{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T14:04:33Z","timestamp":1749737073643},"reference-count":0,"publisher":"IOS Press","isbn-type":[{"value":"9781643684062","type":"print"},{"value":"9781643684079","type":"electronic"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,7,21]]},"abstract":"<jats:p>In this chapter, we present a research line whose aim is to represent neural networks in \u0141ukasiewicz Infinitely-valued Logic (\u0141\u221e) so that one might reason about such neural networks by means of logical machinery. We focus on a class of neural networks with inputs and outputs in the unit interval [0,1] concentrating the study on the ones that compute piecewise linear functions, which is not a strong constraint since such functions may densely approximate any continuous function. For that, we introduce the concept of representation modulo satisfiability, that enlarges the representational power of \u0141\u221e. We derive an algorithm for building such representations, which terminates in polynomial time as long as the input is given in a suitable format. As applications of representation modulo satisfiability, we proceed by showing how properties of neural networks, such as reachability and robustness, may be encoded in \u0141\u221e, giving rise to formal verification techniques. Finally, we present a case study where some formal verifications are performed in real-world neural network.<\/jats:p>","DOI":"10.3233\/faia230160","type":"book-chapter","created":{"date-parts":[[2023,8,4]],"date-time":"2023-08-04T06:45:02Z","timestamp":1691131502000},"source":"Crossref","is-referenced-by-count":2,"title":["Chapter 28. Effective Reasoning over Neural Networks Using \u0141ukasiewicz Logic"],"prefix":"10.3233","author":[{"given":"Sandro","family":"Preto","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Statistics, University of S\u00e3o Paulo"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcelo","family":"Finger","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Statistics, University of S\u00e3o Paulo"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"7437","container-title":["Frontiers in Artificial Intelligence and Applications","Compendium of Neurosymbolic Artificial Intelligence"],"original-title":[],"link":[{"URL":"https:\/\/ebooks.iospress.nl\/pdf\/doi\/10.3233\/FAIA230160","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,4]],"date-time":"2023-08-04T06:45:03Z","timestamp":1691131503000},"score":1,"resource":{"primary":{"URL":"https:\/\/ebooks.iospress.nl\/doi\/10.3233\/FAIA230160"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,21]]},"ISBN":["9781643684062","9781643684079"],"references-count":0,"URL":"https:\/\/doi.org\/10.3233\/faia230160","relation":{},"ISSN":["0922-6389","1879-8314"],"issn-type":[{"value":"0922-6389","type":"print"},{"value":"1879-8314","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,21]]}}}