, but generally they are maps between the outcome space of causal models which in some sense preserve the interventional distributions.

There is a question of bidirectionality, discussed through these papers (eg in Compositional abstraction error and a category of causal models), which precisely mirrors the distinction between lenses and charts in Categorical Systems Theory. Namely, in order to interpret high-level interventions as low-level interventions, we need a map "backwards" (this is muddled by the fact that the set of inputs, possible interventions, is usually equal to the set of outputs in this case).

Abstractions of Kripke Modal Transition Systems

A KMTS is a "partially-defined" transition system, and we are interested in binary relations from one such to another which have the property of being "directed bisimulation" with respect to the partiality - i.e, morally speaking, it would be a bisimulation if we replaced one system with a "more specific" one. See eg Automatic Abstraction Using Generalized Model Checking

References

Reference Compositional abstraction error and a category of causal models [rischel_compositional_2021]

Abstract

Interventional causal models describe several joint distributions over some variables used to describe a system, one for each intervention setting. They provide a formal recipe for how to move between the different joint distributions and make predictions about the variables upon intervening on the system. Yet, it is difficult to formalise how we may change the underlying variables used to describe the system, say moving from fine-grained to coarse-grained variables. Here, we argue that compositionality is a desideratum for such model transformations and the associated errors: When abstracting a reference model M iteratively, first obtaining M' and then further simplifying that to obtain M'', we expect the composite transformation from M to M'' to exist and its error to be bounded by the errors incurred by each individual transformation step. Category theory, the study of mathematical objects via compositional transformations between them, offers a natural language to develop our framework for model transformations and abstractions. We introduce a category of finite interventional causal models and, leveraging theory of enriched categories, prove the desired compositionality properties for our framework. 

@misc{rischel_compositional_2021,
 author = {Rischel, Eigil F. and Weichwald, Sebastian},
 date = {2021-08-05},
 doi = {10.48550/arXiv.2103.15758},
 eprint = {2103.15758 [cs, math, stat]},
 eprinttype = {arxiv},
 keywords = {Computer Science - Artificial Intelligence, Computer Science - Logic in Computer Science, Computer Science - Machine Learning, Mathematics - Category Theory, Statistics - Machine Learning},
 number = {{arXiv}:2103.15758},
 publisher = {{arXiv}},
 title = {Compositional Abstraction Error and a Category of Causal Models},
 url = {http://arxiv.org/abs/2103.15758},
 urldate = {2024-05-25}
}

Reference Approximate causal abstraction [beckers_approximate_2019]

 
@article{beckers_approximate_2019,
 abstract = {Scientific models describe natural phenomena at different levels of abstraction. Abstract descriptions can provide the basis for interventions on the system and explanation of observed phenomena at a level of granularity that is coarser than the most fundamental account of the system. Beckers and Halpern (2019), building on work of Rubenstein et al. (2017), developed an account of abstraction for causal models that is exact. Here we extend this account to the more realistic case where an abstract causal model offers only an approximation of the underlying system. We show how the resulting account handles the discrepancy that can arise between low- and highlevel causal models of the same system, and in the process provide an account of how one causal model approximates another, a topic of independent interest. Finally, we extend the account of approximate abstractions to probabilistic causal models, indicating how and where uncertainty can enter into an approximate abstraction.},
 author = {Beckers, Sander and Eberhardt, Frederick and Halpern, Joseph Y.},
 date = {2019-06-29},
 doi = {10.48550/arXiv.1906.11583},
 eprint = {1906.11583},
 eprinttype = {arxiv},
 file = {Full Text:/home/eigil/Zotero/storage/WIY34UXR/Beckers et al. - 2019 - Approximate Causal Abstraction.pdf:application/pdf},
 journaltitle = {{arXiv}:1906.11583 [cs]},
 keywords = {Computer Science - Artificial Intelligence},
 langid = {english},
 title = {Approximate Causal Abstraction},
 url = {http://arxiv.org/abs/1906.11583},
 urldate = {2021-04-04}
}

Reference Causal feature learning: An overview [chalupka_causal_2017]

 
@article{chalupka_causal_2017,
 abstract = {Causal feature learning ({CFL}) (Chalupka et al., Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence. {AUAI} Press, Edinburgh, pp 181–190, 2015) is a causal inference framework rooted in the language of causal graphical models (Pearl J, Reasoning and inference. Cambridge University Press, Cambridge, 2009; Spirtes et al., Causation, Prediction, and Search. Massachusetts Institute of Technology, Massachusetts, 2000), and computational mechanics (Shalizi, {PhD} thesis, University of Wisconsin at Madison, 2001). {CFL} is aimed at discovering high-level causal relations from low-level data, and at reducing the experimental effort to understand confounding among the high-level variables. We first review the scientific motivation for {CFL}, then present a detailed introduction to the framework, laying out the definitions and algorithmic steps. A simple example illustrates the techniques involved in the learning steps and provides visual intuition. Finally, we discuss the limitations of the current framework and list a number of open problems.},
 author = {Chalupka, Krzysztof and Eberhardt, Frederick and Perona, Pietro},
 date = {2017-01},
 doi = {10.1007/s41237-016-0008-2},
 file = {mscChalupka2017_Article_CausalFeatureLearningAnOvervie.pdf:/home/eigil/Dropbox/Matematik/Tekster/mscChalupka2017_Article_CausalFeatureLearningAnOvervie.pdf:application/pdf},
 issn = {0385-7417, 1349-6964},
 journaltitle = {Behaviormetrika},
 langid = {english},
 number = {1},
 pages = {137--164},
 shortjournal = {Behaviormetrika},
 shorttitle = {Causal feature learning},
 title = {Causal feature learning: an overview},
 url = {http://link.springer.com/10.1007/s41237-016-0008-2},
 urldate = {2021-04-04},
 volume = {44}
}

Reference Visual causal feature learning [chalupka_visual_2015]

 
@article{chalupka_visual_2015,
 abstract = {We provide a rigorous definition of the visual cause of a behavior that is broadly applicable to the visually driven behavior in humans, animals, neurons, robots and other perceiving systems. Our framework generalizes standard accounts of causal learning to settings in which the causal variables need to be constructed from micro-variables. We prove the Causal Coarsening Theorem, which allows us to gain causal knowledge from observational data with minimal experimental effort. The theorem provides a connection to standard inference techniques in machine learning that identify features of an image that correlate with, but may not cause, the target behavior. Finally, we propose an active learning scheme to learn a manipulator function that performs optimal manipulations on the image to automatically identify the visual cause of a target behavior. We illustrate our inference and learning algorithms in experiments based on both synthetic and real data.},
 author = {Chalupka, Krzysztof and Perona, Pietro and Eberhardt, Frederick},
 date = {2015-06-04},
 doi = {10.48550/arXiv.1412.2309},
 file = {msc109.pdf:/home/eigil/Dropbox/Matematik/Tekster/msc109.pdf:application/pdf},
 langid = {english},
 pages = {10},
 title = {Visual Causal Feature Learning}
}

Reference Multi-level cause-effect systems [chalupka_multi-level_2015]

 
@article{chalupka_multi-level_2015,
 abstract = {We present a domain-general account of causation that applies to settings in which macro-level causal relations between two systems are of interest, but the relevant causal features are poorly understood and have to be aggregated from vast arrays of micro-measurements. Our approach generalizes that of Chalupka et al. (2015) to the setting in which the macro-level effect is not specified. We formalize the connection between micro- and macro-variables in such situations and provide a coherent framework describing causal relations at multiple levels of analysis. We present an algorithm that discovers macro-variable causes and effects from microlevel measurements obtained from an experiment. We further show how to design experiments to discover macro-variables from observational micro-variable data. Finally, we show that under specific conditions, one can identify multiple levels of causal structure. Throughout the article, we use a simulated neuroscience multiunit recording experiment to illustrate the ideas and the algorithms.},
 author = {Chalupka, Krzysztof and Perona, Pietro and Eberhardt, Frederick},
 date = {2015},
 doi = {10.48550/arXiv.1512.07942},
 file = {Full Text:/home/eigil/Zotero/storage/VSCBF4YS/Chalupka et al. - Multi-Level Cause-Effect Systems.pdf:application/pdf},
 langid = {english},
 pages = {13},
 title = {Multi-Level Cause-Effect Systems}
}

Reference Automatic Abstraction Using Generalized Model Checking [godefroid-jagadeesan-automatic-abstraction]

Abstract

  • 2002
  • Patrice Godefroid, Radha Jagadeesan

Generalized model checking is a framework for reasoning about partial state spaces of concurrent reactive systems. The state space of a system is only partial" (partially known) when a full state-space exploration is not computationally tractable, or when abstraction techniques are used to simplify the system's representation. In the context of automatic abstraction, generalized model checking means checking whether there exists a concretization of an abstraction that satis es a temporal logic formula. In this paper, we show how generalized model checking can extend existing automatic abstraction techniques (such as predicate abstraction) for model checking concurrent/reactive programs and yield the three following improvements: (1) any temporal logic formula can be checked (not just universal properties as with traditional conservative abstractions), (2) correctness proofs and counter-examples are both guaranteed to be sound, and (3) veri cation results can be more precise. We study the cost needed to improve precision by presenting new upper and lower bounds for the complexity of generalized model checking in the size of the abstraction. 

 @inbook
{Godefroid_Jagadeesan_2002, address={Berlin, Heidelberg}, series={Lecture Notes in Computer Science}, title={Automatic Abstraction Using Generalized Model Checking}, volume={2404}, ISBN={978-3-540-43997-4}, url={http://link.springer.com/10.1007/3-540-45657-0_11}, DOI={10.1007/3-540-45657-0_11}, booktitle={Computer Aided Verification}, publisher={Springer Berlin Heidelberg}, author={Godefroid, Patrice and Jagadeesan, Radha}, editor={Brinksma, Ed and Larsen, Kim Guldstrand}, year={2002}, pages={137–151}, collection={Lecture Notes in Computer Science},language={en} }

Reference Categorical Systems Theory [myers-cst]

 
 @misc
{myers-cst, title={Categorical Systems Theory}, author={Myers, David Jaz},language={en} , url={http://davidjaz.com/Papers/DynamicalBook.pdf}}