[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81562-en":3,"doc-seo-81562-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81562,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Data Provenance as Automatic Differentiation","Data Provenance as Automatic Differentiation proposes a semantic framework that treats program provenance using the same construction as automatic differentiation: the program’s “derivative” is modeled as a linear map between spaces of approximations of inputs and outputs. Scalars come from a commutative semiring of dependency information, so the semiring determines the provenance notion. Over Booleans, Jacobians yield dynamic dependency analysis via forward/backward composition; more generally, distributive lattices and adjunction recover conjugate and Galois slicing relationships. Results are formalized in Agda.","arXiv :2511 .09203v2 [ cs .PL] 10 Jul 2026  \nData Provenance as Automatic Differentiation  \nROBERT ATKEY, University of Strathclyde, UK ROLY PERERA∗ , University of Cambridge, UK  \nAutomatic differentiation (AD) computes the derivative of a program alongside the program itself, as a linear map between tangent spaces, propagated forwards or backwards along an execution. We present a semantic framework that models data provenance via the same construction: taking scalars from a commutative semiring of dependency information rather than the real numbers, the derivative of a program becomes a linear map between spaces of approximations ofits input and output. The choice of semiring determines the notion of provenance. Over the two-element Boolean algebra, the Jacobian of a program records which input positions each output position may depend on, and composing Jacobians forwards or backwards is dependency analysis in the manner of forward-and reverse-mode AD. More generally, over distributive lattices the Jacobian and its transpose propagate dependency information forwards and backwards as a conjugate pair of maps; when the lattice is a Boolean algebra, the two directions are moreover related by adjunction, recovering an approach called Galois slicing. We interpret a higher-order total functional language in this framework, prove that every program of first-order type denotes such a Jacobian, and instantiate the semiring to obtain dependency tracking (Booleans), automatic differentiation (reals), and quantitative interval provenance (the tropical semiring) as examples. All results are formalised in Agda.  \n1 Introduction  \nTo audit any computational process, we need robust and well-founded notions of provenance to track how data are used. This allows us to answer questions like “Where did these data come from?”,“Why are these data in the output?” and “How were these data computed?”. Provenance tracking has a wide range of applications, from debugging and program comprehension [Buneman et al. 1995; Cheney et al. 2007] to improving reproducibility and transparency in scientific workflows [Kontogiannis 2008]. Program slicing, first proposed by Weiser [1981], is a collection of techniques for provenance tracking that attempts to take a run of a program and areas of interest in the output, and turn them into the subset of the input and the program that were responsible for generating those specific outputs.  \nUnderlying all of these questions is a more basic one: how does the output of a program respond when its input changes? For programs that compute with real numbers, calculus provides a canonical answer: the derivative, a linear map between tangent spaces describing how the output varies as the input varies, with automatic differentiation (AD) being a computational technique for computing such derivatives alongside the program itself [Elliott 2018; Siskind and Pearlmutter 2008; Vákárand Smeding 2022] . For data provenance and dependency analysis, where the inputs are database tuples or elements of a data structure, the question is often framed in a more qualitative way: if we perturb the input at a given position, can the output change at all? Posing this question one input at a time yields a vector of Boolean partial “derivatives”, and for a program with several outputs, a Boolean Jacobian: a matrix whose entry at (􀀹, 􀀸) records whether output position 􀀹 may depend on input position 􀀸. Consider multiplication: at the point (􀁇, 􀁾), the usual partial derivative 􀁭(􀁇 · 􀁾)/􀁭􀁇 is 􀁾, and its Boolean counterpart records that the output can vary with 􀁇 only when 􀁾 ≠ 0. Like their numerical counterparts, Boolean Jacobians compose by matrix multiplication, with conjunction and disjunction playing the role of multiplication and addition. This paper presentsan approach to dynamic dependency analysis where such Jacobians are evaluated alongside the  \n∗ Also with University of Bristol.  \nAuthors’ Contact Information: Robert Atkey, [robert.atkey@s","cbCaisbEL1cTO1m1","https://ap.wps.com/l/cbCaisbEL1cTO1m1","pdf",747179,1,25,"English","en",105,"# Introduction\n## Dependencies as Derivatives","[{\"question\":\"What does the paper mean by modeling data provenance as automatic differentiation?\",\"answer\":\"It treats provenance as a derivative-like construction: the derivative of a program becomes a linear map between approximation spaces of inputs and outputs, propagated forwards or backwards during execution. Instead of real numbers, the framework uses scalars from a semiring encoding dependency information.\"},{\"question\":\"How does the choice of semiring affect the resulting provenance analysis?\",\"answer\":\"The semiring determines what “provenance” means. Using the two-element Boolean algebra yields Boolean Jacobians that track which input positions each output position may depend on. Other distributive lattices provide conjugate forward/backward propagation, and a tropical semiring example gives quantitative interval provenance.\"},{\"question\":\"How are forward and backward dependency computations related in the framework?\",\"answer\":\"For distributive lattices, the Jacobian and its transpose form a conjugate pair propagating dependency information forwards and backwards. When the lattice is a Boolean algebra, the two directions relate via adjunction, which recovers an approach called Galois slicing.\"}]",1784174333,63,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"data-provenance-as-automatic-differentiation","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/data-provenance-as-automatic-differentiation/81562/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does the paper mean by modeling data provenance as automatic differentiation?","Question",{"text":74,"@type":75},"It treats provenance as a derivative-like construction: the derivative of a program becomes a linear map between approximation spaces of inputs and outputs, propagated forwards or backwards during execution. Instead of real numbers, the framework uses scalars from a semiring encoding dependency information.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the choice of semiring affect the resulting provenance analysis?",{"text":79,"@type":75},"The semiring determines what “provenance” means. Using the two-element Boolean algebra yields Boolean Jacobians that track which input positions each output position may depend on. Other distributive lattices provide conjugate forward/backward propagation, and a tropical semiring example gives quantitative interval provenance.",{"name":81,"@type":72,"acceptedAnswer":82},"How are forward and backward dependency computations related in the framework?",{"text":83,"@type":75},"For distributive lattices, the Jacobian and its transpose form a conjugate pair propagating dependency information forwards and backwards. 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