{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/operational-calculus-for-differentiable","title":"Operational Calculus for Differentiable Programming","arxiv_id":"1610.07690","date":"2016-10-25","proceeding":null,"authors":["Žiga Sajovic","Martin Vuk"],"abstract":"In this work we present a theoretical model for differentiable programming.\nWe construct an algebraic language that encapsulates formal semantics of\ndifferentiable programs by way of Operational Calculus. The algebraic nature of\nOperational Calculus can alter the properties of the programs that are\nexpressed within the language and transform them into their solutions.\n  In our model programs are elements of programming spaces and viewed as maps\nfrom the virtual memory space to itself. Virtual memory space is an algebra of\nprograms, an algebraic data structure one can calculate with. We define the\noperator of differentiation ($\\partial$) on programming spaces and, using its\npowers, implement the general shift operator and the operator of program\ncomposition. We provide the formula for the expansion of a differentiable\nprogram into an infinite tensor series in terms of the powers of $\\partial$. We\nexpress the operator of program composition in terms of the generalized shift\noperator and $\\partial$, which implements a differentiable composition in the\nlanguage. Such operators serve as abstractions over the tensor series algebra,\nas main actors in our language.\n  We demonstrate our models usefulness in differentiable programming by using\nit to analyse iterators, deriving fractional iterations and their iterating\nvelocities, and explicitly solve the special case of ReduceSum.","url_abs":"http://arxiv.org/abs/1610.07690v6","url_pdf":"http://arxiv.org/pdf/1610.07690v6.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"operational-calculus-for-differentiable","repo_url":"https://github.com/zigasajovic/dCpp","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}