A categorical semantics of fuzzy concepts in conceptual spaces. (English) Zbl 1530.18013
Kishida, Kohei (ed.), Proceedings of the fourth international conference on applied category theory 2021, ACT 2021, Cambridge, UK, July 12–16, 2021. Waterloo: Open Publishing Association (OPA). Electron. Proc. Theor. Comput. Sci. (EPTCS) 372, 306-322 (2022).
Summary: We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within Gärdenfors’ framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these are the most general choice satisfying a criterion due to Gärdenfors and which are well-behaved compositionally. We then generalise these to define the category of log-concave probabilistic channels between convex spaces, which allows one to model fuzzy reasoning with noisy inputs, and provides a novel example of a Markov category.
For the entire collection see [Zbl 1522.68034].
For the entire collection see [Zbl 1522.68034].
References:
| [1] | Janet Aisbett & Greg Gibbon (2001): A general formulation of conceptual spaces as a meso level represen-tation. Artificial Intelligence 133(1-2), pp. 189-232, doi:10.1016/s0004-3702(01)00144-8. · Zbl 0992.68195 · doi:10.1016/s0004-3702(01)00144-8 |
| [2] | Lucas Bechberger & Kai-Uwe Kühnberger (2017): A thorough formalization of conceptual spaces. In: Joint German/Austrian Conference on Artificial Intelligence (Künstliche Intelligenz), Springer, pp. 58-71, doi:10.1007/978-3-319-67190-1 5. · Zbl 1498.68308 · doi:10.1007/978-3-319-67190-1_5 |
| [3] | Joe Bolt, Bob Coecke, Fabrizio Genovese, Martha Lewis, Dan Marsden & Robin Piedeleu (2019): Interact-ing conceptual spaces I: Grammatical composition of concepts. In: Conceptual Spaces: Elaborations and Applications, Springer, pp. 151-181, doi:10.1007/978-3-030-12800-5 9. · doi:10.1007/978-3-030-12800-5_9 |
| [4] | Kenta Cho & Bart Jacobs (2019): Disintegration and Bayesian inversion via string diagrams. Mathematical Structures in Computer Science 29(7), pp. 938-971, doi:10.1017/s0960129518000488. · Zbl 1452.18008 · doi:10.1017/s0960129518000488 |
| [5] | Bob Coecke (2021): The Mathematics of Text Structure. In: Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics, Springer International Publishing, pp. 181-217, doi:10.1007/978-3-030-66545-6 6. · Zbl 1482.81006 · doi:10.1007/978-3-030-66545-6_6 |
| [6] | Bob Coecke, Fabrizio Genovese, Martha Lewis, Dan Marsden & Alex Toumi (2018): Generalized relations in linguistics & cognition. Theoretical Computer Science 752, pp. 104-115. · Zbl 1408.91187 |
| [7] | Bob Coecke & Martha Lewis (2015): A compositional explanation of the ’pet fish’phenomenon. In: Interna-tional Symposium on Quantum Interaction, Springer, pp. 179-192, doi:10.1007/978-3-319-28675-4 14. · Zbl 1435.81025 · doi:10.1007/978-3-319-28675-4_14 |
| [8] | Bob Coecke, Mehrnoosh Sadrzadeh & Stephen Clark (2010): Mathematical foundations for a compositional distributional model of meaning. arXiv preprint arXiv:1003.4394. |
| [9] | Antonin Delpeuch (2020): Autonomization of Monoidal Categories. Electronic Proceedings in Theoretical Computer Science 323, pp. 24-43, doi:10.4204/eptcs.323.3. Available at https://doi.org/10.4204 · Zbl 07453969 · doi:10.4204/eptcs.323.3 |
| [10] | P Erdős & Artur H Stone (1970): On the sum of two Borel sets. Proceedings of the American Mathematical Society 25(2), pp. 304-306, doi:10.2307/2037209. · Zbl 0192.40304 · doi:10.2307/2037209 |
| [11] | Jerry Fodor & Ernest Lepore (1996): The red herring and the pet fish: Why concepts still can’t be prototypes. Cognition 58(2), pp. 253-270, doi:10.1016/0010-0277(95)00694-X. · doi:10.1016/0010-0277(95)00694-X |
| [12] | Tobias Fritz (2020): A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Advances in Mathematics 370, p. 107239, doi:10.1016/j.aim.2020.107239. · Zbl 1505.60004 · doi:10.1016/j.aim.2020.107239 |
| [13] | Tobias Fritz, Tomáš Gonda, Paolo Perrone & Eigil Fjeldgren Rischel (2020): Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability. arXiv preprint arXiv:2010.07416. |
| [14] | Peter Gärdenfors (1993): The emergence of meaning. Linguistics and Philosophy 16(3), pp. 285-309. |
| [15] | Peter Gärdenfors (2004): Conceptual spaces: The geometry of thought. MIT press, doi:10.7551/mitpress/2076.001.0001. · doi:10.7551/mitpress/2076.001.0001 |
| [16] | Peter Gärdenfors (2014): The geometry of meaning: Semantics based on conceptual spaces. MIT press, doi:10.7551/mitpress/9629.001.0001. · Zbl 1303.00038 · doi:10.7551/mitpress/9629.001.0001 |
| [17] | Peter Gärdenfors & Mary-Anne Williams (2001): Reasoning about categories in conceptual spaces. In: IJCAI, pp. 385-392. |
| [18] | Michèle Giry (1982): A categorical approach to probability theory. In: Lecture Notes in Mathematics, Springer Berlin Heidelberg, pp. 68-85, doi:10.1007/bfb0092872. · Zbl 0486.60034 · doi:10.1007/bfb0092872 |
| [19] | Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mo-hamed & Alexander Lerchner (2016): beta-vae: Learning basic visual concepts with a constrained varia-tional framework. |
| [20] | Boaz Klartag & VD Milman (2005): Geometry of log-concave functions and measures. Geometriae Dedicata 112(1), pp. 169-182, doi:10.1007/s10711-004-2462-3. · Zbl 1099.28002 · doi:10.1007/s10711-004-2462-3 |
| [21] | F William Lawvere (1962): The category of probabilistic mappings. preprint. |
| [22] | Martha Lewis & Jonathan Lawry (2016): Hierarchical conceptual spaces for concept combination. Artificial Intelligence 237, pp. 204-227, doi:10.1016/j.artint.2016.04.008. · Zbl 1357.68229 · doi:10.1016/j.artint.2016.04.008 |
| [23] | Chris J Maddison, Andriy Mnih & Yee Whye Teh (2016): The concrete distribution: A continuous relaxation of discrete random variables. arXiv preprint arXiv:1611.00712. |
| [24] | Prakash Panangaden (1998): Probabilistic relations. School of Computer Science Research Reports-University of Birmingham CSR, pp. 59-74. |
| [25] | András Prékopa (1971): Logarithmic concave measures with application to stochastic programming. Acta Scientiarum Mathematicarum 32, pp. 301-316. · Zbl 0235.90044 |
| [26] | John T Rickard, Janet Aisbett & Greg Gibbon (2007): Reformulation of the theory of conceptual spaces. Information Sciences 177(21), pp. 4539-4565, doi:10.1016/j.ins.2007.05.023. · Zbl 1126.68078 · doi:10.1016/j.ins.2007.05.023 |
| [27] | Eleanor H Rosch (1973): Natural categories. Cognitive psychology 4(3), pp. 328-350, doi:10.1016/0010-0285(73)90017-0. · doi:10.1016/0010-0285(73)90017-0 |
| [28] | Adrien Saumard & Jon A Wellner (2014): Log-concavity and strong log-concavity: a review. Statistics surveys 8, p. 45, doi:10.1214/14-ss107. · Zbl 1360.62055 · doi:10.1214/14-ss107 |
| [29] | Peter Selinger (2010): A survey of graphical languages for monoidal categories. In: New structures for physics, Springer, pp. 289-355, doi:10.1007/978-3-642-12821-9 4. · Zbl 1217.18002 · doi:10.1007/978-3-642-12821-9_4 |
| [30] | Vincent Wang (2019): Concept Functionals. SEMSPACE 2019. |
| [31] | Massimo Warglien & Peter Gärdenfors (2013): Semantics, conceptual spaces, and the meeting of minds. Synthese 190(12), pp. 2165-2193, doi:10.1007/s11229-011-9963-z. · doi:10.1007/s11229-011-9963-z |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
